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+{
+    "url": "http://arxiv.org/abs/2405.01029v2",
+    "title": "MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts",
+    "abstract": "Learning to solve vehicle routing problems (VRPs) has garnered much\nattention. However, most neural solvers are only structured and trained\nindependently on a specific problem, making them less generic and practical. In\nthis paper, we aim to develop a unified neural solver that can cope with a\nrange of VRP variants simultaneously. Specifically, we propose a multi-task\nvehicle routing solver with mixture-of-experts (MVMoE), which greatly enhances\nthe model capacity without a proportional increase in computation. We further\ndevelop a hierarchical gating mechanism for the MVMoE, delivering a good\ntrade-off between empirical performance and computational complexity.\nExperimentally, our method significantly promotes zero-shot generalization\nperformance on 10 unseen VRP variants, and showcases decent results on the\nfew-shot setting and real-world benchmark instances. We further conduct\nextensive studies on the effect of MoE configurations in solving VRPs, and\nobserve the superiority of hierarchical gating when facing out-of-distribution\ndata. The source code is available at:\nhttps://github.com/RoyalSkye/Routing-MVMoE.",
+    "authors": "Jianan Zhou, Zhiguang Cao, Yaoxin Wu, Wen Song, Yining Ma, Jie Zhang, Chi Xu",
+    "published": "2024-05-02",
+    "updated": "2024-05-06",
+    "primary_cat": "cs.AI",
+    "cats": [
+        "cs.AI",
+        "cs.LG"
+    ],
+    "label": "Original Paper",
+    "paper_cat": "Mixture AND of AND Experts",
+    "gt": "Learning to solve vehicle routing problems (VRPs) has garnered much\nattention. However, most neural solvers are only structured and trained\nindependently on a specific problem, making them less generic and practical. In\nthis paper, we aim to develop a unified neural solver that can cope with a\nrange of VRP variants simultaneously. Specifically, we propose a multi-task\nvehicle routing solver with mixture-of-experts (MVMoE), which greatly enhances\nthe model capacity without a proportional increase in computation. We further\ndevelop a hierarchical gating mechanism for the MVMoE, delivering a good\ntrade-off between empirical performance and computational complexity.\nExperimentally, our method significantly promotes zero-shot generalization\nperformance on 10 unseen VRP variants, and showcases decent results on the\nfew-shot setting and real-world benchmark instances. We further conduct\nextensive studies on the effect of MoE configurations in solving VRPs, and\nobserve the superiority of hierarchical gating when facing out-of-distribution\ndata. The source code is available at:\nhttps://github.com/RoyalSkye/Routing-MVMoE.",
+    "main_content": "Introduction Vehicle routing problems (VRPs) are a class of canonical combinatorial optimization problems (COPs) in operation research and computer science, with a wide spectrum of 1College of Computing and Data Science, Nanyang Technological University, Singapore 2School of Computing and Information Systems, Singapore Management University, Singapore 3Department of Information Systems, Eindhoven University of Technology, The Netherlands 4Institute of Marine Science and Technology, Shandong University, China 5Singapore Institute of Manufacturing Technology (SIMTech), Agency for Science, Technology and Research (A*STAR), Singapore. Correspondence to: Yaoxin Wu <y.wu2@tue.nl>. Proceedings of the 41 st International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s). applications in logistics (Cattaruzza et al., 2017), transportation (Wu et al., 2023), and manufacturing (Zhang et al., 2023). The intrinsic NP-hard nature makes VRPs exponentially expensive to be solved by exact solvers. As an alternative, heuristic solvers deliver suboptimal solutions within reasonable time, but need substantial domain expertise to be designed for each problem. Recently, learning to solve VRPs has received much attention (Bengio et al., 2021; Bogyrbayeva et al., 2024), with fruitful neural solvers being developed. Most of them apply deep neural networks to learn solution construction policies via various training paradigms (e.g., reinforcement learning (RL)). Besides gaining decent performance, they are characterized by less computational overhead and domain expertise than conventional solvers. However, prevailing neural solvers still need network structures tailored and trained independently for each specific VRP, instigating prohibitive training overhead and less practicality when facing multiple VRPs. In this paper, we aim to develop a unified neural solver, which can be trained for solving a range of VRP variants simultaneously, and has decent zero-shot generalization capability on unseen VRPs. A few recent works explore similar problem settings. Wang & Yu (2023) applies multi-armed bandits to solve multiple VRPs, while Lin et al. (2024) adapts the model pretrained on one base VRP to target VRPs by efficient fine-tuning. They fail to achieve zero-shot generalization to unseen VRPs due to the dependence on networks structured for predetermined problem variants. Liu et al. (2024) empowers the neural solver with such generalizability by the compositional zero-shot learning (Ruis et al., 2021), which treats VRP variants as different combinations of a set of underlying attributes and uses a shared network to learn their representations. However, it still leverages existing network structure proposed for simple VRPs, which is limited by its model capacity and empirical performance. Motivated by the recent advance of large language models (LLMs) (Kaplan et al., 2020; Floridi & Chiriatti, 2020; Touvron et al., 2023), we propose a multi-task VRP solver with mixture-of-experts (MVMoE). Typically, a mixture-ofexpert (MoE) layer replaces a feed-forward network (FFN) with several \"experts\" in a Transformer-based model, which are a group of FFNs with respective trainable parameters. An input to the MoE layer is routed to specific expert(s) by a gating network, and only parameters in selected expert(s) 1 arXiv:2405.01029v2  [cs.AI]  6 May 2024 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts are activated (i.e., conditional computation (Jacobs et al., 1991; Jordan & Jacobs, 1994)). In this manner, partially activated parameters can effectively enhance the model capacity without a proportional increase in computation, making the training and deployment of LLMs viable. Therefore, towards a more generic and powerful neural solver, we first propose an MoE-based neural VRP solver, and present a hierarchical gating mechanism for a good trade-off between empirical performance and computational complexity. We choose the setting from Liu et al. (2024) as a test bed due to its potential to solve an exponential number of new VRP variants as any combination of the underlying attributes. Our contributions are summarized as follows. 1) We propose a unified neural solver MVMoE to solve multiple VRPs, which first brings MoEs into the study of COPs. The sole MVMoE can be trained on diverse VRP variants, and facilitate a strong zero-shot generalization capability on unseen VRPs. 2) We develop a hierarchical gating mechanism for MVMoE to attain a favorable balance between empirical performance and computational overhead. Surprisingly, it exhibits much stronger out-of-distribution generalization capability than the base gating. 3) Extensive experiments demonstrate that MVMoE significantly improves the zeroshot generalization against baselines on 10 unseen VRP variants, and achieves decent results on the few-shot setting and real-world instances. We further provide extensive studies on the effect of MoE configurations (such as the position of MoEs, the number of experts, and the gating mechanism) on the zero-shot generalization performance. 2. Related Work Neural VRP Solvers. Two mainstreams exist in literature on learning to solve VRPs: 1) Construction-based solvers, which learn policies to construct solutions in an end-to-end manner. Vinyals et al. (2015) proposes Pointer Network to estimate the optimal solution to the traveling salesman problem (TSP) in an autoregressive way. The follow-up works apply RL to explore better approximate solutions to TSP (Bello et al., 2017) and capacitated vehicle routing problem (CVRP) (Nazari et al., 2018). Kool et al. (2018) proposes an attention-based model (AM) that uses Transformer to solve a series of VRPs independently. By leveraging the symmetry property in solutions, Kwon et al. (2020) proposes the policy optimization with multiple optima (POMO) to further promote the performance in solving TSP and CVRP. Other construction-based solvers are often developed on top of AM and POMO (Kwon et al., 2021; Li et al., 2021a; Kim et al., 2022; Berto et al., 2023; Chen et al., 2023; Grinsztajn et al., 2023; Chalumeau et al., 2023; Hottung et al., 2024). Besides the autoregressive manner, several works construct a heatmap to solve VRPs in a non-autoregressive manner (Joshi et al., 2019; Fu et al., 2021; Kool et al., 2022; Qiu et al., 2022; Sun & Yang, 2023; Min et al., 2023; Ye et al., 2023; Kim et al., 2024). 2) Improvement-based solvers, which learn policies to iteratively refine an initial solution until a termination condition is satisfied. The policies are often trained in contexts of classic local search (Croes, 1958; Shaw, 1998) or specialized heuristic solvers (Helsgaun, 2017) for obtaining more efficient or effective search components (Chen & Tian, 2019; Lu et al., 2020; Hottung & Tierney, 2020; d O Costa et al., 2020; Wu et al., 2021; Xin et al., 2021; Hudson et al., 2022; Zhou et al., 2023a; Ma et al., 2023). In general, constructionbased solvers can efficiently achieve desired performance, whereas improvement-based solvers have the potential to deliver better solutions given prolonged inference time. Recent research uncovers the deficient generalization capability of neural solvers, which suffer from drastic performance decrement on unseen data (Joshi et al., 2021). Previous works mainly focus on the cross-size generalization (Fu et al., 2021; Hou et al., 2023; Son et al., 2023; Luo et al., 2023; Drakulic et al., 2023) or cross-distribution generalization (Zhang et al., 2022; Geisler et al., 2022; Bi et al., 2022; Jiang et al., 2023) or both (Manchanda et al., 2022; Zhou et al., 2023b; Wang et al., 2024) on a single problem. In this paper, we step further to explore the generalization across different VRP variants (Wang & Yu, 2023; Liu et al., 2024; Lin et al., 2024). Mixture-of-Experts. The original idea of MoEs was proposed three decades ago (Jacobs et al., 1991; Jordan & Jacobs, 1994). In early concepts, the expert was defined as an entire neural network, and hence MoEs was similar to an ensemble of neural networks. Eigen et al. (2013) launchs the era when researchers start applying MoEs as components of neural networks. As an early success of MoEs applied in large neural networks, Shazeer et al. (2017) introduces the sparsely-gated MoEs in language modeling and machine translation, achieving state-of-the-art results at the time with only minor losses in computational efficiency. Follow-up works mainly focus on improving the gating mechanism (Lewis et al., 2021; Roller et al., 2021; Zuo et al., 2022; Zhou et al., 2022; Puigcerver et al., 2024; Xue et al., 2024) or applications to other domains (Lepikhin et al., 2020; Riquelme et al., 2021; Fedus et al., 2022b). We refer interested readers to Yuksel et al. (2012); Fedus et al. (2022a) for a comprehensive survey. 3. Preliminaries In this section, we first present the definition of CVRP, and then introduce its variants featured by additional constraints. Afterwards, we delineate recent construction-based neural solvers for VRPs (Kool et al., 2018; Kwon et al., 2020). VRP Variants. We define a CVRP instance of size n over 2 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts -0.5 +0.2 -0.3 +0.1 Linehaul Backhaul Depot [0, 3.0] (O) (B) (L) (TW) 0.2 0.2 0.2 0.1 0.3 < L [1.0, 1.5] [1.4, 1.8] [2.0, 2.4] [2.4, 2.8] Figure 1. Illustrations of sub-tours with various constraints: open route (O), backhaul (B), duration limit (L), and time window (TW). a graph G = {V, E}, where V includes a depot node v0 and customer nodes {vi}n i=1, and E includes edges e(vi, vj) between node vi and vj(i \u0338= j). Each customer node is associated with a demand \u03b4i, and a capacity limit Q is set for each vehicle. The solution (i.e., tour) \u03c4 is represented as a sequence of nodes, consisting of multiple sub-tours. Each sub-tour represents that a vehicle starts from the depot, visits a subset of customer nodes and returns to the depot. The solution is feasible if each customer node is visited exactly once, and the total demand in each sub-tour does not exceed the capacity limit Q. We consider the Euclidean space with the cost function c(\u00b7) defined as the total length of the tour. The objective is to find the optimal tour \u03c4 \u2217with the minimal cost: \u03c4 \u2217= arg min\u03c4\u2208\u03a6 c(\u03c4|G), where \u03a6 is the discrete search space that contains all feasible tours. On top of CVRP (featured by the capacity constraint (C)), several VRP variants involve additional practical constraints. 1) Open Route (O): The vehicle does not need to return to the depot v0 after visiting customers; 2) Backhaul (B): The demand \u03b4i is positive in CVRP, representing a vehicle unloads goods at the customer node. In practice, a customer can have a negative demand, requiring a vehicle to load goods. We name the customer nodes with \u03b4i > 0 as linehauls and the ones with \u03b4i < 0 as backhauls. Hence, VRP with backhaul allows the vehicle traverses linehauls and backhauls in a mixed manner, without strict precedence between them; 3) Duration Limit (L): To maintain a reasonable workload, the cost (i.e., length) of each route is upper bounded by a predefined threshold; 4) Time Window (TW): Each node vi \u2208V is associated with a time window [ei, li] and a service time si. A vehicle must start serving customer vi in the time slot from ei to li. If the vehicle arrives earlier than ei, it has to wait until ei. All vehicles must return to the depot v0 no later than l0. The aforementioned constraints are illustrated in Fig. 1. By combining them, we can obtain 16 typical VRP variants, which are summarized in Table 3. Note that the combination is not a trivial addition of different constraints. For example, when the open route is coupled with the time window, the vehicle does not need to return to the depot, and hence the constraint imposed by l0 at the depot is relaxed. We present more details of VRP variants and the associated data generation process in Appendix A. Learning to Solve VRPs. Typical neural solvers (Kool et al., 2018; Kwon et al., 2020) parameterize the solution construction policy by an attention-based neural network \u03c0\u03b8, which is trained to generate a solution in an autoregressive way. The feasibility of the generated solution is guaranteed by the masking mechanism during decoding. Without loss of generality, we consider RL training paradigm, wherein the solution construction process is formulated as a Markov Decision Process (MDP). Given an input instance, the encoder processes it and attains all node embeddings, which, with the context representation of the constructed partial tour, represent the current state. The decoder takes them as inputs and outputs the probabilities of valid nodes (i.e., actions) to be selected. After a complete solution \u03c4 is constructed, its probability can be factorized via the chain rule such that p\u03b8(\u03c4|G) = QT t=1 p\u03b8(\u03c0(t) \u03b8 |\u03c0(<t) \u03b8 , G), where \u03c0(t) \u03b8 and \u03c0(<t) \u03b8 denote the selected node and constructed partial tour at step t, and T is the number of total steps. The reward is defined as the negative tour length, i.e., R = \u2212c(\u03c4|G). Given a baseline function b(\u00b7) for training stability, the policy network \u03c0\u03b8 is often trained by REINFORCE (Williams, 1992) algorithm, which applies estimated gradients of the expected reward to optimize the policy as below, \u2207\u03b8La(\u03b8|G) = Ep\u03b8(\u03c4|G)[(c(\u03c4) \u2212b(G))\u2207\u03b8 log p\u03b8(\u03c4|G)]. (1) 4. Methodology In this section, we present the multi-task VRP solver with MoEs (MVMoE), and introduce the gating mechanism. Without loss of generality, we aim to learn a constructionbased neural solver (Kool et al., 2018; Kwon et al., 2020) for tackling VRP variants with the five constraints introduced in Section 3. The structure of MVMoE is illustrated in Fig. 2. 4.1. Multi-Task VRP Solver with MoEs Multi-Task VRP Solver. Given an instance of a specific VRP variant, the static features of each node vi are expressed by Si = {yi, \u03b4i, ei, li}, where yi, \u03b4i, ei, li denote the coordinate, demand, start and end time of the time window, respectively. The encoder takes these static node features as inputs, and outputs d-dimensional node embeddings hi. At the tth decoding step, the decoder takes as input the node embeddings and a context representation, including the embedding of the last selected node and dynamic features Dt = {ct, tt, lt, ot}, where ct, tt, lt, ot denote the remaining capacity of the vehicle, the current time, the length of the current partial route, and the presence indicator of the open route, respectively. Thereafter, the decoder outputs the probability distribution of nodes, from which a valid node is selected and appended to the partial solution. A complete solution is constructed in an autoregressive manner by iterating the decoding process. 3 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts Multi-Head Attention with MoE Node Embedding V Q Probability  Update & Loop Solution N x Multi-Head Self-Attention Add & Normalize Add & Normalize Encoder Linear MoE Dot-Product & Softmax Dynamic Feature Last Node Decoder K Static Feature MoE Layer Top 2 -\" $& = ['&, )&, *&, +&] -% -) -* \u210e\" \u210e% \u210e) \u210e* \u210e) \u210e\" \u210e\" \u210e% \u210e* /+ 0+ 1+ 2+ %\" %# %$ %) -(!)# 0#(!) 3 0'(!) -(!)' MOEs(*) & Figure 2. The model structure of MVMoE. [Green part]: Given an input instance, the encoder and decoder output node embeddings and probabilities of nodes to be selected, respectively. The gray nodes are masked to satisfy problem-specific constraints for feasibility. The node with a deeper color denote a later node embedding. [Yellow part]: In an MoE layer, where we take the (node-level) input-choice Top2 gating as an example, the input x (i.e., node) is routed to two experts that derive the two largest probabilities from the gating network G. In each training step, we randomly select a VRP variant, and train the neural network to solve associated instances in a batch. In this way, MVMoE is able to learn a unified policy that can tackle different VRP tasks. If only a subset of static or dynamic features are involved in the current selected VRP variant, the other features are padded to the default values (e.g., zeros). For example, given a CVRP instance, the static features of the ith customer node are S(C) i = {yi, \u03b4i, 0, 0}, and the dynamic features at the tth decoding step are D(C) t = {ct, 0, lt, 0}. In summary, motivated by the fact that different VRP variants may include some common attributes (e.g., coordinate, demand), we define the static and dynamic features as the union set of attributes that exist in all VRP variants. By training on a few VRP variants with these attributes, the policy network has the potential to solve unseen variants, which are characterized by different combinations of these attributes, i.e., the zero-shot generalization capability (Liu et al., 2024). Mixture-of-Experts. Typically, an MoE layer consists of 1) m experts {E1, E2, . . . , Em}, each of which is a linear layer or FFN with independent trainable parameters, and 2) a gating network G parameterized by WG, which decides how the inputs are distributed to experts. Given a single input x, G(x) and Ej(x) denote the output of the gating network (i.e., an m-dimensional vector), and the output of the jth expert, respectively. In light of this, the output of an MoE layer is calculated as, MoE(x) = m X j=1 G(x)jEj(x). (2) Intuitively, a sparse vector G(x) only activates a small subset of experts with partial model parameters, and hence saves the computation. Typically, a TopK operator can achieve such sparsity by only keeping the K-largest values while setting others as the negative infinity. In this case, the gating network calculates the output as G(x) = Softmax(TopK(x \u00b7 WG)). Given the fact that larger sparse models do not always lead to better performance (Zuo et al., 2022), it is crucial yet tricky to design effective and efficient gating mechanisms to endow each expert being sufficiently trained, given enough training data. To this effect, some works have been put forward in language and vision domains, such as designing an auxiliary loss (Shazeer et al., 2017) or formulating it as a linear assignment problem (Lewis et al., 2021) in pursuit of the load balancing. MVMoE. By integrating the above parts, we obtain the multi-task VRP solver with MoEs. The overall model structure is shown in Fig. 2, where we employ MoEs in both the encoder and decoder. In specific, we substitute MoEs for the FFN layer in the encoder, and substitute MoEs for the final linear layer of multi-head attention in the decoder. We refer more details of the structure of MVMoE to Appendix B. We empirically find our design is effective in generating high-quality solutions, and especially employing MoEs in the decoder tends to exert a greater influence on performance (see Section 5.2). We jointly optimize all trainable parameters \u0398, with the objective formulated as follows, min \u0398 L = La + \u03b1Lb, (3) where La denotes the original loss function of the VRP solver (e.g., the REINFORCE loss used to train the policy for solving VRP variants in Eq. (1)), Lb denotes the loss function associated with MoEs (e.g., the auxiliary loss used to ensure load balancing in Eq. (19) in Appendix B), and \u03b1 is a hyperparameter to control its strength. 4.2. Gating Mechanism We mainly consider the node-level (or token-level) gating, by which each node is routed independently to experts.1 In 1In addition, we also investigate another two gating levels, i.e., instance-level and problem-level gating, which are presented in Section 5.2 and Appendix B. 4 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts 2.31 0.45 0.74 0.65 1.43 -0.68 1.20 -1.02 0.98 -0.53 0.76 1.97 -0.35 1.26 0.87 0.36 2.31 0.45 0.74 0.65 1.43 -0.68 1.20 -1.02 0.98 -0.53 0.76 1.97 -0.35 1.26 0.87 0.36 !! !\" !# !$ \"! \"\" \"# \"$ !! !\" !# !$ \"! \"\" \"# \"$ Figure 3. An illustration of the score matrix and gating algorithm. Left panel: Input-choice gating. Right panel: Expert-choice gating. The selected experts or nodes are in color. The arrow marks the dimension, along which the TopK experts or nodes are selected. each MoE layer, the extra computation originates from the forward pass of the gating network and the distribution of nodes to the selected experts. While employing MoEs in the decoder can significantly improve the performance, the number of decoding steps T increases as the problem size n scales up. It suggests that compared to the encoder with a fixed number of gating steps N (\u226aT), applying MoEs in the decoder may substantially increase the computational complexity. In light of this, we propose a hierarchical gating mechanism to make the better use of MoEs in the decoder for gaining a good trade-off between empirical performance and computational complexity. Next, we detail the nodelevel and hierarchical gating mechanism. Node-Level Gating. The node-level gating routes inputs at the granularity of nodes. Let d denote the hidden dimension and WG \u2208Rd\u00d7m denote trainable parameters of the gating network in MVMoE. Given a batch of inputs X \u2208RI\u00d7d, where I is the total number of nodes (i.e., batch size B \u00d7 problem scale n), each node is routed to the selected experts based on the score matrix H = (X \u00b7 WG) \u2208RI\u00d7m predicted by the gating network. We illustrate an example of the score matrix in Fig. 3, where xi denotes the ith node, and Ej denotes the jth expert in the node-level gating. In this paper, we mainly consider two popular gating algorithms (Shazeer et al., 2017; Zhou et al., 2022): 1) Inputchoice gating: Each node selects TopK experts based on H. Typically, K is set to 1 or 2 to retain a reasonable computational complexity. The input-choice gating is illustrated in the left panel of Fig. 3, where each node is routed to two experts with the largest scores (i.e., Top2). However, this method cannot guarantee load balancing. An expert may receive much more nodes than the others, resulting in a dominant expert while leaving others underfitting. To address this issue, most works employ an auxiliary loss to equalize quantities of nodes sent to different experts during training. Here we use the importance & load loss (Shazeer et al., 2017) as Lb in Eq. (3) to mitigate load imbalance (see Appendix B). 2) Expert-choice gating: Each expert selects Base Gating Top 2 Dense Layer Sparse Layer If Else Hierarchical Gating &\" &# ' %\" %# %$ %) %\" %# %) %$ 3 3 & Figure 4. A base gating (i.e., the input-choice gating with K = 2) and its hierarchical gating counterpart. In the latter, the gating network G1 routes inputs to the sparse layer ({G2, E1, E2, E3, E4}) or the dense layer D. If the sparse layer is chosen, the gating network G2 routes nodes to experts accoring to the base gating. TopK nodes based on H. Typically, K is set to I\u00d7\u03b2 m , where \u03b2 is the capacity factor reflecting the average number of experts utilized by a node. The expert-choice gating is illustrated in the right panel of Fig. 3, where each expert selects two nodes with the largest scores given \u03b2 = 2. While this gating algorithm explicitly ensures load balancing, some nodes may not be chosen by any expert. We refer more details of the above gating algorithms to Appendix B. Hierarchical Gating. In the VRP domain, it is computationally expensive to employ MoEs in each decoding step, since 1) the number of decoding steps T increases as the problem size n rises; 2) the problem-specific feasibility constraints must be satisfied during decoding. To tackle the challenges, we propose to employ MoEs only in partial decoding steps. Accordingly, we present a hierarchical gating, which learns to effectively and efficiently utilize MoEs during decoding. We illustrate the proposed hierarchical gating in Fig. 4. An MoE layer with the hierarchical gating includes two gating networks {G1, G2}, m experts {E1, E2, . . . , Em}, and a dense layer D (e.g., a linear layer). Given a batch of inputs X \u2208RI\u00d7d, the hierarchical gating routes them in two stages. In the first stage, G1 decides to distribute inputs X to either the sparse or dense layer according to the problem-level representation X1. In specific, we obtain X1 by applying the mean pooling along the first dimension of X, and process it to obtain the score matrix H1 = (X1 \u00b7 WG1) \u2208R1\u00d72. Then, we route the batch of inputs X to the sparse or dense layer by sampling from the probability distribution G1(X) = Softmax(H1). Here we employ the problem-level gating in G1 for the generality and efficiency of the hierarchical gating (see Appendix D for further discussions). In the second stage, if X is routed to the sparse layer, the gating network G2 is activated to route nodes to experts on the node-level by using aforementioned gating algorithms (e.g., the input-choice gating). Otherwise, X is routed to the dense layer D and transformed into D(X) \u2208RI\u00d7d. In summary, the hierarchical gating learns to output G1(X)0 Pm j=1 G2(X)jEj(X) or G1(X)1D(X) 5 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts Table 1. Performance on 1K test instances of trained VRPs. * represents 0.000%, with which the gaps are computed. Method n = 50 n = 100 Method n = 50 n = 100 Obj. Gap Time Obj. Gap Time Obj. Gap Time Obj. Gap Time CVRP HGS 10.334 * 4.6m 15.504 * 9.1m VRPTW HGS 14.509 * 8.4m 24.339 * 19.6m LKH3 10.346 0.115% 9.9m 15.590 0.556% 18.0m LKH3 14.607 0.664% 5.5m 24.721 1.584% 7.8m OR-Tools 10.540 1.962% 10.4m 16.381 5.652% 20.8m OR-Tools 14.915 2.694% 10.4m 25.894 6.297% 20.8m OR-Tools (x10) 10.418 0.788% 1.7h 15.935 2.751% 3.5h OR-Tools (x10) 14.665 1.011% 1.7h 25.212 3.482% 3.5h POMO 10.418 0.806% 3s 15.734 1.488% 9s POMO 14.940 2.990% 3s 25.367 4.307% 11s POMO-MTL 10.437 0.987% 3s 15.790 1.846% 9s POMO-MTL 15.032 3.637% 3s 25.610 5.313% 11s MVMoE/4E 10.428 0.896% 4s 15.760 1.653% 11s MVMoE/4E 14.999 3.410% 4s 25.512 4.903% 12s MVMoE/4E-L 10.434 0.955% 4s 15.771 1.728% 10s MVMoE/4E-L 15.013 3.500% 3s 25.519 4.927% 11s OVRP LKH3 6.511 0.198% 4.5m 9.828 * 5.3m VRPL LKH3 10.571 0.790% 7.8m 15.771 * 16.0m OR-Tools 6.531 0.495% 10.4m 10.010 1.806% 20.8m OR-Tools 10.677 1.746% 10.4m 16.496 4.587% 20.8m OR-Tools (x10) 6.498 * 1.7h 9.842 0.122% 3.5h OR-Tools (x10) 10.495 * 1.7h 16.004 1.444% 3.5h POMO 6.609 1.685% 2s 10.044 2.192% 8s POMO 10.491 -0.008% 2s 15.785 0.093% 9s POMO-MTL 6.671 2.634% 2s 10.169 3.458% 8s POMO-MTL 10.513 0.201% 2s 15.846 0.479% 9s MVMoE/4E 6.655 2.402% 3s 10.138 3.136% 10s MVMoE/4E 10.501 0.092% 3s 15.812 0.261% 10s MVMoE/4E-L 6.665 2.548% 3s 10.145 3.214% 9s MVMoE/4E-L 10.506 0.131% 3s 15.821 0.323% 10s VRPB OR-Tools 8.127 0.989% 10.4m 12.185 2.594% 20.8m OVRPTW OR-Tools 8.737 0.592% 10.4m 14.635 1.756% 20.8m OR-Tools (x10) 8.046 * 1.7h 11.878 * 3.5h OR-Tools (x10) 8.683 * 1.7h 14.380 * 3.5h POMO 8.149 1.276% 2s 11.993 0.995% 7s POMO 8.891 2.377% 3s 14.728 2.467% 10s POMO-MTL 8.182 1.684% 2s 12.072 1.674% 7s POMO-MTL 8.987 3.470% 3s 15.008 4.411% 10s MVMoE/4E 8.170 1.540% 3s 12.027 1.285% 9s MVMoE/4E 8.964 3.210% 4s 14.927 3.852% 11s MVMoE/4E-L 8.176 1.605% 3s 12.036 1.368% 8s MVMoE/4E-L 8.974 3.322% 4s 14.940 3.941% 10s based on both problem-level and node-level representations. Overall, the hierarchical gating improves the computational efficiency with a minor loss on the empirical performance. To balance the efficiency and performance of MVMoE, we use the base gating in the encoder and its hierarchical gating counterpart in the decoder. Note that the hierarchical gating is applicable to different gating algorithms, such as the input-choice gating (Shazeer et al., 2017) and expert-choice gating (Zhou et al., 2022). We also explore a more advanced gating algorithm (Puigcerver et al., 2024) for reducing the number of routed nodes and thus the computational complexity. But its empirical performance is unsatisfactory in the VRP domain (see Section 5.3). 5. Experiments In this section, we empirically verify the superiority of the proposed MVMoE, and provide insights into the application of MoEs to solve VRPs. We consider 16 VRP variants with five constraints. Due to page limit, we present more experimental results in Appendix C. All experiments are conducted on a machine with NVIDIA Ampere A100-80GB GPU cards and AMD EPYC 7513 CPU at 2.6GHz. Baselines. Traditional solvers: We employ HGS (Vidal, 2022) to solve CVRP and VRPTW instances with default hyperparameters (i.e., the maximum number of iterations without improvement is 20000). We run LKH3 (Helsgaun, 2017) to solve CVRP, OVRP, VRPL and VRPTW instances with 10000 trails and 1 run. OR-Tools (Furnon & Perron, 2023) is an open source solver for complex optimization problems. It is more versatile than LKH and HGS, and can solve all 16 VRP variants considered in this paper. We use the parallel cheapest insertion as the first solution strategy, and use the guided local search as the local search strategy in OR-Tools. For n = 50/100, we set the search time limit as 20s/40s to solve an instance, and also provide its results given 200s/400s (i.e., OR-Tools (x10)). For all traditional solvers, we use them to solve 32 instances in parallel on 32 CPU cores following Kool et al. (2018). Neural solvers: We compare our method to POMO (Kwon et al., 2020) and POMO-MTL (Liu et al., 2024). While POMO is trained on each single VRP, POMO-MTL is trained on multiple VRPs by multi-task learning. Note that POMO-MTL is the dense model counterpart of MVMoE, which is structured by dense layers (e.g., FFNs) rather than sparse MoEs. In specific, POMO-MTL and MVMoE/4E possess 1.25M and 3.68M parameters, but they activate a similar number of parameters for each single input. Training. We follow most setups in (Kwon et al., 2020). 1) For all neural solvers: Adam optimizer is used with the learning rate of 1e \u22124, the weight decay of 1e \u22126, and the batch size of 128. The model is trained for 5000 epochs, with each containing 20000 training instances (i.e., 100M training instances in total). The learning rate is decayed by 10 for the last 10% training instances. We consider two problem scales n \u2208{50, 100} during training, according to (Liu et al., 2024). 2) For multi-task solvers: The training problem set includes CVRP, OVRP, VRPB, VRPL, VRPTW, and OVRPTW (see Appendix C.1 for further discussions). In each batch of training, we randomly sample a problem from the set and generate its instances. Please refer to Appendix A for details of the generation procedure. 3) For our method: We employ m = 4 experts with K = \u03b2 = 2 in each MoE layer, and set the the weight \u03b1 of the auxiliary loss Lb as 0.01. The default gating mechanism of MVMOE/4E is the node-level input-choice gating in both the encoder 6 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts Table 2. Zero-shot generalization on 1K test instances of unseen VRPs. * represents 0.000%, with which the gaps are computed. Method n = 50 n = 100 Method n = 50 n = 100 Obj. Gap Time Obj. Gap Time Obj. Gap Time Obj. Gap Time OVRPB OR-Tools 5.764 0.332% 10.4m 8.522 1.852% 20.8m OVRPL OR-Tools 6.522 0.480% 10.4m 9.966 1.783% 20.8m OR-Tools (x10) 5.745 * 1.7h 8.365 * 3.5h OR-Tools (x10) 6.490 * 1.7h 9.790 * 3.5h POMO-MTL 6.116 6.430% 2s 8.979 7.335% 8s POMO-MTL 6.668 2.734% 2s 10.126 3.441% 9s MVMoE/4E 6.092 5.999% 3s 8.959 7.088% 9s MVMoE/4E 6.650 2.454% 3s 10.097 3.148% 10s MVMoE/4E-L 6.122 6.522% 3s 8.972 7.243% 9s MVMoE/4E-L 6.659 2.597% 3s 10.106 3.244% 9s VRPBL OR-Tools 8.131 1.254% 10.4m 12.095 2.586% 20.8m VRPBTW OR-Tools 15.053 1.857% 10.4m 26.217 2.858% 20.8m OR-Tools (x10) 8.029 * 1.7h 11.790 * 3.5h OR-Tools (x10) 14.771 * 1.7h 25.496 * 3.5h POMO-MTL 8.188 1.971% 2s 11.998 1.793% 8s POMO-MTL 16.055 8.841% 3s 27.319 7.413% 10s MVMoE/4E 8.172 1.776% 3s 11.945 1.346% 9s MVMoE/4E 16.022 8.600% 4s 27.236 7.078% 11s MVMoE/4E-L 8.180 1.872% 3s 11.960 1.473% 9s MVMoE/4E-L 16.041 8.745% 4s 27.265 7.190% 10s VRPLTW OR-Tools 14.815 1.432% 10.4m 25.823 2.534% 20.8m OVRPBL OR-Tools 5.771 0.549% 10.4m 8.555 2.459% 20.8m OR-Tools (x10) 14.598 * 1.7h 25.195 * 3.5h OR-Tools (x10) 5.739 * 1.7h 8.348 * 3.5h POMO-MTL 14.961 2.586% 3s 25.619 1.920% 12s POMO-MTL 6.104 6.306% 2s 8.961 7.343% 8s MVMoE/4E 14.937 2.421% 4s 25.514 1.471% 13s MVMoE/4E 6.076 5.843% 3s 8.942 7.115% 9s MVMoE/4E-L 14.953 2.535% 4s 25.529 1.545% 12s MVMoE/4E-L 6.104 6.310% 3s 8.957 7.300% 9s OVRPBTW OR-Tools 8.758 0.927% 10.4m 14.713 2.268% 20.8m OVRPLTW OR-Tools 8.728 0.656% 10.4m 14.535 1.779% 20.8m OR-Tools (x10) 8.675 * 1.7h 14.384 * 3.5h OR-Tools (x10) 8.669 * 1.7h 14.279 * 3.5h POMO-MTL 9.514 9.628% 3s 15.879 10.453% 10s POMO-MTL 8.987 3.633% 3s 14.896 4.374% 11s MVMoE/4E 9.486 9.308% 4s 15.808 9.948% 11s MVMoE/4E 8.966 3.396% 4s 14.828 3.903% 12s MVMoE/4E-L 9.515 9.630% 3s 15.841 10.188% 10s MVMoE/4E-L 8.974 3.488% 4s 14.839 3.971% 10s VRPBLTW OR-Tools 14.890 1.402% 10.4m 25.979 2.518% 20.8m OVRPBLTW OR-Tools 8.729 0.624% 10.4m 14.496 1.724% 20.8m OR-Tools (x10) 14.677 * 1.7h 25.342 * 3.5h OR-Tools (x10) 8.673 * 1.7h 14.250 * 3.5h POMO-MTL 15.980 9.035% 3s 27.247 7.746% 11s POMO-MTL 9.532 9.851% 3s 15.738 10.498% 10s MVMoE/4E 15.945 8.775% 4s 27.142 7.332% 12s MVMoE/4E 9.503 9.516% 4s 15.671 10.009% 11s MVMoE/4E-L 15.963 8.915% 4s 27.177 7.473% 11s MVMoE/4E-L 9.518 9.682% 4s 15.706 10.263% 10s 0 2 4 6 8 10 Epoch 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 Gap (%) VRPBLTW POMO-MTL MVMoE/4E MVMoE/4E-L 0 2 4 6 8 10 Epoch 9.0 9.2 9.4 9.6 9.8 Gap (%) OVRPBLTW POMO-MTL MVMoE/4E MVMoE/4E-L Figure 5. Few-shot generalization on unseen VRPs. and decoder layers. MVMoE/4E-L is a computationally light version that replaces the input-choice gating with its hierarchical gating counterpart in the decoder. Inference. For all neural solvers, we use greedy rollout with x8 instance augmentation following Kwon et al. (2020). We report the average results (i.e., objective values and gaps) over the test dataset that contains 1K instances, and the total time to solve the entire test dataset. The gaps are computed with respect to the results of the best-performing traditional VRP solvers (i.e., * in Tables 1 and 2). 5.1. Empirical Results Performance on Trained VRPs. We evaluate all methods on 6 trained VRPs and gather all results in Table 1. The single-task neural solver (i.e., POMO) achieves better performance than multi-task neural solvers on each single problem, since it is restructured and retrained on each VRP independently. However, its average performance over all trained VRPs is quite inferior as shown in Table 4 in Appendix C, since each trained POMO is overfitted to a specific VRP. For example, the average performance of POMO solely trained on CVRP is 16.815%, while POMO-MTL and MVMoE/4E achieve 2.102% and 1.925%, respectively. Notably, our neural solvers consistently outperform POMO-MTL. MVMoE/4E performs slightly better than MVMoE/4E-L at the expense of more computation. Despite that, MVMoE/4E-L exhibits stronger out-of-distribution generalization capability than MVMoE/4E (see Tables 7 and 8 in Appendix C). Generalization on Unseen VRPs. We evaluate multi-task solvers on 10 unseen VRP variants. 1) Zero-shot generalization: We directly test the trained solvers on unseen VRPs. The results in Table 2 reveal that the proposed MVMoE significantly outperforms POMO-MTL across all VRP variants. 2) Few-shot generalization: We also consider the few-shot setting on n = 50, where a trained solver is fine-tuned on the target VRP using 10K instances (0.01% of total training instances) in each epoch. Without loss of generality, we conduct experiments on VRPBLTW and OVRPBLTW following the training setups. The results in Fig. 5 showcase MVMoE generalizes more favorably than POMO-MTL. 5.2. Ablation on MoEs Here we explore the effect of different MoE settings on the zero-shot generalization of neural solvers, and provide insights on how to effectively apply MoEs to solve VRPs. Due to the fast convergence, we reduce the number of epochs to 2500 on VRPs of the size n = 50, while leaving other setups unchanged. We set MVMoE/4E as the default baseline, 7 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts None Raw Enc_Every_Two Enc_Last_Two Enc_All Dec Enc_All_Dec Position of MoEs 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Gap (%) (a) POMO-MTL_50M MVMoE/4E_50M MVMoE/8E_50M MVMoE/16E_50M MVMoE/8E_100M MVMoE/16E_200M Number of Experts 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 Gap (%) (b) POMO-MTL Node-Input Node-Expert Node-Random Instance-Input Instance-Expert Instance-Random Problem-Random Gating Mechanism 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Gap (%) (c) POMO-MTL Input_Choice Soft MoEs Neural Hierarchical Heuristic Hierarchical Advanced Gating 5.7 5.8 5.9 6.0 6.1 6.2 6.3 Gap (%) (d) POMO-MTL Input_Choice Soft MoEs Neural Hierarchical Heuristic Hierarchical Training Efficiency 0 20 40 60 80 100 120 140 160 Time (hr) (e) n = 50 n = 100 Figure 6. Left three panels: The effect of MoE settings on the average zero-shot generalization performance (a) the position of MoEs; (b) the number of experts; (c) the gating mechanism. Right two panels: Further analyses (d) average zero-shot generalization performance of each method employing various gating algorithms in the decoder; (e) training efficiency of each gating algorithm. and ablate on different components of MoEs below. Position of MoEs. We consider three positions to apply MoEs in neural solvers: 1) Raw feature processing (Raw): The linear layer, which projects raw features into initial embeddings, are replaced by MoEs. 2) Encoder (Enc): The FFN in an encoder layer is replaced by MoEs. Typically, MoEs are widely used in every-two or last-two layers (i.e., every or last two layers with even indices \u2113\u2208[0, N \u2212 1]) (Riquelme et al., 2021). Besides, we further attempt to use MoEs in all encoder layers. 3) Decoder (Dec): The final linear layer of the multi-head attention is replaced by MoEs in the decoder. We show the average performance over 10 unseen VRPs in Fig. 6(a). The results reveal that applying MoEs at the shallow layer (e.g., Raw) may worsen the model performance, while using MoEs in all encoder layers (Enc_All) or decoder (Dec) can benefit the zero-shot generalization. Therefore, in this paper, we employ MoEs in both encoder and decoder to pursue a strong unified model architecture to solve various VRPs. Number of Experts. We increase the number of experts in each MoE layer to 8 and 16, and compare the derived MVMoE/8E/16E models to MVMoE/4E. We first train all models using the same number (50M) of instances. After that, we also train MVMoE/8E/16E with more data and computation to explore potential better results, based on the scaling laws (Kaplan et al., 2020). In specific, we provide MVMoE/8E/16E with more data by using larger batch sizes, which linearly scale up against the number of experts (i.e., MVMoE/4E/8E/16E are trained on 50M/100M/200M instances with batch sizes 128/256/512, respectively). The results in Fig. 6(b) show that increasing the number of experts with more training data further unleashes the power of MVMoE, indicating the efficacy of MoEs in solving VRPs. Gating Mechanism. We investigate the effect of different gating levels and algorithms, including three levels (i.e., node-level, instance-level and problem-level) and three algorithms (i.e., input-choice, expert-choice and random gatings), with their details presented in Appendix B. As shown in Fig. 6(c), the node-level input-choice gating performs the best, while the node-level expert-choice gating performs the worst. Interestingly, we observe that the expert-choice gating in the decoder makes MVMoE hard to be optimized. It may suggest that each gating algorithm could have its most suitable position to serve MoEs. However, after an attempt to tune this configuration (i.e., by using MoEs only in the encoder), its performance is still inferior to the baseline, with an average gap of 7.190% on unseen VRPs. 5.3. Additional Results We further provide experiments and discussions on more advanced gating algorithms, training efficiency, benchmark performance, and scalability. We refer readers to more empirical results (e.g., sensitivity analyses) in Appendix C. Advanced Gating. Besides the input-choice and expertchoice gating algorithms evaluated above, we further consider soft MoEs (Puigcerver et al., 2024), which is a recent advanced gating algorithm. Specifically, it performs an implicit soft assignment by distributing K slots (i.e., convex combinations of all inputs) to each expert, rather than a hard assignment between inputs and experts as done by the conventional sparse and discrete gating networks. Since only K (e.g., 1 or 2) slots are distributed to each expert, it can save much computation. We train MVMoE on n = 50 by using node-level soft MoEs in the decoder, following training setups. We also show the result of employing heuristic (random) hierarchical gating in the decoder. However, their results are unsatisfactory as shown in Fig. 6(d). Training Efficiency. Fig. 6(e) shows the training time of employing each gating algorithm in the decoder, combining with their results reported in Fig. 6(d), demonstrating the efficacy of the proposed hierarchical gating in reducing the training overhead with only minor losses in performance. Benchmark Performance. We further evaluate the out-ofdistribution (OOD) generalization performance of all neural solvers on CVRPLIB benchmark instances. Detailed results can be found in Tables 7 and 8 in Appendix C. Surprisingly, we observe that MVMoE/4E performs poorly on large-scale 8 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts instances (e.g., n > 500). It may be caused by the generalization issue of sparse MoEs when transferring to new distributions or domains, which is still an open question in the MoE literature (Fedus et al., 2022a). In contrast, MVMoE/4E-L mostly outperforms MVMoE/4E, demonstrating more favourable potential of the hierarchical gating in promoting the OOD generalization capability. It is worth noting that all neural solvers are only trained on the simple uniformly distributed instances with the size n = 100. Embracing more varied problem sizes (cross-size) and attribute distributions (cross-distribution) into the multi-task training (cross-problem) may further consolidate their performance. Scalability. Given that supervised learning based approaches appear to be more scalable than RL-based approaches in the current literature, we try to build upon a more scalable method, i.e., LEHD (Luo et al., 2023). Concretely, we train a dense model LEHD and a light sparse model with 4 experts LEHD/4E-L on CVRP. The training setups are kept the same as Luo et al. (2023), except that we train all models for only 20 epochs for the training efficiency. We use the hierarchical MoE in each decoder layer of LEHD/4E-L. The results are shown in Table 8, which demonstrates the potential of MoE as a general idea that can further benefit recent scalable methods. Moreover, during the solution construction process, recent works (Drakulic et al., 2023; Gao et al., 2023) typically constrain the search space within a neighborhood of the currently selected node, which is shown to be effective in handling large-scale instances. Integrating MVMoE with these simple yet effective techniques may further improve large-scale performance. 6.",
+    "additional_graph_info": {
+        "graph": [
+            [
+                "Jianan Zhou",
+                "Yaoxin Wu"
+            ],
+            [
+                "Jianan Zhou",
+                "Yining Ma"
+            ],
+            [
+                "Yining Ma",
+                "Yeow Meng Chee"
+            ],
+            [
+                "Yining Ma",
+                "Jingwen Li"
+            ],
+            [
+                "Yining Ma",
+                "Qadeer Khan"
+            ]
+        ],
+        "node_feat": {
+            "Jianan Zhou": [
+                {
+                    "url": "http://arxiv.org/abs/2405.01029v2",
+                    "title": "MVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts",
+                    "abstract": "Learning to solve vehicle routing problems (VRPs) has garnered much\nattention. However, most neural solvers are only structured and trained\nindependently on a specific problem, making them less generic and practical. In\nthis paper, we aim to develop a unified neural solver that can cope with a\nrange of VRP variants simultaneously. Specifically, we propose a multi-task\nvehicle routing solver with mixture-of-experts (MVMoE), which greatly enhances\nthe model capacity without a proportional increase in computation. We further\ndevelop a hierarchical gating mechanism for the MVMoE, delivering a good\ntrade-off between empirical performance and computational complexity.\nExperimentally, our method significantly promotes zero-shot generalization\nperformance on 10 unseen VRP variants, and showcases decent results on the\nfew-shot setting and real-world benchmark instances. We further conduct\nextensive studies on the effect of MoE configurations in solving VRPs, and\nobserve the superiority of hierarchical gating when facing out-of-distribution\ndata. The source code is available at:\nhttps://github.com/RoyalSkye/Routing-MVMoE.",
+                    "authors": "Jianan Zhou, Zhiguang Cao, Yaoxin Wu, Wen Song, Yining Ma, Jie Zhang, Chi Xu",
+                    "published": "2024-05-02",
+                    "updated": "2024-05-06",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.LG"
+                    ],
+                    "main_content": "Introduction Vehicle routing problems (VRPs) are a class of canonical combinatorial optimization problems (COPs) in operation research and computer science, with a wide spectrum of 1College of Computing and Data Science, Nanyang Technological University, Singapore 2School of Computing and Information Systems, Singapore Management University, Singapore 3Department of Information Systems, Eindhoven University of Technology, The Netherlands 4Institute of Marine Science and Technology, Shandong University, China 5Singapore Institute of Manufacturing Technology (SIMTech), Agency for Science, Technology and Research (A*STAR), Singapore. Correspondence to: Yaoxin Wu <y.wu2@tue.nl>. Proceedings of the 41 st International Conference on Machine Learning, Vienna, Austria. PMLR 235, 2024. Copyright 2024 by the author(s). applications in logistics (Cattaruzza et al., 2017), transportation (Wu et al., 2023), and manufacturing (Zhang et al., 2023). The intrinsic NP-hard nature makes VRPs exponentially expensive to be solved by exact solvers. As an alternative, heuristic solvers deliver suboptimal solutions within reasonable time, but need substantial domain expertise to be designed for each problem. Recently, learning to solve VRPs has received much attention (Bengio et al., 2021; Bogyrbayeva et al., 2024), with fruitful neural solvers being developed. Most of them apply deep neural networks to learn solution construction policies via various training paradigms (e.g., reinforcement learning (RL)). Besides gaining decent performance, they are characterized by less computational overhead and domain expertise than conventional solvers. However, prevailing neural solvers still need network structures tailored and trained independently for each specific VRP, instigating prohibitive training overhead and less practicality when facing multiple VRPs. In this paper, we aim to develop a unified neural solver, which can be trained for solving a range of VRP variants simultaneously, and has decent zero-shot generalization capability on unseen VRPs. A few recent works explore similar problem settings. Wang & Yu (2023) applies multi-armed bandits to solve multiple VRPs, while Lin et al. (2024) adapts the model pretrained on one base VRP to target VRPs by efficient fine-tuning. They fail to achieve zero-shot generalization to unseen VRPs due to the dependence on networks structured for predetermined problem variants. Liu et al. (2024) empowers the neural solver with such generalizability by the compositional zero-shot learning (Ruis et al., 2021), which treats VRP variants as different combinations of a set of underlying attributes and uses a shared network to learn their representations. However, it still leverages existing network structure proposed for simple VRPs, which is limited by its model capacity and empirical performance. Motivated by the recent advance of large language models (LLMs) (Kaplan et al., 2020; Floridi & Chiriatti, 2020; Touvron et al., 2023), we propose a multi-task VRP solver with mixture-of-experts (MVMoE). Typically, a mixture-ofexpert (MoE) layer replaces a feed-forward network (FFN) with several \"experts\" in a Transformer-based model, which are a group of FFNs with respective trainable parameters. An input to the MoE layer is routed to specific expert(s) by a gating network, and only parameters in selected expert(s) 1 arXiv:2405.01029v2  [cs.AI]  6 May 2024 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts are activated (i.e., conditional computation (Jacobs et al., 1991; Jordan & Jacobs, 1994)). In this manner, partially activated parameters can effectively enhance the model capacity without a proportional increase in computation, making the training and deployment of LLMs viable. Therefore, towards a more generic and powerful neural solver, we first propose an MoE-based neural VRP solver, and present a hierarchical gating mechanism for a good trade-off between empirical performance and computational complexity. We choose the setting from Liu et al. (2024) as a test bed due to its potential to solve an exponential number of new VRP variants as any combination of the underlying attributes. Our contributions are summarized as follows. 1) We propose a unified neural solver MVMoE to solve multiple VRPs, which first brings MoEs into the study of COPs. The sole MVMoE can be trained on diverse VRP variants, and facilitate a strong zero-shot generalization capability on unseen VRPs. 2) We develop a hierarchical gating mechanism for MVMoE to attain a favorable balance between empirical performance and computational overhead. Surprisingly, it exhibits much stronger out-of-distribution generalization capability than the base gating. 3) Extensive experiments demonstrate that MVMoE significantly improves the zeroshot generalization against baselines on 10 unseen VRP variants, and achieves decent results on the few-shot setting and real-world instances. We further provide extensive studies on the effect of MoE configurations (such as the position of MoEs, the number of experts, and the gating mechanism) on the zero-shot generalization performance. 2. Related Work Neural VRP Solvers. Two mainstreams exist in literature on learning to solve VRPs: 1) Construction-based solvers, which learn policies to construct solutions in an end-to-end manner. Vinyals et al. (2015) proposes Pointer Network to estimate the optimal solution to the traveling salesman problem (TSP) in an autoregressive way. The follow-up works apply RL to explore better approximate solutions to TSP (Bello et al., 2017) and capacitated vehicle routing problem (CVRP) (Nazari et al., 2018). Kool et al. (2018) proposes an attention-based model (AM) that uses Transformer to solve a series of VRPs independently. By leveraging the symmetry property in solutions, Kwon et al. (2020) proposes the policy optimization with multiple optima (POMO) to further promote the performance in solving TSP and CVRP. Other construction-based solvers are often developed on top of AM and POMO (Kwon et al., 2021; Li et al., 2021a; Kim et al., 2022; Berto et al., 2023; Chen et al., 2023; Grinsztajn et al., 2023; Chalumeau et al., 2023; Hottung et al., 2024). Besides the autoregressive manner, several works construct a heatmap to solve VRPs in a non-autoregressive manner (Joshi et al., 2019; Fu et al., 2021; Kool et al., 2022; Qiu et al., 2022; Sun & Yang, 2023; Min et al., 2023; Ye et al., 2023; Kim et al., 2024). 2) Improvement-based solvers, which learn policies to iteratively refine an initial solution until a termination condition is satisfied. The policies are often trained in contexts of classic local search (Croes, 1958; Shaw, 1998) or specialized heuristic solvers (Helsgaun, 2017) for obtaining more efficient or effective search components (Chen & Tian, 2019; Lu et al., 2020; Hottung & Tierney, 2020; d O Costa et al., 2020; Wu et al., 2021; Xin et al., 2021; Hudson et al., 2022; Zhou et al., 2023a; Ma et al., 2023). In general, constructionbased solvers can efficiently achieve desired performance, whereas improvement-based solvers have the potential to deliver better solutions given prolonged inference time. Recent research uncovers the deficient generalization capability of neural solvers, which suffer from drastic performance decrement on unseen data (Joshi et al., 2021). Previous works mainly focus on the cross-size generalization (Fu et al., 2021; Hou et al., 2023; Son et al., 2023; Luo et al., 2023; Drakulic et al., 2023) or cross-distribution generalization (Zhang et al., 2022; Geisler et al., 2022; Bi et al., 2022; Jiang et al., 2023) or both (Manchanda et al., 2022; Zhou et al., 2023b; Wang et al., 2024) on a single problem. In this paper, we step further to explore the generalization across different VRP variants (Wang & Yu, 2023; Liu et al., 2024; Lin et al., 2024). Mixture-of-Experts. The original idea of MoEs was proposed three decades ago (Jacobs et al., 1991; Jordan & Jacobs, 1994). In early concepts, the expert was defined as an entire neural network, and hence MoEs was similar to an ensemble of neural networks. Eigen et al. (2013) launchs the era when researchers start applying MoEs as components of neural networks. As an early success of MoEs applied in large neural networks, Shazeer et al. (2017) introduces the sparsely-gated MoEs in language modeling and machine translation, achieving state-of-the-art results at the time with only minor losses in computational efficiency. Follow-up works mainly focus on improving the gating mechanism (Lewis et al., 2021; Roller et al., 2021; Zuo et al., 2022; Zhou et al., 2022; Puigcerver et al., 2024; Xue et al., 2024) or applications to other domains (Lepikhin et al., 2020; Riquelme et al., 2021; Fedus et al., 2022b). We refer interested readers to Yuksel et al. (2012); Fedus et al. (2022a) for a comprehensive survey. 3. Preliminaries In this section, we first present the definition of CVRP, and then introduce its variants featured by additional constraints. Afterwards, we delineate recent construction-based neural solvers for VRPs (Kool et al., 2018; Kwon et al., 2020). VRP Variants. We define a CVRP instance of size n over 2 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts -0.5 +0.2 -0.3 +0.1 Linehaul Backhaul Depot [0, 3.0] (O) (B) (L) (TW) 0.2 0.2 0.2 0.1 0.3 < L [1.0, 1.5] [1.4, 1.8] [2.0, 2.4] [2.4, 2.8] Figure 1. Illustrations of sub-tours with various constraints: open route (O), backhaul (B), duration limit (L), and time window (TW). a graph G = {V, E}, where V includes a depot node v0 and customer nodes {vi}n i=1, and E includes edges e(vi, vj) between node vi and vj(i \u0338= j). Each customer node is associated with a demand \u03b4i, and a capacity limit Q is set for each vehicle. The solution (i.e., tour) \u03c4 is represented as a sequence of nodes, consisting of multiple sub-tours. Each sub-tour represents that a vehicle starts from the depot, visits a subset of customer nodes and returns to the depot. The solution is feasible if each customer node is visited exactly once, and the total demand in each sub-tour does not exceed the capacity limit Q. We consider the Euclidean space with the cost function c(\u00b7) defined as the total length of the tour. The objective is to find the optimal tour \u03c4 \u2217with the minimal cost: \u03c4 \u2217= arg min\u03c4\u2208\u03a6 c(\u03c4|G), where \u03a6 is the discrete search space that contains all feasible tours. On top of CVRP (featured by the capacity constraint (C)), several VRP variants involve additional practical constraints. 1) Open Route (O): The vehicle does not need to return to the depot v0 after visiting customers; 2) Backhaul (B): The demand \u03b4i is positive in CVRP, representing a vehicle unloads goods at the customer node. In practice, a customer can have a negative demand, requiring a vehicle to load goods. We name the customer nodes with \u03b4i > 0 as linehauls and the ones with \u03b4i < 0 as backhauls. Hence, VRP with backhaul allows the vehicle traverses linehauls and backhauls in a mixed manner, without strict precedence between them; 3) Duration Limit (L): To maintain a reasonable workload, the cost (i.e., length) of each route is upper bounded by a predefined threshold; 4) Time Window (TW): Each node vi \u2208V is associated with a time window [ei, li] and a service time si. A vehicle must start serving customer vi in the time slot from ei to li. If the vehicle arrives earlier than ei, it has to wait until ei. All vehicles must return to the depot v0 no later than l0. The aforementioned constraints are illustrated in Fig. 1. By combining them, we can obtain 16 typical VRP variants, which are summarized in Table 3. Note that the combination is not a trivial addition of different constraints. For example, when the open route is coupled with the time window, the vehicle does not need to return to the depot, and hence the constraint imposed by l0 at the depot is relaxed. We present more details of VRP variants and the associated data generation process in Appendix A. Learning to Solve VRPs. Typical neural solvers (Kool et al., 2018; Kwon et al., 2020) parameterize the solution construction policy by an attention-based neural network \u03c0\u03b8, which is trained to generate a solution in an autoregressive way. The feasibility of the generated solution is guaranteed by the masking mechanism during decoding. Without loss of generality, we consider RL training paradigm, wherein the solution construction process is formulated as a Markov Decision Process (MDP). Given an input instance, the encoder processes it and attains all node embeddings, which, with the context representation of the constructed partial tour, represent the current state. The decoder takes them as inputs and outputs the probabilities of valid nodes (i.e., actions) to be selected. After a complete solution \u03c4 is constructed, its probability can be factorized via the chain rule such that p\u03b8(\u03c4|G) = QT t=1 p\u03b8(\u03c0(t) \u03b8 |\u03c0(<t) \u03b8 , G), where \u03c0(t) \u03b8 and \u03c0(<t) \u03b8 denote the selected node and constructed partial tour at step t, and T is the number of total steps. The reward is defined as the negative tour length, i.e., R = \u2212c(\u03c4|G). Given a baseline function b(\u00b7) for training stability, the policy network \u03c0\u03b8 is often trained by REINFORCE (Williams, 1992) algorithm, which applies estimated gradients of the expected reward to optimize the policy as below, \u2207\u03b8La(\u03b8|G) = Ep\u03b8(\u03c4|G)[(c(\u03c4) \u2212b(G))\u2207\u03b8 log p\u03b8(\u03c4|G)]. (1) 4. Methodology In this section, we present the multi-task VRP solver with MoEs (MVMoE), and introduce the gating mechanism. Without loss of generality, we aim to learn a constructionbased neural solver (Kool et al., 2018; Kwon et al., 2020) for tackling VRP variants with the five constraints introduced in Section 3. The structure of MVMoE is illustrated in Fig. 2. 4.1. Multi-Task VRP Solver with MoEs Multi-Task VRP Solver. Given an instance of a specific VRP variant, the static features of each node vi are expressed by Si = {yi, \u03b4i, ei, li}, where yi, \u03b4i, ei, li denote the coordinate, demand, start and end time of the time window, respectively. The encoder takes these static node features as inputs, and outputs d-dimensional node embeddings hi. At the tth decoding step, the decoder takes as input the node embeddings and a context representation, including the embedding of the last selected node and dynamic features Dt = {ct, tt, lt, ot}, where ct, tt, lt, ot denote the remaining capacity of the vehicle, the current time, the length of the current partial route, and the presence indicator of the open route, respectively. Thereafter, the decoder outputs the probability distribution of nodes, from which a valid node is selected and appended to the partial solution. A complete solution is constructed in an autoregressive manner by iterating the decoding process. 3 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts Multi-Head Attention with MoE Node Embedding V Q Probability  Update & Loop Solution N x Multi-Head Self-Attention Add & Normalize Add & Normalize Encoder Linear MoE Dot-Product & Softmax Dynamic Feature Last Node Decoder K Static Feature MoE Layer Top 2 -\" $& = ['&, )&, *&, +&] -% -) -* \u210e\" \u210e% \u210e) \u210e* \u210e) \u210e\" \u210e\" \u210e% \u210e* /+ 0+ 1+ 2+ %\" %# %$ %) -(!)# 0#(!) 3 0'(!) -(!)' MOEs(*) & Figure 2. The model structure of MVMoE. [Green part]: Given an input instance, the encoder and decoder output node embeddings and probabilities of nodes to be selected, respectively. The gray nodes are masked to satisfy problem-specific constraints for feasibility. The node with a deeper color denote a later node embedding. [Yellow part]: In an MoE layer, where we take the (node-level) input-choice Top2 gating as an example, the input x (i.e., node) is routed to two experts that derive the two largest probabilities from the gating network G. In each training step, we randomly select a VRP variant, and train the neural network to solve associated instances in a batch. In this way, MVMoE is able to learn a unified policy that can tackle different VRP tasks. If only a subset of static or dynamic features are involved in the current selected VRP variant, the other features are padded to the default values (e.g., zeros). For example, given a CVRP instance, the static features of the ith customer node are S(C) i = {yi, \u03b4i, 0, 0}, and the dynamic features at the tth decoding step are D(C) t = {ct, 0, lt, 0}. In summary, motivated by the fact that different VRP variants may include some common attributes (e.g., coordinate, demand), we define the static and dynamic features as the union set of attributes that exist in all VRP variants. By training on a few VRP variants with these attributes, the policy network has the potential to solve unseen variants, which are characterized by different combinations of these attributes, i.e., the zero-shot generalization capability (Liu et al., 2024). Mixture-of-Experts. Typically, an MoE layer consists of 1) m experts {E1, E2, . . . , Em}, each of which is a linear layer or FFN with independent trainable parameters, and 2) a gating network G parameterized by WG, which decides how the inputs are distributed to experts. Given a single input x, G(x) and Ej(x) denote the output of the gating network (i.e., an m-dimensional vector), and the output of the jth expert, respectively. In light of this, the output of an MoE layer is calculated as, MoE(x) = m X j=1 G(x)jEj(x). (2) Intuitively, a sparse vector G(x) only activates a small subset of experts with partial model parameters, and hence saves the computation. Typically, a TopK operator can achieve such sparsity by only keeping the K-largest values while setting others as the negative infinity. In this case, the gating network calculates the output as G(x) = Softmax(TopK(x \u00b7 WG)). Given the fact that larger sparse models do not always lead to better performance (Zuo et al., 2022), it is crucial yet tricky to design effective and efficient gating mechanisms to endow each expert being sufficiently trained, given enough training data. To this effect, some works have been put forward in language and vision domains, such as designing an auxiliary loss (Shazeer et al., 2017) or formulating it as a linear assignment problem (Lewis et al., 2021) in pursuit of the load balancing. MVMoE. By integrating the above parts, we obtain the multi-task VRP solver with MoEs. The overall model structure is shown in Fig. 2, where we employ MoEs in both the encoder and decoder. In specific, we substitute MoEs for the FFN layer in the encoder, and substitute MoEs for the final linear layer of multi-head attention in the decoder. We refer more details of the structure of MVMoE to Appendix B. We empirically find our design is effective in generating high-quality solutions, and especially employing MoEs in the decoder tends to exert a greater influence on performance (see Section 5.2). We jointly optimize all trainable parameters \u0398, with the objective formulated as follows, min \u0398 L = La + \u03b1Lb, (3) where La denotes the original loss function of the VRP solver (e.g., the REINFORCE loss used to train the policy for solving VRP variants in Eq. (1)), Lb denotes the loss function associated with MoEs (e.g., the auxiliary loss used to ensure load balancing in Eq. (19) in Appendix B), and \u03b1 is a hyperparameter to control its strength. 4.2. Gating Mechanism We mainly consider the node-level (or token-level) gating, by which each node is routed independently to experts.1 In 1In addition, we also investigate another two gating levels, i.e., instance-level and problem-level gating, which are presented in Section 5.2 and Appendix B. 4 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts 2.31 0.45 0.74 0.65 1.43 -0.68 1.20 -1.02 0.98 -0.53 0.76 1.97 -0.35 1.26 0.87 0.36 2.31 0.45 0.74 0.65 1.43 -0.68 1.20 -1.02 0.98 -0.53 0.76 1.97 -0.35 1.26 0.87 0.36 !! !\" !# !$ \"! \"\" \"# \"$ !! !\" !# !$ \"! \"\" \"# \"$ Figure 3. An illustration of the score matrix and gating algorithm. Left panel: Input-choice gating. Right panel: Expert-choice gating. The selected experts or nodes are in color. The arrow marks the dimension, along which the TopK experts or nodes are selected. each MoE layer, the extra computation originates from the forward pass of the gating network and the distribution of nodes to the selected experts. While employing MoEs in the decoder can significantly improve the performance, the number of decoding steps T increases as the problem size n scales up. It suggests that compared to the encoder with a fixed number of gating steps N (\u226aT), applying MoEs in the decoder may substantially increase the computational complexity. In light of this, we propose a hierarchical gating mechanism to make the better use of MoEs in the decoder for gaining a good trade-off between empirical performance and computational complexity. Next, we detail the nodelevel and hierarchical gating mechanism. Node-Level Gating. The node-level gating routes inputs at the granularity of nodes. Let d denote the hidden dimension and WG \u2208Rd\u00d7m denote trainable parameters of the gating network in MVMoE. Given a batch of inputs X \u2208RI\u00d7d, where I is the total number of nodes (i.e., batch size B \u00d7 problem scale n), each node is routed to the selected experts based on the score matrix H = (X \u00b7 WG) \u2208RI\u00d7m predicted by the gating network. We illustrate an example of the score matrix in Fig. 3, where xi denotes the ith node, and Ej denotes the jth expert in the node-level gating. In this paper, we mainly consider two popular gating algorithms (Shazeer et al., 2017; Zhou et al., 2022): 1) Inputchoice gating: Each node selects TopK experts based on H. Typically, K is set to 1 or 2 to retain a reasonable computational complexity. The input-choice gating is illustrated in the left panel of Fig. 3, where each node is routed to two experts with the largest scores (i.e., Top2). However, this method cannot guarantee load balancing. An expert may receive much more nodes than the others, resulting in a dominant expert while leaving others underfitting. To address this issue, most works employ an auxiliary loss to equalize quantities of nodes sent to different experts during training. Here we use the importance & load loss (Shazeer et al., 2017) as Lb in Eq. (3) to mitigate load imbalance (see Appendix B). 2) Expert-choice gating: Each expert selects Base Gating Top 2 Dense Layer Sparse Layer If Else Hierarchical Gating &\" &# ' %\" %# %$ %) %\" %# %) %$ 3 3 & Figure 4. A base gating (i.e., the input-choice gating with K = 2) and its hierarchical gating counterpart. In the latter, the gating network G1 routes inputs to the sparse layer ({G2, E1, E2, E3, E4}) or the dense layer D. If the sparse layer is chosen, the gating network G2 routes nodes to experts accoring to the base gating. TopK nodes based on H. Typically, K is set to I\u00d7\u03b2 m , where \u03b2 is the capacity factor reflecting the average number of experts utilized by a node. The expert-choice gating is illustrated in the right panel of Fig. 3, where each expert selects two nodes with the largest scores given \u03b2 = 2. While this gating algorithm explicitly ensures load balancing, some nodes may not be chosen by any expert. We refer more details of the above gating algorithms to Appendix B. Hierarchical Gating. In the VRP domain, it is computationally expensive to employ MoEs in each decoding step, since 1) the number of decoding steps T increases as the problem size n rises; 2) the problem-specific feasibility constraints must be satisfied during decoding. To tackle the challenges, we propose to employ MoEs only in partial decoding steps. Accordingly, we present a hierarchical gating, which learns to effectively and efficiently utilize MoEs during decoding. We illustrate the proposed hierarchical gating in Fig. 4. An MoE layer with the hierarchical gating includes two gating networks {G1, G2}, m experts {E1, E2, . . . , Em}, and a dense layer D (e.g., a linear layer). Given a batch of inputs X \u2208RI\u00d7d, the hierarchical gating routes them in two stages. In the first stage, G1 decides to distribute inputs X to either the sparse or dense layer according to the problem-level representation X1. In specific, we obtain X1 by applying the mean pooling along the first dimension of X, and process it to obtain the score matrix H1 = (X1 \u00b7 WG1) \u2208R1\u00d72. Then, we route the batch of inputs X to the sparse or dense layer by sampling from the probability distribution G1(X) = Softmax(H1). Here we employ the problem-level gating in G1 for the generality and efficiency of the hierarchical gating (see Appendix D for further discussions). In the second stage, if X is routed to the sparse layer, the gating network G2 is activated to route nodes to experts on the node-level by using aforementioned gating algorithms (e.g., the input-choice gating). Otherwise, X is routed to the dense layer D and transformed into D(X) \u2208RI\u00d7d. In summary, the hierarchical gating learns to output G1(X)0 Pm j=1 G2(X)jEj(X) or G1(X)1D(X) 5 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts Table 1. Performance on 1K test instances of trained VRPs. * represents 0.000%, with which the gaps are computed. Method n = 50 n = 100 Method n = 50 n = 100 Obj. Gap Time Obj. Gap Time Obj. Gap Time Obj. Gap Time CVRP HGS 10.334 * 4.6m 15.504 * 9.1m VRPTW HGS 14.509 * 8.4m 24.339 * 19.6m LKH3 10.346 0.115% 9.9m 15.590 0.556% 18.0m LKH3 14.607 0.664% 5.5m 24.721 1.584% 7.8m OR-Tools 10.540 1.962% 10.4m 16.381 5.652% 20.8m OR-Tools 14.915 2.694% 10.4m 25.894 6.297% 20.8m OR-Tools (x10) 10.418 0.788% 1.7h 15.935 2.751% 3.5h OR-Tools (x10) 14.665 1.011% 1.7h 25.212 3.482% 3.5h POMO 10.418 0.806% 3s 15.734 1.488% 9s POMO 14.940 2.990% 3s 25.367 4.307% 11s POMO-MTL 10.437 0.987% 3s 15.790 1.846% 9s POMO-MTL 15.032 3.637% 3s 25.610 5.313% 11s MVMoE/4E 10.428 0.896% 4s 15.760 1.653% 11s MVMoE/4E 14.999 3.410% 4s 25.512 4.903% 12s MVMoE/4E-L 10.434 0.955% 4s 15.771 1.728% 10s MVMoE/4E-L 15.013 3.500% 3s 25.519 4.927% 11s OVRP LKH3 6.511 0.198% 4.5m 9.828 * 5.3m VRPL LKH3 10.571 0.790% 7.8m 15.771 * 16.0m OR-Tools 6.531 0.495% 10.4m 10.010 1.806% 20.8m OR-Tools 10.677 1.746% 10.4m 16.496 4.587% 20.8m OR-Tools (x10) 6.498 * 1.7h 9.842 0.122% 3.5h OR-Tools (x10) 10.495 * 1.7h 16.004 1.444% 3.5h POMO 6.609 1.685% 2s 10.044 2.192% 8s POMO 10.491 -0.008% 2s 15.785 0.093% 9s POMO-MTL 6.671 2.634% 2s 10.169 3.458% 8s POMO-MTL 10.513 0.201% 2s 15.846 0.479% 9s MVMoE/4E 6.655 2.402% 3s 10.138 3.136% 10s MVMoE/4E 10.501 0.092% 3s 15.812 0.261% 10s MVMoE/4E-L 6.665 2.548% 3s 10.145 3.214% 9s MVMoE/4E-L 10.506 0.131% 3s 15.821 0.323% 10s VRPB OR-Tools 8.127 0.989% 10.4m 12.185 2.594% 20.8m OVRPTW OR-Tools 8.737 0.592% 10.4m 14.635 1.756% 20.8m OR-Tools (x10) 8.046 * 1.7h 11.878 * 3.5h OR-Tools (x10) 8.683 * 1.7h 14.380 * 3.5h POMO 8.149 1.276% 2s 11.993 0.995% 7s POMO 8.891 2.377% 3s 14.728 2.467% 10s POMO-MTL 8.182 1.684% 2s 12.072 1.674% 7s POMO-MTL 8.987 3.470% 3s 15.008 4.411% 10s MVMoE/4E 8.170 1.540% 3s 12.027 1.285% 9s MVMoE/4E 8.964 3.210% 4s 14.927 3.852% 11s MVMoE/4E-L 8.176 1.605% 3s 12.036 1.368% 8s MVMoE/4E-L 8.974 3.322% 4s 14.940 3.941% 10s based on both problem-level and node-level representations. Overall, the hierarchical gating improves the computational efficiency with a minor loss on the empirical performance. To balance the efficiency and performance of MVMoE, we use the base gating in the encoder and its hierarchical gating counterpart in the decoder. Note that the hierarchical gating is applicable to different gating algorithms, such as the input-choice gating (Shazeer et al., 2017) and expert-choice gating (Zhou et al., 2022). We also explore a more advanced gating algorithm (Puigcerver et al., 2024) for reducing the number of routed nodes and thus the computational complexity. But its empirical performance is unsatisfactory in the VRP domain (see Section 5.3). 5. Experiments In this section, we empirically verify the superiority of the proposed MVMoE, and provide insights into the application of MoEs to solve VRPs. We consider 16 VRP variants with five constraints. Due to page limit, we present more experimental results in Appendix C. All experiments are conducted on a machine with NVIDIA Ampere A100-80GB GPU cards and AMD EPYC 7513 CPU at 2.6GHz. Baselines. Traditional solvers: We employ HGS (Vidal, 2022) to solve CVRP and VRPTW instances with default hyperparameters (i.e., the maximum number of iterations without improvement is 20000). We run LKH3 (Helsgaun, 2017) to solve CVRP, OVRP, VRPL and VRPTW instances with 10000 trails and 1 run. OR-Tools (Furnon & Perron, 2023) is an open source solver for complex optimization problems. It is more versatile than LKH and HGS, and can solve all 16 VRP variants considered in this paper. We use the parallel cheapest insertion as the first solution strategy, and use the guided local search as the local search strategy in OR-Tools. For n = 50/100, we set the search time limit as 20s/40s to solve an instance, and also provide its results given 200s/400s (i.e., OR-Tools (x10)). For all traditional solvers, we use them to solve 32 instances in parallel on 32 CPU cores following Kool et al. (2018). Neural solvers: We compare our method to POMO (Kwon et al., 2020) and POMO-MTL (Liu et al., 2024). While POMO is trained on each single VRP, POMO-MTL is trained on multiple VRPs by multi-task learning. Note that POMO-MTL is the dense model counterpart of MVMoE, which is structured by dense layers (e.g., FFNs) rather than sparse MoEs. In specific, POMO-MTL and MVMoE/4E possess 1.25M and 3.68M parameters, but they activate a similar number of parameters for each single input. Training. We follow most setups in (Kwon et al., 2020). 1) For all neural solvers: Adam optimizer is used with the learning rate of 1e \u22124, the weight decay of 1e \u22126, and the batch size of 128. The model is trained for 5000 epochs, with each containing 20000 training instances (i.e., 100M training instances in total). The learning rate is decayed by 10 for the last 10% training instances. We consider two problem scales n \u2208{50, 100} during training, according to (Liu et al., 2024). 2) For multi-task solvers: The training problem set includes CVRP, OVRP, VRPB, VRPL, VRPTW, and OVRPTW (see Appendix C.1 for further discussions). In each batch of training, we randomly sample a problem from the set and generate its instances. Please refer to Appendix A for details of the generation procedure. 3) For our method: We employ m = 4 experts with K = \u03b2 = 2 in each MoE layer, and set the the weight \u03b1 of the auxiliary loss Lb as 0.01. The default gating mechanism of MVMOE/4E is the node-level input-choice gating in both the encoder 6 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts Table 2. Zero-shot generalization on 1K test instances of unseen VRPs. * represents 0.000%, with which the gaps are computed. Method n = 50 n = 100 Method n = 50 n = 100 Obj. Gap Time Obj. Gap Time Obj. Gap Time Obj. Gap Time OVRPB OR-Tools 5.764 0.332% 10.4m 8.522 1.852% 20.8m OVRPL OR-Tools 6.522 0.480% 10.4m 9.966 1.783% 20.8m OR-Tools (x10) 5.745 * 1.7h 8.365 * 3.5h OR-Tools (x10) 6.490 * 1.7h 9.790 * 3.5h POMO-MTL 6.116 6.430% 2s 8.979 7.335% 8s POMO-MTL 6.668 2.734% 2s 10.126 3.441% 9s MVMoE/4E 6.092 5.999% 3s 8.959 7.088% 9s MVMoE/4E 6.650 2.454% 3s 10.097 3.148% 10s MVMoE/4E-L 6.122 6.522% 3s 8.972 7.243% 9s MVMoE/4E-L 6.659 2.597% 3s 10.106 3.244% 9s VRPBL OR-Tools 8.131 1.254% 10.4m 12.095 2.586% 20.8m VRPBTW OR-Tools 15.053 1.857% 10.4m 26.217 2.858% 20.8m OR-Tools (x10) 8.029 * 1.7h 11.790 * 3.5h OR-Tools (x10) 14.771 * 1.7h 25.496 * 3.5h POMO-MTL 8.188 1.971% 2s 11.998 1.793% 8s POMO-MTL 16.055 8.841% 3s 27.319 7.413% 10s MVMoE/4E 8.172 1.776% 3s 11.945 1.346% 9s MVMoE/4E 16.022 8.600% 4s 27.236 7.078% 11s MVMoE/4E-L 8.180 1.872% 3s 11.960 1.473% 9s MVMoE/4E-L 16.041 8.745% 4s 27.265 7.190% 10s VRPLTW OR-Tools 14.815 1.432% 10.4m 25.823 2.534% 20.8m OVRPBL OR-Tools 5.771 0.549% 10.4m 8.555 2.459% 20.8m OR-Tools (x10) 14.598 * 1.7h 25.195 * 3.5h OR-Tools (x10) 5.739 * 1.7h 8.348 * 3.5h POMO-MTL 14.961 2.586% 3s 25.619 1.920% 12s POMO-MTL 6.104 6.306% 2s 8.961 7.343% 8s MVMoE/4E 14.937 2.421% 4s 25.514 1.471% 13s MVMoE/4E 6.076 5.843% 3s 8.942 7.115% 9s MVMoE/4E-L 14.953 2.535% 4s 25.529 1.545% 12s MVMoE/4E-L 6.104 6.310% 3s 8.957 7.300% 9s OVRPBTW OR-Tools 8.758 0.927% 10.4m 14.713 2.268% 20.8m OVRPLTW OR-Tools 8.728 0.656% 10.4m 14.535 1.779% 20.8m OR-Tools (x10) 8.675 * 1.7h 14.384 * 3.5h OR-Tools (x10) 8.669 * 1.7h 14.279 * 3.5h POMO-MTL 9.514 9.628% 3s 15.879 10.453% 10s POMO-MTL 8.987 3.633% 3s 14.896 4.374% 11s MVMoE/4E 9.486 9.308% 4s 15.808 9.948% 11s MVMoE/4E 8.966 3.396% 4s 14.828 3.903% 12s MVMoE/4E-L 9.515 9.630% 3s 15.841 10.188% 10s MVMoE/4E-L 8.974 3.488% 4s 14.839 3.971% 10s VRPBLTW OR-Tools 14.890 1.402% 10.4m 25.979 2.518% 20.8m OVRPBLTW OR-Tools 8.729 0.624% 10.4m 14.496 1.724% 20.8m OR-Tools (x10) 14.677 * 1.7h 25.342 * 3.5h OR-Tools (x10) 8.673 * 1.7h 14.250 * 3.5h POMO-MTL 15.980 9.035% 3s 27.247 7.746% 11s POMO-MTL 9.532 9.851% 3s 15.738 10.498% 10s MVMoE/4E 15.945 8.775% 4s 27.142 7.332% 12s MVMoE/4E 9.503 9.516% 4s 15.671 10.009% 11s MVMoE/4E-L 15.963 8.915% 4s 27.177 7.473% 11s MVMoE/4E-L 9.518 9.682% 4s 15.706 10.263% 10s 0 2 4 6 8 10 Epoch 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 Gap (%) VRPBLTW POMO-MTL MVMoE/4E MVMoE/4E-L 0 2 4 6 8 10 Epoch 9.0 9.2 9.4 9.6 9.8 Gap (%) OVRPBLTW POMO-MTL MVMoE/4E MVMoE/4E-L Figure 5. Few-shot generalization on unseen VRPs. and decoder layers. MVMoE/4E-L is a computationally light version that replaces the input-choice gating with its hierarchical gating counterpart in the decoder. Inference. For all neural solvers, we use greedy rollout with x8 instance augmentation following Kwon et al. (2020). We report the average results (i.e., objective values and gaps) over the test dataset that contains 1K instances, and the total time to solve the entire test dataset. The gaps are computed with respect to the results of the best-performing traditional VRP solvers (i.e., * in Tables 1 and 2). 5.1. Empirical Results Performance on Trained VRPs. We evaluate all methods on 6 trained VRPs and gather all results in Table 1. The single-task neural solver (i.e., POMO) achieves better performance than multi-task neural solvers on each single problem, since it is restructured and retrained on each VRP independently. However, its average performance over all trained VRPs is quite inferior as shown in Table 4 in Appendix C, since each trained POMO is overfitted to a specific VRP. For example, the average performance of POMO solely trained on CVRP is 16.815%, while POMO-MTL and MVMoE/4E achieve 2.102% and 1.925%, respectively. Notably, our neural solvers consistently outperform POMO-MTL. MVMoE/4E performs slightly better than MVMoE/4E-L at the expense of more computation. Despite that, MVMoE/4E-L exhibits stronger out-of-distribution generalization capability than MVMoE/4E (see Tables 7 and 8 in Appendix C). Generalization on Unseen VRPs. We evaluate multi-task solvers on 10 unseen VRP variants. 1) Zero-shot generalization: We directly test the trained solvers on unseen VRPs. The results in Table 2 reveal that the proposed MVMoE significantly outperforms POMO-MTL across all VRP variants. 2) Few-shot generalization: We also consider the few-shot setting on n = 50, where a trained solver is fine-tuned on the target VRP using 10K instances (0.01% of total training instances) in each epoch. Without loss of generality, we conduct experiments on VRPBLTW and OVRPBLTW following the training setups. The results in Fig. 5 showcase MVMoE generalizes more favorably than POMO-MTL. 5.2. Ablation on MoEs Here we explore the effect of different MoE settings on the zero-shot generalization of neural solvers, and provide insights on how to effectively apply MoEs to solve VRPs. Due to the fast convergence, we reduce the number of epochs to 2500 on VRPs of the size n = 50, while leaving other setups unchanged. We set MVMoE/4E as the default baseline, 7 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts None Raw Enc_Every_Two Enc_Last_Two Enc_All Dec Enc_All_Dec Position of MoEs 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Gap (%) (a) POMO-MTL_50M MVMoE/4E_50M MVMoE/8E_50M MVMoE/16E_50M MVMoE/8E_100M MVMoE/16E_200M Number of Experts 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 Gap (%) (b) POMO-MTL Node-Input Node-Expert Node-Random Instance-Input Instance-Expert Instance-Random Problem-Random Gating Mechanism 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 Gap (%) (c) POMO-MTL Input_Choice Soft MoEs Neural Hierarchical Heuristic Hierarchical Advanced Gating 5.7 5.8 5.9 6.0 6.1 6.2 6.3 Gap (%) (d) POMO-MTL Input_Choice Soft MoEs Neural Hierarchical Heuristic Hierarchical Training Efficiency 0 20 40 60 80 100 120 140 160 Time (hr) (e) n = 50 n = 100 Figure 6. Left three panels: The effect of MoE settings on the average zero-shot generalization performance (a) the position of MoEs; (b) the number of experts; (c) the gating mechanism. Right two panels: Further analyses (d) average zero-shot generalization performance of each method employing various gating algorithms in the decoder; (e) training efficiency of each gating algorithm. and ablate on different components of MoEs below. Position of MoEs. We consider three positions to apply MoEs in neural solvers: 1) Raw feature processing (Raw): The linear layer, which projects raw features into initial embeddings, are replaced by MoEs. 2) Encoder (Enc): The FFN in an encoder layer is replaced by MoEs. Typically, MoEs are widely used in every-two or last-two layers (i.e., every or last two layers with even indices \u2113\u2208[0, N \u2212 1]) (Riquelme et al., 2021). Besides, we further attempt to use MoEs in all encoder layers. 3) Decoder (Dec): The final linear layer of the multi-head attention is replaced by MoEs in the decoder. We show the average performance over 10 unseen VRPs in Fig. 6(a). The results reveal that applying MoEs at the shallow layer (e.g., Raw) may worsen the model performance, while using MoEs in all encoder layers (Enc_All) or decoder (Dec) can benefit the zero-shot generalization. Therefore, in this paper, we employ MoEs in both encoder and decoder to pursue a strong unified model architecture to solve various VRPs. Number of Experts. We increase the number of experts in each MoE layer to 8 and 16, and compare the derived MVMoE/8E/16E models to MVMoE/4E. We first train all models using the same number (50M) of instances. After that, we also train MVMoE/8E/16E with more data and computation to explore potential better results, based on the scaling laws (Kaplan et al., 2020). In specific, we provide MVMoE/8E/16E with more data by using larger batch sizes, which linearly scale up against the number of experts (i.e., MVMoE/4E/8E/16E are trained on 50M/100M/200M instances with batch sizes 128/256/512, respectively). The results in Fig. 6(b) show that increasing the number of experts with more training data further unleashes the power of MVMoE, indicating the efficacy of MoEs in solving VRPs. Gating Mechanism. We investigate the effect of different gating levels and algorithms, including three levels (i.e., node-level, instance-level and problem-level) and three algorithms (i.e., input-choice, expert-choice and random gatings), with their details presented in Appendix B. As shown in Fig. 6(c), the node-level input-choice gating performs the best, while the node-level expert-choice gating performs the worst. Interestingly, we observe that the expert-choice gating in the decoder makes MVMoE hard to be optimized. It may suggest that each gating algorithm could have its most suitable position to serve MoEs. However, after an attempt to tune this configuration (i.e., by using MoEs only in the encoder), its performance is still inferior to the baseline, with an average gap of 7.190% on unseen VRPs. 5.3. Additional Results We further provide experiments and discussions on more advanced gating algorithms, training efficiency, benchmark performance, and scalability. We refer readers to more empirical results (e.g., sensitivity analyses) in Appendix C. Advanced Gating. Besides the input-choice and expertchoice gating algorithms evaluated above, we further consider soft MoEs (Puigcerver et al., 2024), which is a recent advanced gating algorithm. Specifically, it performs an implicit soft assignment by distributing K slots (i.e., convex combinations of all inputs) to each expert, rather than a hard assignment between inputs and experts as done by the conventional sparse and discrete gating networks. Since only K (e.g., 1 or 2) slots are distributed to each expert, it can save much computation. We train MVMoE on n = 50 by using node-level soft MoEs in the decoder, following training setups. We also show the result of employing heuristic (random) hierarchical gating in the decoder. However, their results are unsatisfactory as shown in Fig. 6(d). Training Efficiency. Fig. 6(e) shows the training time of employing each gating algorithm in the decoder, combining with their results reported in Fig. 6(d), demonstrating the efficacy of the proposed hierarchical gating in reducing the training overhead with only minor losses in performance. Benchmark Performance. We further evaluate the out-ofdistribution (OOD) generalization performance of all neural solvers on CVRPLIB benchmark instances. Detailed results can be found in Tables 7 and 8 in Appendix C. Surprisingly, we observe that MVMoE/4E performs poorly on large-scale 8 \fMVMoE: Multi-Task Vehicle Routing Solver with Mixture-of-Experts instances (e.g., n > 500). It may be caused by the generalization issue of sparse MoEs when transferring to new distributions or domains, which is still an open question in the MoE literature (Fedus et al., 2022a). In contrast, MVMoE/4E-L mostly outperforms MVMoE/4E, demonstrating more favourable potential of the hierarchical gating in promoting the OOD generalization capability. It is worth noting that all neural solvers are only trained on the simple uniformly distributed instances with the size n = 100. Embracing more varied problem sizes (cross-size) and attribute distributions (cross-distribution) into the multi-task training (cross-problem) may further consolidate their performance. Scalability. Given that supervised learning based approaches appear to be more scalable than RL-based approaches in the current literature, we try to build upon a more scalable method, i.e., LEHD (Luo et al., 2023). Concretely, we train a dense model LEHD and a light sparse model with 4 experts LEHD/4E-L on CVRP. The training setups are kept the same as Luo et al. (2023), except that we train all models for only 20 epochs for the training efficiency. We use the hierarchical MoE in each decoder layer of LEHD/4E-L. The results are shown in Table 8, which demonstrates the potential of MoE as a general idea that can further benefit recent scalable methods. Moreover, during the solution construction process, recent works (Drakulic et al., 2023; Gao et al., 2023) typically constrain the search space within a neighborhood of the currently selected node, which is shown to be effective in handling large-scale instances. Integrating MVMoE with these simple yet effective techniques may further improve large-scale performance. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2305.19587v2",
+                    "title": "Towards Omni-generalizable Neural Methods for Vehicle Routing Problems",
+                    "abstract": "Learning heuristics for vehicle routing problems (VRPs) has gained much\nattention due to the less reliance on hand-crafted rules. However, existing\nmethods are typically trained and tested on the same task with a fixed size and\ndistribution (of nodes), and hence suffer from limited generalization\nperformance. This paper studies a challenging yet realistic setting, which\nconsiders generalization across both size and distribution in VRPs. We propose\na generic meta-learning framework, which enables effective training of an\ninitialized model with the capability of fast adaptation to new tasks during\ninference. We further develop a simple yet efficient approximation method to\nreduce the training overhead. Extensive experiments on both synthetic and\nbenchmark instances of the traveling salesman problem (TSP) and capacitated\nvehicle routing problem (CVRP) demonstrate the effectiveness of our method. The\ncode is available at: https://github.com/RoyalSkye/Omni-VRP.",
+                    "authors": "Jianan Zhou, Yaoxin Wu, Wen Song, Zhiguang Cao, Jie Zhang",
+                    "published": "2023-05-31",
+                    "updated": "2023-06-20",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Combinatorial optimization problems (COPs) are of great importance in computer science and operation research. The exact methods suffer from the scalability issue due to the NP-hardness, while the heuristic ones need substantial handcrafted rules and domain expertise for each specific problem. Recently, the neural method, which leverages machine learning (ML) to automatically learn or discover heuristics for a wide range of COPs, has gained much attention (Bengio et al., 2021). By exploiting the underlying pattern among a 1School of Computer Science and Engineering, Nanyang Technological University, Singapore 2Department of Information Systems, Eindhoven University of Technology, The Netherlands 3Institute of Marine Science and Technology, Shandong University, China 4School of Computing and Information Systems, Singapore Management University, Singapore. Correspondence to: Yaoxin Wu <wyxacc@hotmail.com>, Wen Song <wensong@email.sdu.edu.cn>. Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s). group of COP instances, the neural method has the potential to reduce computational efforts while achieving desirable solution quality. In this paper, we focus on vehicle routing problems (VRPs), which is a class of canonical NP-hard COPs with wide applications in transportation (Garaix et al., 2010; Zhou et al., 2023; Wu et al., 2023) and logistics (Cattaruzza et al., 2017; Konstantakopoulos et al., 2022). The neural methods for VRPs usually employ advanced deep models (e.g., pointer network (Vinyals et al., 2015), attention mechanism (Vaswani et al., 2017) and graph neural network (Bresson & Laurent, 2017)) to learn heuristics for route construction or improvement with supervised learning or reinforcement learning. They have achieved competitive or even superior performance to the conventional heuristics. However, most of the neural methods are trained and tested on the same task with a fixed size and distribution, and thus suffer from poor generalization. For example, the popular attention-based models (Kool et al., 2018; Kwon et al., 2020) are only trained and tested on instances of fixed size (e.g., 100), with node coordinates sampled from the uniform distribution. The performance of the learned heuristic drastically decreases when it is applied to an unseen task during training (see POMO* in Table 1). This generalization issue severely hinders the application of these neural methods in practice. A simple measure for improvement is to train them on diverse data. However, covering the whole problem space is intractable (e.g., the \"Catch-22\" for NP-hard problems (Yehuda et al., 2020)). Thus how to effectively learn from diverse data for VRPs is still a challenging problem. While some attempts have been made to tackle the generalization issue of neural methods for VRPs, most of them solely focus on either size (Lisicki et al., 2020; Bdeir et al., 2022; Kim et al., 2022b) or distribution (Zhang et al., 2022; Jiang et al., 2022; Wang et al., 2022; Geisler et al., 2022; Bi et al., 2022). We argue that it is more realistic to simultaneously consider the generalization of size and distribution, since the real-world VRP instances (e.g., TSPLIB (Reinelt, 1991) and CVRPLIB (Uchoa et al., 2017)) may vary in both. As a promising work, Manchanda et al. (2022) handles this challenging setting by exploiting a meta-learning technique, i.e., Reptile (Nichol et al., 2018), which only relies on the first-order derivatives for training. However, it is still far from satisfaction in that, 1) Reptile is less sample efficient as it typically needs multiple inner-loop updates to incorporate 1 arXiv:2305.19587v2  [cs.LG]  20 Jun 2023 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems information from higher-order derivatives (see Appendix A) to achieve desirable performance (see Appendix D.2). Moreover, theoretical and empirical evidences of its effectiveness are only demonstrated on the few-shot supervised learning setting (Nichol et al., 2018) rather than the reinforcement learning setting, the latter of which is more favored by the neural methods for VRPs; 2) it simply selects the training task by randomly sampling from the task set in each iteration, which overlooks the training dynamics and fails to fully make use of the diverse data information. In this paper, we tackle the omni-generalization issue of neural VRP methods by training on diverse tasks, each of which relates to a unique size and distribution. According to the No Free Lunch Theorems of Machine Learning (Wolpert & Macready, 1997), it is unrealistic to train a one-size-fitsall model that could perform well on any task. Therefore, we also resort to meta-learning (Vilalta & Drissi, 2002; Hospedales et al., 2021) to learn a good initialized model for fine-tuning afterwards, while bypassing the limitations in Manchanda et al. (2022). Specifically, we propose a generic meta-learning framework, which is model-agnostic and compatible with any model trained with gradient updates. It learns a good initialization of model parameters by performing meta-training on tasks, which are adaptively selected from the training task set via a hierarchical scheduler. The trained model is able to efficiently adapt to new tasks only using limited data during inference. Despite being effective, it needs the second-order derivatives to perform meta-model updates, making it computationally expensive when training on instances of large sizes. Therefore, we further develop a simple yet efficient approximation method by early stopping the usage of second-order derivatives, and only leveraging the first-order ones afterwards. Our contributions are summarized as follows. 1) We study a challenging yet realistic setting for neural VRP methods by considering the omni-generalization across both size and distribution. 2) We propose a generic meta-learning framework, where we leverage a second-order technique to enable effective learning of an initialized model with the capability of efficient adaptation to new tasks only using limited data during inference. 3) To reduce the meta-training cost, we develop a simple yet efficient approximation method, which performs comparable to the one with full secondorder derivatives. 4) We evaluate the effectiveness of our method on the traveling salesman problem (TSP) and capacitated vehicle routing problem (CVRP) by meta-training POMO (Kwon et al., 2020) and L2D (Li et al., 2021). The experimental results demonstrate that our method could improve the omni-generalization of the base models even on the zero-shot setting. We also observe consistent superiority on the few-shot settings, and on the classical benchmark datasets such as TSPLIB and CVRPLIB. 2. Related Work Neural Methods for VRPs. Most of recent neural methods for VRPs could be divided into two categories: 1) Learning Construction Heuristics: the solution is constructed sequentially or in a one-shot manner by the learned heuristic without iterative modifications. Vinyals et al. (2015) proposes the Pointer Network (Ptr-Net) to solve TSP with supervised learning. Subsequent works train Ptr-Net using reinforcement learning to solve TSP (Bello et al., 2017) and CVRP (Nazari et al., 2018). Kool et al. (2018) introduces the attention model (AM) based on the Transformer architecture (Vaswani et al., 2017) to solve a wide range of COPs including TSP and CVRP. Kwon et al. (2020) further proposes the policy optimization with multiple optima (POMO), which improves upon AM by exploiting solution symmetries. Besides Ptr-Net and attention-based models, graph neural networks are also exploited to solve VRPs (Dai et al., 2017; Joshi et al., 2019). Other works have also been proposed to improve upon them (Ma et al., 2019; Kwon et al., 2021; Xin et al., 2021a; Kim et al., 2022a). 2) Learning Improvement Heuristics: an initial complete solution is iteratively refined by the learned heuristic until a termination condition is satisfied. In this line of research, the classical local search methods (e.g., 2-opt (Croes, 1958) or large neighborhood search (LNS) (Shaw, 1998)) or (part of) specialized heuristic solvers for VRPs (e.g., Lin-KernighanHelsgaun (LKH) (Helsgaun, 2000; 2017)) are usually exploited (Chen & Tian, 2019; Lu et al., 2020; Hottung & Tierney, 2020; d O Costa et al., 2020; Wu et al., 2021; Wang et al., 2021; Ma et al., 2021; Xin et al., 2021b; Kim et al., 2021; Hudson et al., 2022). In general, the improvement heuristics could achieve better performance than the construction ones, but at the expense of much longer inference time. Besides VRPs, neural methods have also been applied to solve other COPs, such as the job shop scheduling problem (JSSP) (Zhang et al., 2020), maximal independent set (MIS) (Dai et al., 2017; Ahn et al., 2020) and boolean satisfiability (SAT) (Selsam et al., 2019). In this paper, we mainly focus on the neural methods for VRPs. Generalization of Neural Methods. Previous works mainly train and test on instances with the same size and distribution, which results in poor generalization performance (Joshi et al., 2021; Liu et al., 2022). Recently, some works attempt to improve the generalization capability of neural methods for VRPs, which could be further divided into two classes: 1) Size Generalization: the aim is to generalize the learned heuristics to instances smaller or larger1 than the training ones. Lisicki et al. (2020) trains a model on multiple sizes via the curriculum learning strategy (Bengio et al., 2009). Kim et al. (2022b) and Bdeir et al. (2022) 1The community cares more about the larger instances since they are usually much harder than the smaller ones. 2 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems study the size generalization upon the attention-based models by improving their structures, such as incorporating a scale conditioned network and a sparse dynamic attention, respectively. We also would like to mention another line of research that deals with the scalability, which is less related to the generalization issue studied in this paper. Specifically, they mainly rely on the heat-map (Joshi et al., 2019; Fu et al., 2021; Qiu et al., 2022; Sun & Yang, 2023) or decomposition (Li et al., 2021; Hou et al., 2023) so as to scale up to larger instances. 2) Distribution Generalization: the aim is to generalize the learned heuristics to instances sampled from various (unseen) distributions. Some works augment the training instances with diverse distributions either by jointly training an instance generator (Wang et al., 2022) or adversarial training (Zhang et al., 2022; Geisler et al., 2022). Other than the data perspective, Jiang et al. (2022) and Bi et al. (2022) exploit the group distributionally robust optimization (Sagawa et al., 2020) and knowledge distillation (Hinton et al., 2015) to improve the distribution generalization, respectively. Although some efforts have been made either for the size or distribution generalization, rare works consider both of them. Manchanda et al. (2022) first studies this setting using Reptile (Nichol et al., 2018). It gradually updates the metamodel towards the task-specific model on each task such that the learned meta-model could serve as a good initialization for fine-tuning during inference. In this paper, we also focus on this challenging yet realistic problem setting. 3. Preliminaries In this section, we introduce the problem statement for VRPs and the Markov Decision Process (MDP) formulation for constructing solutions to VRP instances. Without loss of generality, we define a VRP instance of size n over a graph G = {V, E}, where vi \u2208V (V = {vi}n i=1) represents the node (e.g., customer in TSP) and e(vi, vj) \u2208E represents the edge between node vi and vj. The solution (i.e., tour) \u03c4 is represented as a sequence of nodes in V. In this paper, we consider the Euclidean VRPs with the cost function c(\u00b7) defined as the total length of the tour. The objective of VRPs is to find the optimal tour \u03c4 \u2217with the minimal cost: \u03c4 \u2217= arg min \u03c4\u2208S c(\u03c4|G), (1) where S is the discrete search space that contains all the feasible tours, subject to the problem-specific constraints. Specifically, a feasible tour in TSP should visit each (customer) node exactly once and return to the starting node in the end. On top of TSP, a depot node v0 is introduced in CVRP, where each customer node is featured by a demand \u03b4i, and a capacity limit Q is set for each vehicle. A tour in CVRP consists of multiple sub-tours, each of which represents a vehicle starting from the depot, visiting a subset of nodes in V and returning to the depot. It is feasible if each customer node is visited exactly once and the total demand in each sub-tour does not exceed the capacity limit Q. Neural construction methods formulate the solving process of a VRP instance G as a MDP, where they typically parameterize the policy by an encoder-decoder structured neural network to learn node selection for constructing a solution. The encoder in the policy network outputs a global representation of the instance, which, with the representation of the context (e.g., the partial tour in construction), captures the current state. The decoder takes the global and context representations as inputs to compute the probabilities of nodes (i.e., actions) to be visited. The node is selected sequentially until a complete tour \u03c4 is constructed. Hence, the probability of the tour is factorized via the chain rule as: p\u03b8(\u03c4|G) = T Y t=1 p\u03b8(\u03c0\u03b8(t)|\u03c0\u03b8(< t), G), (2) where \u03c0\u03b8(t) and \u03c0\u03b8(< t) are the selected node and the current partial solution at time step t, respectively; T denotes the number of total steps. The reward is defined as the negative cost of a tour, i.e., R = \u2212c(\u03c4|G). To train the policy network, the REINFORCE algorithm (Williams, 1992) is commonly used to estimate the gradient of the expected reward L(\u03b8|G) = Ep\u03b8(\u03c4|G)c(\u03c4) such that: \u2207\u03b8L(\u03b8|G) = Ep\u03b8(\u03c4|G)[(c(\u03c4) \u2212b(G))\u2207\u03b8 log p\u03b8(\u03c4|G)], (3) where b(\u00b7) is a baseline function to reduce gradient variance. 4. Methodology In this section, we introduce a generic meta-learning framework to tackle the omni-generalizable issue of neural methods across both size and distribution in VRPs. We further develop a simple yet efficient approximation method to promote the training efficiency. Without loss of generality, we present our method by taking the meta-training of POMO (Kwon et al., 2020) on TSP as an example. 4.1. Meta-Learning Framework We define a task T (n, d) \u223cp(T ) as a class of instances with the same size n \u2208N and distribution d \u2208D, where N = [nmin, nmax] is the range of problem sizes; D = {Dj}|D| j=1 is a set of distributions; p(T ) is the underlying task distribution. For notation simplicity, we use Ti to represent a task. Inspired by MAML (Finn et al., 2017), we propose a generic meta-learning framework to improve the generalization capability of neural methods for VRPs, which is model-agnostic and compatible with any model trained with gradient updates. The framework is illustrated in Figure 1, and we elaborate its key components below. 3 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems Meta-Training . . . . . . Training Task Set Inner-loop Optimization Task-Specific  Model ... Eval Inference Tour Fine-Tune New Task Hierarchical Task Scheduler Initialize Solver Softmax with T ... Meta-Model Outer-loop Optimization ... + ... ... ... ... Sample Task Select (n, d) Initialize ... . . . ! \" \u2460 \u2461 ! ! \"\" \"! (\") \"! (%) \"! (&) # #\u2032 \"% (&) \"' (&) \"\" \u2217 \"!! (\") \"!! (%) \"!! (&) \u2112' \u2112% &% &' ! !! Figure 1. The illustration of the proposed framework. The training task set consists of tasks with diverse sizes n \u223cN and distributions d \u223cD. In each iteration, the hierarchical task scheduler adaptively selects a batch of tasks {Ti}B i=1 for meta-training, which consists of a pair of inner-loop and outer-loop optimization. During inference, given a new task Ti\u2032, the trained meta-model \u03b8\u2217 0 is used to initialize the task-specific model \u03b8(0) i\u2032 , which is then adapted to Ti\u2032 by further taking K gradient steps with a limited number of instances. Meta-Training. In the meta-learning framework, we aim to train a meta-model \u03b80, which as a good initialized model can be efficiently adapted to new tasks during inference. Formally, we define the meta-objective as follows: \u03b8\u2217 0 = arg min \u03b80 ETi\u223cp(T )EG\u223cTiLi(\u03b8(K) i |G), (4) where \u03b8(K) i is the fine-tuned model after K gradient updates of \u03b80 on the task Ti; Li is the loss function on the task Ti. In this paper, we use the same loss function (e.g., reinforcement loss) for different tasks. To directly optimize this objective, the meta-training procedure comprises the inner-loop and outer-loop optimization at each iteration. The meta-training pseudocode is presented in Algorithm 1. Inner-loop optimization: It optimizes a task-specific model iteratively, which is similar to the fine-tuning stage during inference. Specifically, given a task Ti \u223cp(T ), we initialize the task-specific model by the meta-model, i.e., \u03b8(0) i \u2190\u03b80 (line 5), and adapt it to Ti by performing K gradient update steps on the training instances (line 6-11). The gradient of the loss function Li at the kth step is computed as follows: \u2207\u03b8(k\u22121) i Li(\u03b8(k\u22121) i ) \u21901 M M X m=1 \u2207\u03b8(k\u22121) i Li(\u03b8(k\u22121) i |Gm), (5) where \u2207\u03b8(k\u22121) i Li(\u03b8(k\u22121) i |Gm) can be estimated by the REINFORCE algorithm as in Eq. (3). Outer-loop optimization: It optimizes the meta-model with the objective in Eq. (4). Concretely, for each task Ti, the few-shot generalization performance of the task-specific model \u03b8(K) i is evaluated on the validation instances. The meta-gradient (line 13) is obtained as follows: \u2207\u03b80Li(\u03b8(K) i ) \u21901 M M X m=1 \u2207\u03b8(K) i Li(\u03b8(K) i |G\u2032 m) \u00b7 \u2202\u03b8(K) i \u2202\u03b80 . (6) After conducting the inner-loop optimization on a batch of tasks {Ti}B i=1, the meta-model \u03b80 is then updated once (as in line 16). Intuitively, the inner-loop optimization serves as the task adaption stage, imitating the fine-tuning process during inference, while the outer-loop optimization updates the meta-model with the objective of maximizing the few-shot generalization performance of the task-specific model \u03b8(K) i . Therefore, after meta-training, we can get a good initialized model \u03b8\u2217 0 with the capability of efficient adaptation to new tasks only using limited data. Note that Eq. (6) needs the second-order derivatives since we expect to get the gradient direction with respect to the meta-model \u03b80. We will discuss and analyze its first-order approximation methods later. Hierarchical Task Scheduler. The aim of a task scheduler is to guide the task selection for the meta-training process, so as to improve the optimization performance. Most existing works (Finn et al., 2017; Raghu et al., 2020; Flennerhag et al., 2022; Manchanda et al., 2022) randomly sample training tasks from the task set with a uniform probability, which assumes all tasks are equally important. It may overlook the training dynamics and cannot make full use of diverse data information. In this paper, we propose a simple yet effective hierarchical task scheduler for VRPs. During training, we first gradually increase the size of training task following a linear scheduler. At the eth iteration of meta-training, we select tasks of the size ne = \u230anmin + min( e Es , 1) \u00b7 (nmax \u2212nmin)\u230b, where Es is the working duration of the scheduler. Then, a probability distribution over the tasks in {T (ne, dj)}|D| j=1 is generated for 4 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems Algorithm 1 Meta-Training for VRPs Input: distribution over tasks p(T ), number of tasks in a minibatch B, number of inner-loop updates K, batch size M, step sizes of inner-loop and outer-loop optimization \u03b1, \u03b2; Output: meta-model \u03b8\u2217 0; 1: Initialize meta-model \u03b80 2: while not done do 3: {Ti, wi}B i=1, \u2190Hierarchical task scheduler 4: for i = 1, . . . , B do 5: Initialize task-specific model \u03b8(0) i \u2190\u03b80 6: for k = 1, . . . , K do 7: { // Inner-loop optimization } 8: Sample training instances {Gm}M m=1 from task Ti 9: Obtain \u2207\u03b8(k\u22121) i Li(\u03b8(k\u22121) i ) using Eq. (5) 10: \u03b8(k) i \u2190\u03b8(k\u22121) i \u2212\u03b1\u2207\u03b8(k\u22121) i Li(\u03b8(k\u22121) i ) 11: end for 12: Sample validation instances {G\u2032 m}M m=1 from task Ti 13: Obtain \u2207\u03b80Li(\u03b8(K) i ) using Eq. (6) 14: end for 15: { // Outer-loop optimization } 16: \u03b80 \u2190\u03b80 \u2212\u03b2 PB i=1 wi\u2207\u03b80Li(\u03b8(K) i ) 17: end while training task selection, based on their hardness. The optimality gap is an appropriate metric to measure the hardness of each task. However, it is intractable to obtain optimal solutions due to the NP-hardness. Here we use a general VRP solver LKH3 (Helsgaun, 2017) to efficiently2 obtain near-optimal solutions for a validation set of instances in a one-shot manner. The validation instances are fixed throughout the meta-training process, and hence we only need to run LKH3 once. To avoid overfitting, we only sample M instances from the validation set to calculate the relative gap gi = 1 M PM m=1 c(\u03c4m)\u2212c(\u00af \u03c4m) c(\u00af \u03c4m) for each task Ti, where \u03c4m and \u00af \u03c4m are solutions to the mth instance constructed by the current meta-model and LKH3, respectively. Accoringly, the probability of selecting each task Ti is defined as: wi = exp(gi/\u03b7) P|D| j=1 exp(gj/\u03b7) , (7) where \u03b7 is the temperature to control the entropy of probability distribution, from which the task is sampled. The initial probability distribution is uniform, and is updated by Eq. (7) periodically (e.g., every 100 iterations). After a batch of tasks {Ti}B i=1 is sampled, we normalize their weights such that PB i=1 wi = 1. Then, the meta-model is optimized with the weighted sum of their losses {wiLi(\u03b8(K) i )}B i=1 (line 16). Inference. During inference, given instances sampled from a new task Ti\u2032, the trained meta-model \u03b8\u2217 0 could be used to approximate solutions in several ways: 1) zero-shot: the solution is directly constructed using the learned policy 2We set the maximum trials of LKH3 to 100 for efficiency. 0 20000 40000 60000 80000 100000 Iteration 20 30 40 50 60 70 80 90 100 Obj. FOMAML Ours-SO Ours 0 10000 20000 30000 40000 50000 Iteration 0.05 0.00 0.05 0.10 0.15 0.20 0.25 Cos Sim. Reptile-First Reptile-Last FOMAML-First FOMAML-Last Figure 2. Left panel: the meta-training curves; Right panel: the cosine similarity between the second-order derivative and its firstorder approximation at the first and last layer of the model. \u03c0\u03b8\u2217 0 with efficient search strategies (e.g., greedy rollout); 2) few-shot: similar to the inner-loop optimization as shown in line 6-11 of Algorithm 1, the task-specific model \u03b8i\u2032 is initialized by \u03b8\u2217 0, and is adapted to the new task Ti\u2032 by taking K gradient steps on a small set of instances (different from test instances) sampled from Ti\u2032. Then the solution is constructed using the adapted policy; 3) active search: the model is adapted to each test instance by learning instancedependent parameters. Bello et al. (2017) first proposes the general active search that iteratively adjusts the model parameters with the objective of increasing the likelihood of constructing high-quality solutions for each instance. However, it is extremely computationally expensive. Hottung et al. (2022) proposes the efficient active search (EAS) by introducing extra instance-specific parameters (e.g., a MLP layer) for each test instance while fixing the original model parameters. We mainly consider the zero-shot and few-shot settings in our experiments (in Section 5), and only use EAS when evaluating on benchmark instances (in Appendix D.3). 4.2. First-Order Approximation The meta-gradient in Eq. (6) involves a gradient through a gradient (i.e., second-order derivative), and therefore it is computational expensive to obtain due to the calculation of Hessian-vector products. To tackle this issue, Finn et al. (2017) proposes a first-order approximation method (i.e., FOMAML) which simply drops the second-order term. The empirical evidence of its effectiveness has been verified in the few-shot supervised learning, while lacking in the more complex reinforcement learning. Specifically, the first-order approximation of meta-model update can be expressed as: \u03b80 \u2190\u03b80 \u2212\u03b2 B X i=1 wi\u2207\u03b8iLi(\u03b8(K) i ). (8) However, as shown in the left panel of Figure 2, we empirically observe that meta-training POMO from scratch with Eq. (8) may induce fluctuating validation performance. The unstable meta-training may be attributed to the deviation of the first-order approximation (Eq. (8)) from the (groundtruth) gradient direction of the second-order term (line 16 in 5 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems Algorithm 1). Intuitively, sign(\u2207\u03b8iLi(\u03b8(K) i )) is the (steepest) descent direction for the task-specific model \u03b8(K) i , but not necessarily the descent direction for the meta-model \u03b80, especially at the early stage of meta-training when the optimization tends to be unstable. To justify our hypothesis, we show the cosine similarities of the gradient directions for meta-model updates between our method and others in the right panel of Figure 2. The detailed experimental setups are presented in Appendix D.1. We observe that sign(\u2207\u03b8iLi(\u03b8(K) i )) cannot approximate sign(\u2207\u03b80Li(\u03b8(K) i )) well (i.e., with a negative cosine similarity) at the early stage of meta-training, while gradually having the positive correlation as the training progresses. Therefore, in order to reduce the computational cost and stabilize the metatraining, we develop a simple yet efficient method by early stopping the usage of second-order derivatives. Specifically, we start the meta-model updates with second-order derivatives, and switch to the first-order ones (i.e., by replacing line 16 with Eq. (8)) when the optimization tends to be stable. Recently, Manchanda et al. (2022) leverages another first-order method called Reptile (Nichol et al., 2018), with the form of the meta-model update as follows: \u03b80 \u2190\u03b80 + \u03b2 B X i=1 wi(\u03b8(K) i \u2212\u03b80). (9) However, it needs multiple inner-loop updates to effectively incorporate information from higher-order derivatives of the loss function so as to achieve satisfactory performance (as shown in Appendix A and D.2), making it less sample efficient. Moreover, similar to Nichol et al. (2018) which observes negative results after applying Reptile to the reinforcement learning setting, we also empirically find its weak performance (see Meta-POMO in Section 5, Appendix D.2 and D.3). In contrast, our method could achieve decent performance only running a single inner-loop update. 5. Experiments To demonstrate the effectiveness of the proposed framework, we apply it to POMO3 (Kwon et al., 2020), which is a strong construction-based neural method. We consider two representative VRP problems (i.e., TSP and CVRP). The details of POMO are introduced in Appendix C. Moreover, we also evaluate the generalizability of our method on L2D (Li et al., 2021) as shown in Section 5.3 and Appendix D.5. Baselines. 1) Traditional VRP solvers: we employ Concorde and LKH3 (Helsgaun, 2017) for solving TSP, and the hybrid genetic search (HGS) (Vidal, 2022) and LKH3 for CVRP. 2) Neural methods: we compare our method with POMO-based methods, including the original POMO (Kwon et al., 2020), AMDKD-POMO (Bi et al., 3https://github.com/yd-kwon/POMO 2022) and Meta-POMO (Manchanda et al., 2022) for TSP and CVRP. AMDKD-POMO is a recent method that improves the cross-distribution generalization of POMO using knowledge distillation. Meta-POMO uses Reptile to improve the generalization performance across both size and distribution. For a fair comparision, we re-train all methods following our training setups. Note that the setting of MetaPOMO is the most relevant to ours. As shown in Appendix D.2, we tune the key hyperparameters (i.e., \u03b2 and K) of Meta-POMO since we empirically find its straightforward adaptation to POMO (i.e., decaying \u03b2 in Table 4) performs poorly. We also show the results of their open-sourced pretrained models (i.e., POMO* and AMDKD-POMO*) in Table 1, with the aim of demonstrating the severe generalization issue of current neural methods rather than the direct comparison. Specifically, POMO* is trained on instances with a fixed size and distribution (i.e., n = 100 with the uniform distribution), and AMDKD-POMO* is adaptively distilling from teacher models trained on fixed-sized instances following different distributions (i.e., n = 100 with the uniform, cluster and mixed distributions). More implementation details are provided in Appendix D.1. Training Setups. We follow most of the setups in Kwon et al. (2020). For our method, Adam optimizer (Kingma & Ba, 2015) is used in both inner-loop and outer-loop optimization, with the weight decay of 1e \u22126. The step sizes (learning rates) are \u03b1 = \u03b2 = 1e \u22124, and decayed by 10 in the last 10% iterations to achieve a faster convergence. The batch size is M =64 (M =32 for instances with sizes larger than 150). The training task set consists of hundreds of (i.e., 341) tasks, with diverse sizes N = [50, 200] and distributions (i.e., uniform (U) and gaussian mixture (GM) distributions). More details about the generation of training and test data are presented in Appendix B. Similar to Finn et al. (2017), we simply set B = K = 1 and empirically observe strong performance. As suggested by Kwon et al. (2020), most of the training is already completed by 200 epochs (i.e., 20M instances) for POMO. We give more instances due to our complicated problem setting. Specifically, we re-train all methods for roughly the same number of instances (i.e., 32M) sampled from our training task set. For example, we re-train POMO for roughly 500K iterations (i.e., gradient updates). For our method, one iteration of meta-training consists of a pair of inner-loop and out-loop optimization, which needs two batches of instances. Therefore, for a fair comparison, we train our method for roughly 250K iterations. For the hierarchical task scheduler, we set \u03b7 = 1 and Es = 225K. It evaluates the hardness and updates the weight of each task every 100 iterations. Due to the training efficiency, we regard meta-training with the first-order approximation (i.e., Ours) as the default method, which uses the second-order derivatives in the first 50K iterations, and switch to the first-order ones afterwards. In 6 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems Table 1. Evaluation on cross-size or distribution generalization. (* marks results derived by open-sourced pretrained models.) Method Cross-Distribution Generalization (1K ins.) Cross-Size Generalization (1K ins.) (200, GM 5 2 ) (200, R) (200, E) (300, U) (300, GM 10 3 ) (300, GM 50 7 ) Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time TSP Concorde 8.78 (0.00%) 0.6m 8.20 (0.00%) 0.5m 8.09 (0.00%) 0.5m 12.95 (0.00%) 1.4m 9.47 (0.00%) 1.2m 5.63 (0.00%) 1.0m LKH3 8.78 (0.00%) 3.3m 8.20 (0.00%) 3.3m 8.09 (0.00%) 3.5m 12.95 (0.00%) 5.9m 9.47 (0.00%) 12.5m 5.63 (0.01%) 18.5m POMO* 9.36 (6.67%) 0.5m 8.41 (2.66%) 0.5m 8.28 (2.35%) 0.5m 13.82 (6.70%) 1.5m 10.73 (13.38%) 1.5m 6.54 (16.04%) 1.5m AMDKD-POMO* 9.05 (2.97%) 0.5m 8.41 (2.57%) 0.5m 8.30 (2.61%) 0.5m 13.97 (7.83%) 1.5m 10.25 (8.22%) 1.5m 6.25 (11.00%) 1.5m POMO 9.01 (2.56%) 0.5m 8.37 (2.13%) 0.5m 8.24 (1.85%) 0.5m 13.54 (4.51%) 1.5m 9.88 (4.27%) 1.5m 5.83 (3.46%) 1.5m AMDKD-POMO 9.10 (3.56%) 0.5m 8.47 (3.32%) 0.5m 8.38 (3.55%) 0.5m 13.74 (6.08%) 1.5m 9.97 (5.30%) 1.5m 6.00 (6.44%) 1.5m Meta-POMO 9.03 (2.78%) 0.5m 8.39 (2.31%) 0.5m 8.25 (2.00%) 0.5m 13.50 (4.23%) 1.5m 9.89 (4.38%) 1.5m 5.80 (2.94%) 1.5m Ours-SO 9.01 (2.59%) 0.5m 8.36 (1.99%) 0.5m 8.23 (1.72%) 0.5m 13.37 (3.22%) 1.5m 9.82 (3.72%) 1.5m 5.78 (2.61%) 1.5m Ours 9.02 (2.71%) 0.5m 8.37 (2.14%) 0.5m 8.24 (1.86%) 0.5m 13.40 (3.42%) 1.5m 9.84 (3.89%) 1.5m 5.79 (2.75%) 1.5m Meta-POMO+FS (K = 1) 9.02 (2.74%) 2.0m 8.38 (2.24%) 2.0m 8.25 (1.92%) 2.0m 13.46 (3.87%) 6.8m 9.86 (4.11%) 6.8m 5.78 (2.64%) 6.8m Meta-POMO+FS (K = 10) 9.02 (2.67%) 15.7m 8.38 (2.17%) 15.7m 8.24 (1.83%) 15.7m 13.42 (3.58%) 0.9h 9.84 (3.91%) 0.9h 5.77 (2.46%) 0.9h Ours-SO+FS (K = 1) 9.01 (2.53%) 2.0m 8.36 (1.95%) 2.0m 8.22 (1.63%) 2.0m 13.35 (3.05%) 6.8m 9.81 (3.57%) 6.8m 5.77 (2.47%) 6.8m Ours+FS (K = 1) 9.01 (2.60%) 2.0m 8.37 (2.05%) 2.0m 8.23 (1.74%) 2.0m 13.37 (3.19%) 6.8m 9.82 (3.69%) 6.8m 5.78 (2.52%) 6.8m CVRP HGS 18.89 (0.00%) 0.7h 19.36 (0.00%) 0.6h 19.45 (0.00%) 0.5h 25.61 (0.00%) 1.0h 22.20 (0.00%) 1.6h 22.11 (0.00%) 0.9h LKH3 19.09 (1.06%) 0.6h 19.55 (0.99%) 0.6h 19.66 (1.04%) 0.5h 25.97 (1.39%) 0.6h 22.46 (1.19%) 0.7h 22.24 (0.59%) 0.6h POMO* 19.93 (5.60%) 0.6m 20.45 (5.74%) 0.6m 20.54 (5.69%) 0.6m 28.72 (12.32%) 1.8m 24.81 (12.00%) 1.8m 24.33 (10.22%) 1.8m AMDKD-POMO* 20.29 (7.59%) 0.6m 20.89 (8.07%) 0.6m 20.96 (7.92%) 0.6m 30.49 (19.17%) 1.8m 25.65 (15.92%) 1.8m 24.41 (10.60%) 1.8m POMO 19.47 (3.12%) 0.6m 19.99 (3.32%) 0.6m 20.12 (3.49%) 0.6m 27.07 (5.74%) 1.8m 23.25 (4.80%) 1.8m 22.80 (3.16%) 1.8m AMDKD-POMO 19.58 (3.69%) 0.6m 20.07 (3.72%) 0.6m 20.20 (3.93%) 0.6m 26.94 (5.20%) 1.8m 23.28 (4.92%) 1.8m 22.82 (3.27%) 1.8m Meta-POMO 19.48 (3.19%) 0.6m 20.01 (3.40%) 0.6m 20.15 (3.65%) 0.6m 26.87 (4.94%) 1.8m 23.09 (4.07%) 1.8m 22.75 (2.93%) 1.8m Ours-SO 19.38 (2.66%) 0.6m 19.91 (2.87%) 0.6m 20.05 (3.13%) 0.6m 26.67 (4.15%) 1.8m 22.93 (3.33%) 1.8m 22.60 (2.23%) 1.8m Ours 19.39 (2.69%) 0.6m 19.91 (2.88%) 0.6m 20.07 (3.21%) 0.6m 26.65 (4.10%) 1.8m 22.93 (3.35%) 1.8m 22.61 (2.27%) 1.8m Meta-POMO+FS (K = 1) 19.43 (2.92%) 2.4m 19.96 (3.13%) 2.4m 20.10 (3.39%) 2.4m 26.71 (4.32%) 8.2m 22.99 (3.64%) 8.2m 22.70 (2.72%) 8.2m Meta-POMO+FS (K = 10) 19.41 (2.83%) 18.8m 19.94 (3.03%) 18.8m 20.08 (3.27%) 18.8m 26.65 (4.07%) 1.1h 22.95 (3.43%) 1.1h 22.67 (2.55%) 1.1h Ours-SO+FS (K = 1) 19.38 (2.65%) 2.4m 19.90 (2.81%) 2.4m 20.03 (3.00%) 2.4m 26.61 (3.93%) 8.2m 22.90 (3.21%) 8.2m 22.58 (2.16%) 8.2m Ours+FS (K = 1) 19.38 (2.66%) 2.4m 19.90 (2.83%) 2.4m 20.04 (3.05%) 2.4m 26.61 (3.92%) 8.2m 22.91 (3.23%) 8.2m 22.59 (2.20%) 8.2m specific, the meta-training with full second-order derivatives (i.e., Ours-SO) needs roughly 5 days and 53GB GPU memory for TSP (6 days and 71GB GPU memory for CVRP), while Ours needs 2.5 days and 17GB GPU memory for TSP (3 days and 25GB GPU memory for CVRP). Inference Setups. For all neural methods, we use the greedy rollout with x8 instance augmentations following Kwon et al. (2020). We report the average results over the test dataset containing 1K instances. The reported time is the total time to solve the entire test dataset. The reported gaps are computed with respect to the traditional VRP solvers (i.e., Concorde for TSP, and HGS for CVRP). Specifically, we evaluate the effectiveness of our method on the zero-shot and few-shot (FS) settings. For the zero-shot setting, the trained model is directly used to construct the solutions. We further evaluate meta-learning based methods (i.e., Meta-POMO, Ours-SO and Ours) on the few-shot setting, where we finetune the meta-model for K iterations only using extra 1K instances sampled from the test task (0.003% of instances used for meta-training). The instances are augmented following Kwon et al. (2020). Note that the instances for fine-tuning are different from the test ones. The Adam optimizer is used with the learning rate of \u03b1 =1e \u22125 and the weight decay of 1e \u22126. Moreover, we further combine our method with EAS (Hottung et al., 2022) when evaluating on benchmark instances (see Appendix D.3). 5.1. Performance Evaluation Below, we demonstrate the effectiveness of our method on synthetic and real-world datasets. For the synthetic data, we evaluate the generalization performance across size, distribution and the both. Note that we conduct t-test (with threshold of 5%) to verify the statistical significance, if the average objectives of two neural methods are close. Cross-Size or Distribution Generalization. We first consider a simple setting where either the cross-size or distribution generalization is evaluated. For the cross-distribution setting, we test on instances of size n = 200 \u2208[50, 200], while following diverse distributions that are unseen during training. Note that we do not strictly choose the test tasks sampled from the presumed training task distribution p(T ) since it only covers a small part of the entire problem space. Therefore, we also evaluate all methods on several complex distributions, e.g., rotation (R) and explosion (E) distributions (Bossek et al., 2019). For the cross-size setting, we evaluate on instances of the size n = 300 / \u2208[50, 200] following distributions used in training. Besides the zeroshot setting, we further compare with another meta-learning based method (i.e., Meta-POMO) on the few-shot setting, where we fine-tune the learned model for K steps only using limited data. The detailed results are shown in Table 1, where we observe that our method can achieve superior performance on both settings. The inferior performance 7 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems Table 2. Evaluation on cross-size and distribution generalization. Method Cross-Size and Distribution Generalization (1K ins.) (300, R) (300, E) (500, R) (500, E) (1000, R) (1000, E) Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time Obj. (Gap) Time TSP Concorde 9.79 (0.00%) 1.2m 9.48 (0.00%) 1.5m 12.39 (0.00%) 5.0m 11.73 (0.00%) 5.8m 17.09 (0.00%) 0.7h 15.66 (0.00%) 0.9h LKH3 9.79 (0.00%) 6.0m 9.48 (0.00%) 6.8m 12.39 (0.00%) 11.8m 11.73 (0.00%) 13.8m 17.09 (0.00%) 0.4h 15.66 (0.00%) 0.5h POMO 10.23 (4.43%) 1.5m 9.88 (4.20%) 1.5m 13.63 (10.00%) 6.0m 12.89 (9.88%) 6.0m 20.74 (21.38%) 0.8h 18.94 (20.97%) 0.8h AMDKD-POMO 10.35 (5.69%) 1.5m 10.06 (6.15%) 1.5m 13.74 (10.85%) 6.0m 13.08 (11.52%) 6.0m 20.73 (21.25%) 0.8h 19.08 (21.85%) 0.8h Meta-POMO 10.22 (4.37%) 1.5m 9.87 (4.14%) 1.5m 13.56 (9.41%) 6.0m 12.84 (9.44%) 6.0m 20.51 (19.97%) 0.8h 18.77 (19.88%) 0.8h Ours-SO 10.14 (3.54%) 1.5m 9.78 (3.13%) 1.5m 13.39 (8.07%) 6.0m 12.64 (7.73%) 6.0m 20.37 (19.20%) 0.8h 18.59 (18.74%) 0.8h Ours 10.16 (3.74%) 1.5m 9.80 (3.35%) 1.5m 13.42 (8.30%) 6.0m 12.66 (7.90%) 6.0m 20.40 (19.36%) 0.8h 18.60 (18.80%) 0.8h Meta-POMO+FS (K = 1) 10.18 (3.96%) 6.8m 9.83 (3.70%) 6.8m 13.34 (7.60%) 0.5h 12.63 (7.66%) 0.5h 19.58 (14.52%) 6.5h 17.92 (14.48%) 6.5h Meta-POMO+FS (K = 10) 10.16 (3.69%) 0.9h 9.80 (3.41%) 0.9h 13.23 (6.75%) 4.1h 12.54 (6.84%) 4.1h \u2013 \u2013 \u2013 \u2013 Ours-SO+FS (K = 1) 10.12 (3.32%) 6.8m 9.76 (2.91%) 6.8m 13.19 (6.45%) 0.5h 12.45 (6.11%) 0.5h 19.53 (14.28%) 6.5h 17.79 (13.65%) 6.5h Ours+FS (K = 1) 10.13 (3.41%) 6.8m 9.77 (3.05%) 6.8m 13.20 (6.52%) 0.5h 12.51 (6.64%) 0.5h 19.53 (14.30%) 6.5h 17.75 (13.38%) 6.5h CVRP HGS 22.40 (0.00%) 1.3h 23.02 (0.00%) 1.3h 26.62 (0.00%) 4.5h 26.89 (0.00%) 4.6h 32.36 (0.00%) 30.9h 32.01 (0.00%) 37.7h LKH3 22.68 (1.28%) 0.7h 23.32 (1.28%) 0.7h 27.06 (1.69%) 0.9h 27.32 (1.61%) 0.9h 33.16 (2.51%) 1.6h 32.78 (2.43%) 1.6h POMO 23.56 (5.30%) 1.8m 24.20 (5.30%) 1.8m 29.06 (9.48%) 6.9m 29.29 (9.29%) 6.9m 39.33 (22.44%) 1.0h 38.63 (21.73%) 1.0h AMDKD-POMO 23.54 (5.18%) 1.8m 24.24 (5.39%) 1.8m 29.06 (9.32%) 6.9m 29.33 (9.29%) 6.9m 39.72 (23.17%) 1.0h 38.86 (21.90%) 1.0h Meta-POMO 23.39 (4.54%) 1.8m 24.08 (4.71%) 1.8m 28.53 (7.34%) 6.9m 28.80 (7.32%) 6.9m 37.46 (16.09%) 0.9h 36.85 (15.52%) 0.9h Ours-SO 23.24 (3.83%) 1.8m 23.93 (4.07%) 1.8m 28.34 (6.60%) 6.7m 28.63 (6.69%) 6.7m 37.30 (15.62%) 0.8h 36.61 (14.83%) 0.8h Ours 23.23 (3.79%) 1.8m 23.94 (4.08%) 1.8m 28.29 (6.41%) 6.7m 28.60 (6.56%) 6.7m 37.02 (14.73%) 0.8h 36.40 (14.15%) 0.8h Meta-POMO+FS (K = 1) 23.29 (4.05%) 8.2m 23.96 (4.20%) 8.2m 28.13 (5.80%) 0.6h 28.43 (5.90%) 0.6h 36.14 (11.93%) 7.5h 35.78 (12.07%) 7.5h Meta-POMO+FS (K = 10) 23.23 (3.79%) 1.1h 23.90 (3.92%) 1.1h 27.95 (5.14%) 4.9h 28.24 (5.19%) 4.7h \u2013 \u2013 \u2013 \u2013 Ours-SO+FS (K = 1) 23.19 (3.61%) 8.2m 23.87 (3.78%) 8.2m 28.03 (5.41%) 0.6h 28.33 (5.52%) 0.6h 35.69 (10.52%) 7.4h 35.40 (10.92%) 7.4h Ours+FS (K = 1) 23.19 (3.59%) 8.2m 23.87 (3.79%) 8.2m 28.01 (5.34%) 0.6h 28.31 (5.44%) 0.6h 35.60 (10.26%) 7.4h 35.25 (10.45%) 7.4h of AMDKD-POMO may be attributed to its design for the trivial problem setting and sample inefficiency. While it is specialized for the cross-distribution generalization, its original problem setting is much easier than ours, with only three distributions on the fixed size (i.e., 100) considered during training. To achieve satisfactory performance, a good pretrained model for each training task is needed, which requires a huge amount of training instances. Cross-Size and Distribution Generalization. We further evaluate all methods on a much more complex setting, where the generalization across both size and distribution is considered. Specifically, we choose the test task with the unseen size n\u2208[300, 500, 1000] and distribution d\u2208[R, E] during training. The results are presented in Table 2, where we observe our method has consistently better performance than baselines. Notably, our method achieves superior results on the large-scale CVRP1000 task with totally unseen distributions, showing a strong omni-generalization capability. Results on Benchmark Datasets. We further evaluate all methods on the well-konwn benchmark datasets TSPLIB (Reinelt, 1991) and CVRPLIB (Set-X (Uchoa et al., 2017) and Set-XML100 (Queiroga et al., 2022)). Detailed results can be found in Appendix D.3, where we observe our method performs well in most cases. 5.2. Analyses In this section, we conduct further analyses, including the ablation studies and few-shot experiments, to demonstrate Table 3. Ablation study on Components. (200, GM 5 2 ) (300, U) (500, R) (1000, E) POMO 3.12% 5.74% 9.48% 21.73% + task scheduler 2.67% 4.44% 7.53% 19.03% + meta-training 3.08% 4.79% 7.02% 15.39% Ours 2.69% 4.10% 6.41% 14.15% the effectiveness and sensitivity of the proposed framework. More ablation studies on hyperparameters, optimizers and normalization layers are presented in Appendix D.4. Ablation Study on Components. In Section 5.1, we have shown the effect of the first-order approximation. Compared with the full second-order method, it could achieve similar or even better zero-shot and few-shot performance, and meanwhile greatly reduce the training complexity. Here, following the training setups presented in Section 5, we further conduct the ablation study on CVRP to demonstrate the benefit of each component in our framework. The results are shown in Table 3, where we observe that the meta-training significantly improves POMO (zero-shot) performance on the large-scale instances, and the task scheduler can further boost the overall performance of the meta-training. Efficient Adaptation. As shown in Table 1 and 2, given the same amount of training instances, our method achieves strong zero-shot performance, which enables efficient adaptation to a new task afterwards. To demonstrate it, we further conduct two experiments on the adaptation to CVRP (500, R) only using 100 and 1000 instances. As shown 8 \fTowards Omni-generalizable Neural Methods for Vehicle Routing Problems 0 2 4 6 8 10 Step 5.5 6.0 6.5 7.0 7.5 Gap (%) Meta-POMO Ours-SO Ours 0 2 4 6 8 10 Step 5.0 5.5 6.0 6.5 7.0 7.5 Gap (%) Meta-POMO Ours-SO Ours Figure 3. Adaptation to CVRP (500, R) test task using (a) 100 instances; (b) 1000 instances. in Figure 3, our methods can be efficiently adapted to the new task, while Meta-POMO needs to run multiple steps to achieve similar few-shot performance (e.g., the dotted line for K = 1) to our method. Moreover, we observe that the number of instances is crucial to the few-shot performance in the reinforcement learning. Meta-POMO may need more instances in order to achieve strong few-shot performance. 5.3. Generalizability To evaluate the generalizability of the proposed framework, we further apply it to L2D (Li et al., 2021), which is an improvement-based method outperforming LKH3 on largescale CVRP instances. Specifically, it decomposes the largescale problem instance into several subproblems, which are selected by a (supervised) learned policy, and uses an existing solver (e.g., LKH3 or HGS) to solve each subproblem. We train the model on the omni-generalization setting, where the training task set consists of various sizes and distributions. The results show that our method could improve the generalization of L2D, demonstrating the effectiveness and generalizability of our method. The detailed training setups and empirical results are shown in Appendix D.5. 6."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2211.00269v1",
+                    "title": "Adversarial Training with Complementary Labels: On the Benefit of Gradually Informative Attacks",
+                    "abstract": "Adversarial training (AT) with imperfect supervision is significant but\nreceives limited attention. To push AT towards more practical scenarios, we\nexplore a brand new yet challenging setting, i.e., AT with complementary labels\n(CLs), which specify a class that a data sample does not belong to. However,\nthe direct combination of AT with existing methods for CLs results in\nconsistent failure, but not on a simple baseline of two-stage training. In this\npaper, we further explore the phenomenon and identify the underlying challenges\nof AT with CLs as intractable adversarial optimization and low-quality\nadversarial examples. To address the above problems, we propose a new learning\nstrategy using gradually informative attacks, which consists of two critical\ncomponents: 1) Warm-up Attack (Warm-up) gently raises the adversarial\nperturbation budgets to ease the adversarial optimization with CLs; 2)\nPseudo-Label Attack (PLA) incorporates the progressively informative model\npredictions into a corrected complementary loss. Extensive experiments are\nconducted to demonstrate the effectiveness of our method on a range of\nbenchmarked datasets. The code is publicly available at:\nhttps://github.com/RoyalSkye/ATCL.",
+                    "authors": "Jianan Zhou, Jianing Zhu, Jingfeng Zhang, Tongliang Liu, Gang Niu, Bo Han, Masashi Sugiyama",
+                    "published": "2022-11-01",
+                    "updated": "2022-11-01",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.CR"
+                    ],
+                    "main_content": "Introduction Deep neural networks are vulnerable to adversarial examples [15], which motivates the development of various defensive methods [30; 21; 24; 2] to mitigate the arisen security issue. As one of the most effective and practical defensive methods, adversarial training (AT) [24] has been widely studied [41; 5; 10]. Speci\ufb01cally, it formulates the problem as min-max optimization, which generates adversarial data during training and minimizes the training loss of the generated adversarial variants. In this way, the models are equipped with adversarial robustness against human-imperceptible perturbations within the neighborhood of inputs. Although AT with perfect supervision (e.g., ordinary labels) has been thoroughly studied by previous works [25; 33; 44; 21; 34; 43; 11], the more common learning scenarios with imperfect supervision [45; 42] has only received limited attention. Complementary labels (CLs) [17], which convey partial label information by identifying a class that a data sample does not belong to, are one of those practical and promising imperfect supervision in weakly supervised learning [3; 27]. Several works [39; 18; 7; 13; 38; 14] recently focused on learning with CLs, where the model trained only with CLs through their designed losses (i.e., complementary loss \u00af \u2113) is expected to predict ordinary labels accurately during inference. Their success illustrates the \u2217Equal contributions. \u2020Corresponding authors: Bo Han (bhanml@comp.hkbu.edu.hk) and Gang Niu (gang.niu.ml@gmail.com). 36th Conference on Neural Information Processing Systems (NeurIPS 2022). arXiv:2211.00269v1  [cs.LG]  1 Nov 2022 \f0 20 40 60 80 100 epoch 0 20 40 60 80 PGD-20 Test Accuracy (%) Oracle Two-stage LOG (a) Failure of direct combination 0 20 40 60 80 100 epoch 5 10 15 20 25 30 35 40 45 PGD-20 Test Accuracy (%) FORWARD FREE NN SCL_NL SCL_EXP EXP LOG (b) Trials of Complementary Losses 0 20 40 60 80 100 epoch 5 10 15 20 25 30 35 40 PGD-20 Test Accuracy (%) LOG-SGD LOG-Adam LOG-Grad-Norm LOG-Grad-Clip LOG-ResNet-18 (c) Trials of Training Tricks Figure 1: (a) in contrast to a simple two-stage baseline, the direct combination of AT with CLs (LOG [13]) results in the worse performance with the failed training curve. The two-stage baseline consists of complementary learning in the \ufb01rst 50 epochs and AT with predicted labels in the last 50 epochs; (b) we try various complementary loss functions while all of them result in consistent experimental failure in AT with CLs; (c) several training alternatives (e.g., optimizer) are tried with no success. We report mean with standard errors for PGD20 test accuracy on Kuzushiji within multiple random trials. The training on the direct combination of AT with CLs is signi\ufb01cantly unstable. possibility of training ordinary classi\ufb01ers even when all the labels given for training are wrong (except that the data and labels are statistically independent). Due to the low acquisition cost and privacy preservability, it has been applied to online learning [19] and medical image segmentation [31]. To push AT towards more practical scenarios, we explore a brand new yet challenging setting, i.e., AT with CLs. In this paper, our interest is how to equip machine learning models with adversarial robustness when all given labels in a dataset are wrong (i.e., CLs), which is of scienti\ufb01c interests to both research areas of weakly supervised learning and adversarial learning. To conduct AT with CLs, a straightforward attempt is to regard \u00af \u2113as the training objective of the min-max optimization [15; 24]. However, the direct combination of AT with existing methods [39; 18; 7; 13] for learning with CLs results in consistent failure, but not on a simple baseline of two-stage training (as shown in Figure 1). To be speci\ufb01c, we mainly consider the single complementary label (SCL) with the uniform assumption [17; 18; 7], where labels other than the true label are chosen as a CL with the same probability. In complementary learning, the complementary losses \u00af \u2113are obtained by certain correction techniques [17; 39], which guarantee the training with \u00af \u2113on complementary datasets achieves similar performance to the training with the ordinary loss \u2113on ordinary datasets. In contrast, when conducting AT with CLs, most experiments (as shown in Figure 1 with different \u00af \u2113and optimizers) consistently show failed training curves except for a simple two-stage baseline, which consists of a natural complementary learning phase and an AT phase using the predicted labels of the \ufb01rst-stage model. The above fact motivated us to further investigate AT with CLs, and we identi\ufb01ed its unique challenges as intractable adversarial optimization and low-quality adversarial examples (as veri\ufb01ed in Figure 2). Speci\ufb01cally, we \ufb01rst theoretically prove that conducting AT with the complementary risk is statistically consistent to that with the ordinary risk when training with the general backward corrected loss [18]. Although the guarantee holds with suf\ufb01cient data, it is not preserved any more with a limited number of CLs given in practice [7] (see Eq. (8) in Section 4.1). Empirically, we analyze the training dynamics of AT with CLs (e.g., gradient directions and norms), and observe that adversarial optimization with CLs through \u00af \u2113suffers from large gradient variance, with the gradient vanishing issue at the early stage of training. Moreover, we also \ufb01nd that the complementary loss as the objective of adversarial optimization fails to generate high-quality adversarial examples due to the less informative nature [17] of CLs and the implicit way of attacking ordinary labels (i.e., attacking P j\u0338=\u00af y p\u03b8(j|x) instead). To mitigate the problems, we propose a new learning strategy using gradually informative attacks (in Algorithm 1) for AT with CLs, which consists of two critical components, i.e., Warm-up Attack (Warmup) and Pseudo-Label Attack (PLA). We introduce the Warm-up to ease the intractable optimization with CLs, which is gradually transferred from natural training to AT by controlling the adversarial budget. We design the PLA to improve the low-quality adversarial generation, which incorporates the informative predictions from the progressively discriminative model itself. Overall, we propose a uni\ufb01ed framework using a \ufb02exible scheduler to adjust two critical components during training. 2 \fOur contributions are summarized as follows. \u2022 Setting level: we study AT with CLs, which is a brand new yet challenging setting. \u2022 Challenge level: we discover the substantial experimental failures in the direct combination of AT with existing methods for learning with CLs (in Figure 1), and provide the underlying insights of the observed phenomenon from both theoretical and empirical perspectives (in Sections 4.1 and 4.2). Speci\ufb01cally, we identify two challenges as intractable adversarial optimization and low-quality adversarial examples. \u2022 Methodology level: we accordingly propose a uni\ufb01ed framework, using Warm-up and PLA with a controllable scheduler to mitigate the problems (in Sections 4.3). \u2022 Experimental level: we conduct extensive experiments to verify the effectiveness of our method, and to understand the bene\ufb01ts of two critical components (in Section 5). 2 Related Work Complementary Learning. Learning with CLs was introduced by [17], where the SCL with the uniform assumption was \ufb01rstly considered. An unbiased risk estimator (URE) was derived by modifying a speci\ufb01c loss function (e.g., one-versus-all or pairwise comparison) that satis\ufb01es a symmetric condition. Continuing with this work, [18] derived a general URE for arbitrary loss functions and models, and further proposed non-negative correction and gradient ascent methods to cope with the over\ufb01tting issue of URE in practice. Although URE has good statistical properties, it has inferior empirical performance due to huge gradient variance. [7] mitigated this problem by proposing a surrogate complementary loss framework, which is a biased risk estimator trading zero bias with reduced variance. Other than the uniform assumption, [39] considered the biased CLs, where labels other than the true label are chosen as a CL with different probabilities due to the annotator\u2019s bias. In addition to the SCL setting, [13] derived an URE of the ordinary risk for multiple complementary labels, and further improved it by minimizing properly chosen upper bounds. Moreover, there are also several works less related to the loss correction in complementary learning, such as introducing regularization [38] or reweighting [14] during the training process. Adversarial Training. As one of the most effective defensive methods, AT has been widely studied to improve the robustness of deep learning models [1; 12; 37; 36; 26; 35; 6]. The standard AT [24] formulated the problem as min-max optimization. [4] proposed CAT to mitigate the catastrophic forgetting and the generalization issues of AT. [41] decomposed the prediction error into the natural and boundary errors, and proposed TRADES to balance the classi\ufb01cation performance between the natural and adversarial examples. [43] resolved the trade off between robustness and accuracy via reweighting adversarial data by their geometrical information to the decision boundary. In contrast to previous works, only limited effort has been made to explore AT with imperfect supervision [27; 18]. [45] explored the interaction of AT with noisy labels (NLs), and found that AT itself is NLs correction since the number of PGD steps could be used to improve sample selection quality. [42] investigated the effects of NLs injection on the adversarial optimization, and proposed NoiLIn to mitigate the robust over\ufb01tting issue based on the empirical observation that NLs are not always detrimental. 3 Preliminaries In this section, we provide preliminaries for ordinary learning, complementary learning and AT. For complementary learning, we consider the SCL setting with the uniform assumption [17; 18; 7]. For AT, we focus on the standard method proposed in [24]. 3.1 Ordinary Learning Let X \u2282Rd be the input space, and Y \u2208[K] := {1, 2, . . . , K}(K > 2) be the label space. The data {(xi, yi)}n i=1 is sampled independently and identically from the joint distribution D with density p(X, Y ), which could be further decomposed into p(X)p(Y |X). We de\ufb01ne a class-probability function \u03b7i(x) = p(Y = i|X = x). The decision and loss functions are de\ufb01ned as g : X \u2192RK and \u2113: Y \u00d7 RK \u2192R+, respectively. The goal of ordinary learning is to train a classi\ufb01er f(x) : X \u2192Y, 3 \fwhich is expected to predict the correct label by arg maxi g(x)i. The ordinary risk is de\ufb01ned as R(g; \u2113) = E(x,y)\u223cD[\u2113(y, g(x))]. (1) 3.2 Complementary Learning In complementary learning, the label space is transformed into \u00af Y, where each ordinary label y is \ufb02ipped into \u00af y \u2208\u00af Y with a class-dependent probability p(\u00af y|y). Then, the data is sampled from a different joint distribution \u00af D with density p(X, \u00af Y ). We assume the transition matrix Q \u2208RK\u00d7K is invertible, where Qij = p( \u00af Y = j|Y = i). With the uniform assumption, it takes 0 on diagonals and 1 K\u22121 on non-diagonals. The class-probability function is then written as \u00af \u03b7i(x) = p( \u00af Y = i|X = x) = X j\u0338=i p( \u00af Y = i|Y = j)p(Y = j|X = x) = X j\u0338=i Qji\u03b7j(x). (2) The complementary risk is de\ufb01ned as \u00af R(g; \u00af \u2113) = E(x,\u00af y)\u223c\u00af D[\u00af \u2113(\u00af y, g(x))], where \u00af \u2113is a complementary loss function. Typically, it can be derived by the loss correction, which is a technique that ensures risk or classi\ufb01er consistency in the weakly supervised setting [27; 17]. The backward correction is a risk consistent algorithm that ensures \u00af R is a statistically consistent risk estimator of the ordinary risk R, while the forward correction is a classi\ufb01er consistent algorithm that guarantees the classi\ufb01er learned from \u00af D using \u00af \u2113converges to the optimal one learned from D using \u2113. In complementary learning, [17; 18; 13] focused on the backward correction that multiplies the loss by Q\u22121, while [39] proposed a forward correction method via multiplying the model prediction by Q. Other methods (e.g., a bounded loss) can also be used to derive complementary losses [7; 13]. The detailed complementary loss functions are provided in Table 2 of Appendix B. General Backward Correction. Among this line of research, a notable work [18] derived an URE of the ordinary risk, without any restriction on the used loss and model. Given the notations borrowed from [18; 7]: \u2113(g(x)) = [\u2113(1, g(x)), \u2113(2, g(x)), . . . , \u2113(K, g(x))] and ei is the one-hot column vector with one on the ith entry, the URE is derived as follows: Proposition 1. The ordinary risk can be expressed in terms of CLs as follows, R(g; \u2113) = E(x,y)\u223cD[\u2113(y, g(x))] = E(x,\u00af y)\u223c\u00af D[\u00af \u2113(\u00af y, g(x))] = \u00af R(g; \u00af \u2113), (3) when \u00af \u2113is rewritten as \u00af \u2113(\u00af y, g(x)) = eT \u00af y (Q\u22121)\u2113(g(x)), (4) With the uniform assumption, \u00af \u2113can be further rewritten as \u00af \u2113(\u00af y, g(x)) = \u2212(K \u22121)\u2113(\u00af y, g(x)) + K X j=1 \u2113(j, g(x)). (5) With this expression, we can obtain an URE of the ordinary risk only from CLs. 3.3 Adversarial Training AT has been a commonly used defensive technique that equips deep learning models with adversarial robustness against imperceptible perturbations within a small neighborhood of input. Here, we consider conducting AT with ordinary labels (e.g., \u2113is the cross-entropy loss). Generally, it can be formulated into a min-max optimization problem: min \u03b8 E(x,y)\u223cD[\u2113(y, g(\u02dc x))], with \u02dc x = arg max\u02dc xi\u2208B\u03f5[x][\u2113(y, g(\u02dc xi))], (6) where B\u03f5[x] = {\u02dc x : \u2225x \u2212\u02dc x\u2225p \u2264\u03f5} is the closed ball centered at input x with radius \u03f5 > 0 under lp-norm threat models, and \u02dc x is the adversarial data found within B\u03f5[x]. Empirically, the solution of inner maximization is approximated by projected gradient descent (PGD) as follows: x(t+1) = \u03a0B\u03f5[x][x(t) + \u03b1sign(\u2207x(t)\u2113(y, g(x(t))))], (7) where \u03b1 is the step size, \u03a0 is the projection operator that projects the adversarial data back to the epsilon ball, x(t) is the adversarial data found at step t, and x(0) is initialized by natural data or natural data with a small Gaussian or uniformly random perturbation. The adversarial data is updated iteratively in the direction of loss maximization until a stop criterion is satis\ufb01ed. 4 \f4 Adversarial Training with Complementary Labels In this section, we \ufb01rst provide some theoretical insights for AT with CLs. Then, we empirically analyze the experimental failure and identify the critical challenges as intractable adversarial optimization and low-quality adversarial examples when conducting AT with corrected complementary losses. Accordingly, we propose a new learning strategy with gradually informative attacks. 4.1 Theoretical Analysis Without loss of generality, below we analyze the general backward correction [18] theoretically, and other popular complementary loss functions on the SCL setting. Proposition 2. For the general backward correction, conducting AT on the complementary risk is equivalent to that on the ordinary risk. However, this is not the case on their empirical risks: min \u03b8 Ex\u223cp(X) max \u02dc x\u2208B\u03f5[x] Ey\u223cp(Y |X=x)[\u2113(y, g(\u02dc x))] = min \u03b8 Ex\u223cp(X) max \u02dc x\u2208B\u03f5[x] E\u00af y\u223cp( \u00af Y |X=x)[\u00af \u2113(\u00af y, g(\u02dc x))], min \u03b8 1 n n X i=1 max \u02dc xi\u2208B\u03f5[xi][\u2113(yi, g(\u02dc xi))] \u0338= min \u03b8 1 n n X i=1 max \u02dc xi\u2208B\u03f5[xi][\u00af \u2113(\u00af yi, g(\u02dc xi))]. (8) The detailed proof is provided in Appendix A.2. In brief, when considering the general backward corrected loss, the maximization induced by AT keeps the statistical properties of the complementary risk, and it still yields a consistent risk estimator. In theory, the adversarial optimization of the complementary risk with the general backward corrected loss is statistically consistent to that of the ordinary risk with the ordinary loss. But in practice, we can only obtain limited CLs (e.g., the sample size of E\u00af y\u223cp( \u00af Y |X=x) is one on the SCL setting), which causes the inconsistency between their empirical risks (i.e., the solutions to their inner maximization are different), and hence leads to a dif\ufb01cult adversarial optimization (as veri\ufb01ed in Appendix D.2). The solution to inner maximization of the complementary risk is equivalent to that of the ordinary risk if and only if maximizing the weighted loss over all candidate CLs, that is E\u00af y\u223cp( \u00af Y |X=x)[\u00af \u2113(\u00af y, g(\u02dc x))]. From the perspective of AT, we expect to \ufb01nd the adversarial examples \u02dc x within the neighborhood of input x that make the model predictions on the ordinary label p\u03b8(y|\u02dc x) as low as possible. When we have ordinary labels, it is quite straightforward to achieve this objective by maximizing the crossentropy \u2113= \u2212log p\u03b8(y|\u02dc x). However, if we further consider recent proposed methods in addition to the general backward correction in complementary learning, most of \u00af \u2113mathematically try to maximize P j\u2208Y\\{\u00af y} p\u03b8(j|\u02dc x) (or equivalently minimize p\u03b8(\u00af y|\u02dc x)) as shown in Table 2. Maximizing \u00af \u2113 leads to the minimization of P j\u2208Y\\{\u00af y} p\u03b8(j|\u02dc x), which cannot guarantee the minimization of p\u03b8(y|\u02dc x). In such a case, the inner maximization may serve as a random attacker without prior knowledge. Hence, corrected complementary losses fail to achieve the objective of AT, and low-quality adversarial examples may be generated, which is detailedly discussed in the following part. 4.2 Empirical Analysis We conduct the experiment on Kuzushiji to justify the previous theoretical insights. For the upper panels of Figure 2, we focus on the issue of intractable adversarial optimization by adversarially training the model using several loss functions separately, with three random trials. For the lower panels of Figure 2, we focus on the issue of low-quality adversarial examples by measuring the different adversarial examples generated by various loss functions, given the same optimization model (i.e., the oracle model, which is trained using AT with ordinary labels). We show the average results over different seeds and training samples, respectively. Without loss of generality, here we only show the results of one complementary loss (i.e., LOG [13]), and our proposed method (i.e., Warm-up+PLA in Section 4.3). The detailed experiment settings and more comprehensive empirical results with other complementary losses and datasets are provided in Appendix C. Intractable Adversarial Optimization. In the outer minimization of AT, we \ufb01rst collect the gradient direction (with respect to the model parameters) of each data in one mini-batch (i.e., 256), and then compute the trace sum of their covariance matrix (the average result for each epoch is shown 5 \f0 20 40 60 80 100 epoch 100 125 150 175 200 225 250 275 Oracle LOG Warm-up+PLA (a) Gradient Direction Variance 0 20 40 60 80 100 epoch 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Oracle LOG Warm-up+PLA (b) (Last Layer) Gradient l2-Norm 0 20 40 60 80 100 epoch 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Oracle LOG Warm-up+PLA (c) (First Layer) Gradient l2-Norm 0 20 40 60 80 100 epoch 0.2 0.4 0.6 0.8 1.0 LOG Warm-up+PLA (d) Cosine Similarity 0 20 40 60 80 100 epoch 20 40 60 80 100 LOG Warm-up+PLA (e) l1-Distance 0 20 40 60 80 100 epoch 0.00 0.05 0.10 0.15 0.20 Oracle LOG Warm-up+PLA (f) p\u03b8(y|x) \u2212p\u03b8(y|\u02dc x) Figure 2: [Intractable Adversarial Optimization] (a) the trace sum of covariance matrix of gradient directions w.r.t. the parameters of the \ufb01rst layer; (b) the l2-norm of stochastic gradients of the last layer; (c) the l2-norm of stochastic gradients of the \ufb01rst layer; [Low-quality Adversarial Examples] (d) the cosine similarity of gradient directions (i.e., sign(\u2207x\u00af \u2113(x, \u00af y; \u03b8))) with the oracle in the inner maximization; (e) the l1-distance of the constructed adversarial examples with the oracle; (f) the difference of model predictions of natural data and adversarial data on the ordinary label. in the upper-left panel of Figure 2). A larger value re\ufb02ects a group of gradient directions with a bigger variance. Note that the result of LOG nearly reaches the maximum trace sum of covariance matrix of the gradient directions, which results in its consistent failure at the early stage of training. We also compute the average l2-norm of stochastic gradients of the last and \ufb01rst layer as shown in the upper-middle and upper-right panels of Figure 2, respectively. Although some gradient signals exist for LOG in the last layer, the gradients back-propagated to the \ufb01rst layer is so small that gradient vanishing problem occurs at the early stage of training. This phenomenon is similar to that observed in the standard AT, which is conjectured as the prevalence of dead neurons caused by a large radius of epsilon ball [23]. Moreover, considering AT with CLs, it is expected that the gradient norm tends to be small since the observed variance of gradient directions in one mini-batch is extremely large. The above observations indicate the adversarial optimization with CLs is dif\ufb01cult, and it is rational to use the method (i.e., our Warm-up+PLA) to ease the optimization. Low-quality Adversarial Examples. In the inner maximization of AT, given the same optimization model, we measure the quality of generated adversarial examples by computing the cosine similarity of gradient directions (with respect to the same data) and the l1-distance of constructed adversarial examples, with the oracle examples generated by ordinary label (in the lower-left and lower-middle panels of Figure 2). A larger cosine similarity and a lower l1-distance indicate a stronger resemblance to the oracle, but not necessarily a stronger attack (or high-quality adversarial examples). Hence, we further compute the difference of model predictions of natural data and adversarial data on the ordinary label during training (in the lower-right panel of Figure 2). A bigger difference explicitly demonstrates a stronger attack. The results show that the complementary loss as the objective of AT with CLs fails to generate high-quality adversarial examples as our analysis in Section 4.1. 4.3 Methodology To address the previous problems, we accordingly propose a new strategy using gradually informative attacks to ease the intractable adversarial optimization and improve the quality of adversarial examples, respectively. Our algorithm is concretely presented in Algorithm 1. Speci\ufb01cally, it consists of Warm6 \fAlgorithm 1 AT with CLs Using Gradually Informative Attacks Input: number of epochs: E, number of batches: B, number of class: K, radius of epsilon ball: \u03f5, number of attack steps: k, step size: \u03b1, hyperparameter for exponential moving average of cashed model predictions: \u03b2; hyperparameter for label weights: \u03b3; Output: robust model \u03b8R; 1: pc \u2190 1 K\u22121(1 \u2212e\u00af y)T for all data 2: for e = 1, . . . , E do 3: \u03b2, \u03b3 \u2190Update hyperparameters 4: \u03f5e, \u03b1e, ke \u2190Adversarial budget scheduler \u25b7Warm-up Attack 5: for b = 1, . . . , B do 6: x, \u00af y \u2190Sample a batch of data 7: p\u03b8(x) \u2190Softmax(\u03b8(x)), pc(x) \u2190\u03b2pc(x) + (1 \u2212\u03b2)p\u03b8(x), pc(\u00af y|x) \u21900 8: \u02c6 y \u2190arg maxj\u0338=\u00af y pc(j|x) 9: \u02dc x \u2190Approximate the solutions of max \u00af \u2113(x, \u00af y; \u03b8) (Eq. 10) by PGD(\u03f5e, \u03b1e, ke) 10: L = \u2212(K \u22121) log(\u03b3 P j\u0338=\u00af y p\u03b8(j|\u02dc x) + (1 \u2212\u03b3)p\u03b8(\u02c6 y|\u02dc x)) \u25b7Pseudo-Label Attack 11: \u03b8 \u2190SGD(\u03b8, \u2207\u03b8L) 12: end for 13: Evaluate(\u03f5, \u03b1, k) 14: end for up Attack and Pseudo-Label Attack, whose dynamics are controlled by a \ufb02exible scheduler: T(a, e, Es) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 a 2 \u00b7 (1 \u2212cos(min( e Es , 1) \u00b7 \u03c0)) Warm-up Attack, a \u00b7 (1 \u2212min( e Es , 1)) Pseudo-Label Attack, (9) where a is the maximum value of the scheduled variable (i.e., \u03f5max = \u03f5 in Warm-up Attack and \u03b3max = 1 in Pseudo-Label Attack), e is the epoch index, and Es is the duration of the scheduler. Both the proposed attacks can be controlled by a \ufb02exible scheduler with the increasing and decreasing trends respectively (e.g., see the example in Figure 3). Below, we describe each component in detail. Warm-up Attack. Adversarial optimization is harder compared with ordinary optimization [23], and CLs further strengthen the degree of hardness as observed in our empirical analysis. The major difference of AT with CLs from the standard AT is the labels used in the adversarial generation. As shown in Figure 2, training with the adversarial examples generated by different labels results in the different gradient variance. Since the adversarial generation on CLs induces large gradient variance which causes the dif\ufb01cult adversarial optimization, we can adjust the adversarial generation to control the variance introduced into the optimization. To be speci\ufb01c, we adopt a dynamic adversarial budget to gradually enhance the degree of adversarial generation. Typically, adversarial budgets consist of the radius of epsilon ball \u03f5, number of attack steps k, and step size \u03b1. Here, we mainly focus on the dynamics of \u03f5 in Eq. (9). The radius of the epsilon ball is increased from 0 to \u03f5 within Es epochs, and is kept constant until the end of adversarial optimization. The number of attack steps is constant (ke = k), and the step size is increased proportionally to \u03f5 (\u03b1e = \u03f5e \u03f5 \u03b1). In this way, the gradually increased adversarial budget can ease the dif\ufb01culty of adversarial optimization. Pseudo-Label Attack. Due to the existence of an adversarial budget scheduler, at the early stage of optimization, simple patterns may be prevalent in adversarial data found within an extremely small epsilon ball, which is helpful for the self-evolvement of a discriminative model. The model tends to assign high con\ufb01dence to the potential ordinary label, while low con\ufb01dence to others. This informative model prediction is a strong supplementary information that is promising for improving the quality of the adversarial generation. Therefore, we propose to optimize the objective consisting of the convex combination outputs of partial labels (labels other than CLs) and predicted ordinary label. This transforms the whole process into an optimization that starts with learning with CLs using certain correction techniques, and gradually moves to learning with predicted labels regarding them as ordinary labels. The loss function can be de\ufb01ned as follows: \u00af \u2113(x, \u00af y; \u03b8) = \u2212(K \u22121) log(\u03b3 X j\u0338=\u00af y p\u03b8(j|x) + (1 \u2212\u03b3)p\u03b8(\u02c6 y|x)), \u02c6 y = arg max j\u0338=\u00af y pc(j|x), (10) where \u03b3 \u2208[0, 1] is a hyperparameter that controls the transition of learning. Recent studies found that adversarial training can degrade the generalization, and a trade-off between natural accuracy 7 \f0 20 40 60 80 100 epoch 0.0 0.2 0.4 0.6 0.8 1.0 e (a) Dynamics of \u03f5e and \u03b3 0 20 40 60 80 100 epoch 0 20 40 60 80 PGD-20 Test Accuracy (%) Warm-up+PLA W/O Warm-up W/O PLA LOG (b) Ablation Study on Kuzushiji 0 20 40 60 80 100 120 epoch 0 10 20 30 40 PGD-20 Test Accuracy (%) Warm-up+PLA W/O Warm-up W/O PLA LOG (c) Ablation Study on CIFAR10 10 20 30 40 50 60 70 80 90 Es 81 82 83 84 85 86 87 PGD-20 Test Accuracy (%) (d) Ablation Study on Es Figure 3: (a) example of the dynamics of \u03f5e and \u03b3 adopted on MNIST/Kuzushiji, where the warmup period is 10 epochs, and Es = 50; (b) ablation study of the two components on Kuzushiji; (c) ablation study of the two components on CIFAR10; (d) PGD20 test accuracy on Kuzushiji w.r.t. the Es. and adversarial robustness exists [32]. To avoid the effect of such degradation and stabilize the optimization (e.g., a drastic shift of model predictions may occur due to the increasing adversarial budget), we use a simple exponential moving average to collect the dynamics of model predictions, and choose the label with the maximum probability as a predicted ordinary label. Speci\ufb01cally, given the model predictions p\u03b8(x) of the current epoch, the cashed model predictions pc(x) is updated as pc(x) = \u03b2pc(x) + (1 \u2212\u03b2)p\u03b8(x), where pc(x) is initialized with (1\u2212e\u00af y)T K\u22121 , pc(\u00af y|x) = 0, and \u03b2 = 0.9. 5 Experiments In this section, we empirically verify the effectiveness of our method on MNIST [22], Kuzushiji [8], CIFAR10 [20] and SVHN [28] datasets, and conduct ablation studies to understand the two attacks. More detailed information and the conducted experiments are presented in Appendix D. Setups. 1) General Setups: for MNIST and Kuzushiji, we train a small convolutional neural network, as used in [41], for 100 epochs3 with batch size of 256. For CIFAR-10 and SVHN, ResNet18 is trained for 120 epochs with batch size of 128. The training sets of CIFAR-10 and SVHN are augmented with random cropping and horizontal \ufb02ipping. 2) Adversarial Training Setups: we focus on the L\u221ethreat model. For MNIST and Kuzushiji, stochastic gradient descent (SGD) is used with learning rate of 0.01 and momentum of 0.9. PGD is used as the approximation method of solving inner maximization, with \u03f5 = 0.3, \u03b1 = 0.01 and k = 40. For CIFAR10 and SVHN, SGD is used with weight decay of 5e-4 and momentum of 0.9. The initial learning rate is 0.01, and decayed by 10 at Epoch 30 and 60 of adversarial optimization. For the L\u221ethreat model, \u03f5 = 8/255, \u03b1 = 2/255 and k = 10. 3) Complementary Learning Setups: following the settings in [18; 13], for MNIST and Kuzushiji, Adam optimizer is used with learning rate of 1e-3 and weight decay of 1e-4. For CIFAR10 and SVHN, SGD is used with weight decay of 5e-4 and momentum of 0.9. The learning rate is linearly increased to 0.01 within the \ufb01rst 5 epochs. More details are provided in Appendix D.1. Baseline. 1). Two-stage method: it \ufb01rst trains a model for 50 epochs with the methods of complementary learning (i.e., LOG, which shows better performance and stability compared to others [13]). Then we re-label the complementary training set using the checkpoint with the best validation performance, and adversarially train another model using the re-labeled training set afterwards. 2). Conducting adversarial training directly with complementary losses, including FORWARD [39], FREE, NN [18], SCL_NL, SCL_EXP [7], EXP and LOG [13]. We evaluate the adversarial robustness using PGD20, CW30 and AutoAttack (AA) [9], and report the results of checkpoint with the best PGD20 test accuracy. All experiments are conducted with multiple random seeds (i.e., 1-3). 5.1 Performance Evaluation The detailed results are reported in Table 1, where Oracle refers to the standard AT with ordinary labels, and it is just given for measuring the performance gap between AT with CLs and ordinary labels. For easy datasets (e.g., MNIST), the complementary losses are possible to obtain satisfying results, while most of them fail on complicated datasets. We would like to mention that the performance of 3Note that AT is nearly 10-40 times [24; 40] slower than the natural training due to the adversarial generation. Here we train 100 epochs which is kept the same as the most literature in AT [24; 41; 5; 29]. 8 \fTable 1: Means (standard deviations) of natural and adversarial test accuracy. Dataset Method Natural PGD CW AA MNIST Oracle 99.46(\u00b10.04) 98.14(\u00b10.04) 97.45(\u00b10.08) 92.53(\u00b10.23) Two-stage 99.07(\u00b10.02) 97.44(\u00b10.23) 96.72(\u00b10.21) 92.06(\u00b10.45) FORWARD [39] 97.22(\u00b11.13) 93.76(\u00b12.38) 92.13(\u00b12.87) 85.41(\u00b13.69) FREE [18] 48.94(\u00b126.04) 38.02(\u00b122.21) 32.81(\u00b119.63) 22.68(\u00b113.91) NN [18] 68.48(\u00b140.40) 66.80(\u00b139.22) 66.25(\u00b138.84) 60.65(\u00b138.04) SCL_NL [7] 93.09(\u00b17.35) 87.07(\u00b112.40) 84.59(\u00b114.51) 75.98(\u00b118.29) SCL_EXP [7] 14.88(\u00b14.99) 14.34(\u00b14.23) 13.58(\u00b13.16) 10.47(\u00b11.25) EXP [13] 10.99(\u00b10.50) 10.99(\u00b10.50) 10.99(\u00b10.50) 10.99(\u00b10.50) LOG [13] 97.16(\u00b10.64) 93.38(\u00b11.25) 91.67(\u00b11.39) 84.88(\u00b12.09) Warm-up+PLA 99.22(\u00b10.02) 97.73(\u00b10.06) 97.11(\u00b10.07) 92.37(\u00b10.19) Kuzushiji Oracle 95.94(\u00b10.15) 90.01(\u00b10.43) 88.06(\u00b10.96) 70.63(\u00b10.48) Two-stage 89.75(\u00b10.42) 82.91(\u00b11.01) 80.21(\u00b11.27) 64.57(\u00b11.79) FORWARD 35.48(\u00b127.96) 29.84(\u00b125.55) 28.09(\u00b124.37) 22.01(\u00b118.98) FREE 16.17(\u00b11.77) 12.01(\u00b10.55) 9.33(\u00b11.50) 4.08(\u00b11.60) NN 10.00(\u00b10.00) 10.00(\u00b10.00) 10.00(\u00b10.00) 8.87(\u00b11.60) SCL_NL 40.83(\u00b124.03) 32.82(\u00b122.88) 29.93(\u00b122.53) 20.86(\u00b119.74) SCL_EXP 10.00(\u00b10.00) 10.00(\u00b10.00) 10.00(\u00b10.00) 8.21(\u00b12.54) EXP 10.00(\u00b10.00) 10.00(\u00b10.00) 10.00(\u00b10.00) 10.00(\u00b10.00) LOG 32.66(\u00b125.50) 26.87(\u00b122.66) 24.90(\u00b121.20) 18.78(\u00b116.95) Warm-up+PLA 91.60(\u00b10.49) 85.88(\u00b10.48) 83.74(\u00b10.35) 68.75(\u00b10.68) CIFAR10 Oracle 78.10(\u00b10.36) 47.35(\u00b10.05) 45.66(\u00b10.22) 43.47(\u00b10.23) Two-stage 64.98(\u00b12.70) 42.48(\u00b10.92) 39.90(\u00b10.67) 38.89(\u00b10.46) FORWARD 15.41(\u00b10.85) 13.58(\u00b10.56) 13.03(\u00b10.34) 12.94(\u00b10.35) FREE 11.60(\u00b10.30) 10.54(\u00b10.23) 10.45(\u00b10.20) 10.32(\u00b10.26) NN 11.79(\u00b10.61) 11.13(\u00b10.56) 11.14(\u00b10.56) 11.10(\u00b10.53) SCL_NL 14.45(\u00b11.07) 13.13(\u00b11.03) 12.84(\u00b11.17) 12.79(\u00b11.18) SCL_EXP 13.49(\u00b11.76) 12.55(\u00b11.39) 12.45(\u00b11.35) 12.38(\u00b11.31) EXP 10.97(\u00b11.01) 10.56(\u00b10.60) 10.38(\u00b10.36) 10.37(\u00b10.34) LOG 11.70(\u00b11.06) 11.04(\u00b10.72) 10.49(\u00b10.59) 10.44(\u00b10.55) Warm-up+PLA 65.88(\u00b11.51) 43.29(\u00b10.70) 41.30(\u00b10.59) 40.28(\u00b10.39) SVHN Oracle 91.95(\u00b10.11) 53.89(\u00b10.11) 50.59(\u00b10.23) 47.05(\u00b10.22) Two-stage 90.62(\u00b10.27) 53.92(\u00b10.33) 50.86(\u00b10.28) 47.31(\u00b10.11) FORWARD 19.59(\u00b10.00) 19.59(\u00b10.00) 19.59(\u00b10.00) 19.68(\u00b10.00) FREE 19.59(\u00b10.00) 19.59(\u00b10.00) 19.59(\u00b10.00) 19.68(\u00b10.00) NN 19.59(\u00b10.00) 19.59(\u00b10.00) 19.59(\u00b10.00) 19.68(\u00b10.00) SCL_NL 19.59(\u00b10.00) 19.59(\u00b10.00) 19.58(\u00b10.00) 19.67(\u00b10.00) SCL_EXP 19.59(\u00b10.00) 19.59(\u00b10.00) 19.58(\u00b10.00) 19.68(\u00b10.00) EXP 19.60(\u00b10.01) 19.60(\u00b10.02) 19.56(\u00b10.03) 19.65(\u00b10.04) LOG 19.59(\u00b10.00) 19.60(\u00b10.02) 19.59(\u00b10.00) 19.68(\u00b10.00) Warm-up+PLA 90.50(\u00b10.16) 54.58(\u00b10.10) 51.04(\u00b10.04) 47.47(\u00b10.13) the two-stage and our methods slightly outperform that of the oracle on SVHN. We assume it may be attributed to the recent observation that noisy datasets (with a small portion of noisy labels) are not always detrimental to AT, and can even further boost the adversarial robustness [42]. 5.2 Ablation Study In this part, we conduct the ablation study to demonstrate the importance of the two components, i.e., Warm-up Attack and Pseudo-Label Attack in our method with the critical hyperparameter. Two Critical Components. We investigate the roles of two components in our method by removing them separately. We report the results within three trials on Kuzushiji and CIFAR10 datasets as shown in the left-middle and right-middle panels of Figure 3. In summary, warm-up attack could stabilize 9 \fthe optimization especially when the model capacity are not suf\ufb01cient large, and pseudo-label attack is crucial for the achieving high adversarial robustness. Speci\ufb01cally, the warm-up attack achieves quite good performance on Kuzushiji, while limited improvements on CIFAR10. This is expected due to the dif\ufb01culty of CIFAR10 without further informative attacks. Pseudo-label attack solves this problem by incorporating the informative model predictions, and hence achieves quite good performance on CIFAR10. However, it has an unstable result on Kusushiji, which is attributed to the limited model capacity. It is of great importance to achieve adversarial robustness given limited model capacity. Therefore, warm-up attack is still a crucial component for stabilizing the optimization. Hyperparameter. We investigate the performance sensitivity to the key hyperparameter Es, which controls the growing speed of informative attacks. We report the results within three trials on Kuzushiji as shown in the right panels of Figure 3. The result shows a similar performance across a wide range of Es, which demonstrates that our method is not sensitive to the tuning of key hyperparameter. 6"
+                }
+            ],
+            "Yaoxin Wu": [
+                {
+                    "url": "http://arxiv.org/abs/2111.03466v1",
+                    "title": "Learning Large Neighborhood Search Policy for Integer Programming",
+                    "abstract": "We propose a deep reinforcement learning (RL) method to learn large\nneighborhood search (LNS) policy for integer programming (IP). The RL policy is\ntrained as the destroy operator to select a subset of variables at each step,\nwhich is reoptimized by an IP solver as the repair operator. However, the\ncombinatorial number of variable subsets prevents direct application of typical\nRL algorithms. To tackle this challenge, we represent all subsets by\nfactorizing them into binary decisions on each variable. We then design a\nneural network to learn policies for each variable in parallel, trained by a\ncustomized actor-critic algorithm. We evaluate the proposed method on four\nrepresentative IP problems. Results show that it can find better solutions than\nSCIP in much less time, and significantly outperform other LNS baselines with\nthe same runtime. Moreover, these advantages notably persist when the policies\ngeneralize to larger problems. Further experiments with Gurobi also reveal that\nour method can outperform this state-of-the-art commercial solver within the\nsame time limit.",
+                    "authors": "Yaoxin Wu, Wen Song, Zhiguang Cao, Jie Zhang",
+                    "published": "2021-11-01",
+                    "updated": "2021-11-01",
+                    "primary_cat": "cs.AI",
+                    "cats": [
+                        "cs.AI",
+                        "cs.LG",
+                        "math.OC"
+                    ],
+                    "main_content": "Introduction Combinatorial optimization problems (COPs) have been widely studied in computer science and operations research, which cover numerous real-world tasks in many \ufb01elds such as communication, transportation and manufacturing [1]. Most COPs are very dif\ufb01cult to solve ef\ufb01ciently due to their NP-hardness. The performance of classic methods, including exact and heuristic algorithms [2], is generally limited by hand-crafted policies that are costly to design, since considerable trial-and-error and domain knowledge are needed. On the other hand, it is common in practice that similar instances with shared structure are frequently solved, and differ only in data that normally follows a distribution [3]. This provides a chance for machine learning to automatically generate heuristics or policies. In doing so, the learned alternatives are expected to save massive manual work in algorithm design, and raise the performance of the algorithm on a class of problems. Recently, a number of works apply deep (reinforcement) learning to automatically design heuristic algorithms, either in constructive or improving fashion. Different from construction heuristics that sequentially extend partial solutions to complete ones [4, 5, 6, 7, 8, 9, 10, 11], learning improvement heuristics can often deliver high solution quality by iteratively reoptimizing an initial solution using \u2217Wen Song is the corresponding author. 35th Conference on Neural Information Processing Systems (NeurIPS 2021). arXiv:2111.03466v1  [cs.AI]  1 Nov 2021 \flocal operations [12, 13, 14]. In this line, some methods are developed under the Large Neighborhood Search (LNS) framework [15, 16], which is a powerful improving paradigm to \ufb01nd near-optimal solutions for COPs. However, the above methods are restricted to speci\ufb01c problem types, and cannot generalize to those from different domains. This motivates the studies of learning to directly solve Integer Programs (IPs), which is very powerful and \ufb02exible in modelling a wide range of COPs. The standard approach to solve IPs is branch-and-bound (B&B) [17], which lies at the core of common solvers such as SCIP, Gurobi, and CPLEX. Thus, most of existing methods improve the performance of a solver on a distribution of instances, by training models for critical search decisions in B&B such as variable and node selection [18, 19, 20]. Nevertheless, these methods are generally limited to small instances and require suf\ufb01cient interface access to the internal solving process of the solvers. This paper mainly tackle the issue that how to improve a solver from externals such that it can \ufb01nd high-quality solutions more quickly? In speci\ufb01c, we propose a high-level, learning based LNS method to solve general IP problems. Based on deep reinforcement learning (RL), we train a policy network as the destroy operator in LNS, which decides a subset of variables in the current solution for reoptimization. Then we use a solver as the repair operator, which solves sub-IPs to reoptimize the destroyed variables. Despite being heuristic, our method can effectively handle the large-scale IP by solving a series of smaller sub-IPs. Moreover, complex interface to the solver\u2019s internal logic is not required. However, the above RL task is challenging, mainly because the action space, i.e., number of variable subsets at each LNS step, is exponentially large. To resolve this issue, we represent all the subsets by factorizing them into binary decisions on each variable, i.e., whether a variable should be destroyed. In doing so, we make it possible to learn a policy to select any subset from large discrete action spaces (at least 21000 candidates in our experiments). To this end, we design a Graph Neural Network (GNN) based policy network that enables learning policies for each variable in parallel, and train it by a customized actor-critic algorithm. A recent work [21] also attempts to learn LNS policy to solve IP problems, and we generalize this framework to enable learning more \ufb02exible and powerful LNS algorithms. One limitation of [21] is that it hypothesizes a constant cardinality of the destroyed variable subset at each LNS step, which is a prede\ufb01ned hyperparameter. However, the number and choice of optimized variables at each step should be adaptive according to instance information and solving status, which is achieved in our method. In doing so, the LNS policies trained by our method signi\ufb01cantly outperform those trained by the method in [21]. We evaluate our method on four NP-hard benchmark problems with SCIP as the repair solver. Extensive results show that our method generally delivers better solutions than SCIP with mere 1/2 or 1/5 of runtime, and signi\ufb01cantly outperforms LNS baselines with the same runtime. These advantages notably persist when the trained policies are directly applied to much larger problems. We also apply our LNS framework to Gurobi, which shows superiority over the solver itself and other baselines. 2 Related work In this section, we brie\ufb02y review existing works related to ours. We \ufb01rst describe two main streams of learning based methods to solve COPs, and then introduce the literature that study RL with large discrete action space, which is also an essential issue we confront in this paper. Learning to solve speci\ufb01c COPs. Quite a few works attempt to learn heuristics to solve certain types of COPs. Compared to construction ones, methods that learn improvement heuristics can often deliver smaller optimality gap, by training policies to iteratively improve the solution. Chen and Tian [12] propose to learn how to locally rewrite a solution; Wu et al. [13] train policies to pick the next solution in local moves; Lu et al. [14] learn to select local operators to reform a solution. These methods are generally limited by simple local operations. A few methods learn more powerful operators under the LNS framework. Hottung and Tierney [15] train an attention model to repair the solution every time it is broken by a prede\ufb01ned destroy operator. Similarly, Gao et al. [16] combine GNN and Recurrent Neural Network to learn a reinsertion operator to repair sequence-based solution. However, all the above methods are limited to speci\ufb01c problem types, e.g., the LNS methods in [15, 16] are designed only for routing problems. In contrast, this paper aims to solve general IP problems with a high-level LNS framework and raise its performance by learning better policies. 2 \fLearning to solve IP problems. Most of learning based methods for IPs aim to improve inner policies of B&B algorithm. For example, He et al. [18] learn to explore nodes in B&B tree by imitating an oracle. Gasse et al. [20] train a GNN model to predict the strong branching rule by imitation learning. Khalil et al. [19] predict the use of primal heuristics by logistic regression. Other components of B&B are also considered to improve its practical performance, such as learning to select cutting planes by Tang et al. [22], predicting local branching at root node by Ding et al. [23] or re\ufb01ning the primal heuristic and branching rule concurrently by Nair et al. [24]. Different from these works, we employ RL to improve practical performance of IP solvers especially for large-scale problems, without much engineering effort on interfacing with inner process of solvers.2 This is also noted in [21], which proposes to combine learning and LNS to solve IPs. However, a major drawback of this method is that the subsets of destroyed variables are assumed to be \ufb01xed-sized, which limits its performance. In contrast, our LNS framework allows picking variable subsets in a more adaptive way, and thus empowers learning broader classes of policies. RL with large action spaces. Learning with large-sized, high-dimensional discrete action spaces is still intricate in current RL research. In this direction, Pazis and Parr [25] use binary format to encode all actions, and learn value functions for each bit. Tavakoli et al. [26] design neural networks with branches to decide on each component of the action. Besides, Dulac-Arnold et al. [27] and Chandak et al. [28] update the action representation by solving continuous control problems so as to alleviate learning complexity via generalization. Other similar works can be found in [29, 30, 31]. However, we note that all these methods are generally designed for tasks with (discretized) continuous action spaces, which are relatively low-dimensional and small-sized (at most 1025 as in [26]). In contrast, action spaces in our scenario are much larger (at least 21000). In this paper, we propose to factorize the action space into binary actions on each dimension, and train the individual policies in parallel through parameter sharing. 3 Preliminaries Integer Program (IP) is a typically de\ufb01ned as arg minx{\u00b5\u22a4x|Ax \u2264b; x \u22650; x \u2208Zn}, where x is a vector of n decision variables; \u00b5 \u2208Rn denotes the vector of objective coef\ufb01cients; the incidence matrix A \u2208Rm\u00d7n and right-hand-side (RHS) vector b \u2208Rm together de\ufb01ne m linear constraints. With the above formulation, the size of an IP problem is generally re\ufb02ected by the number of variables (n) and constraints (m). Large Neighborhood Search (LNS) is a type of improvement heuristics, which iteratively reoptimizes a solution by the destroy and repair operator until certain termination condition is reached [32]. Speci\ufb01cally, the former one breaks part of the solution xt at step t, then the latter one \ufb01xes the broken solution to derive the next solution xt+1. The destroy and repair operator together de\ufb01ne the solution neighborhood N(xt), i.e., solution candidates that can be accessed at time step t. Compared to typical local search heuristics, LNS is more effective in exploring the solution space, since a larger neighborhood is considered at each step [32]. Most of existing LNS methods rely on problem speci\ufb01c operators, e.g., removal and reinsertion for solutions with sequential structure [33, 34]. Instead, we propose a RL based LNS for general IP problems, with the learned destroy operator to select variables for reoptimization at each step. 4 Methodology In this section, we \ufb01rst formulate our LNS framework as a Markov Decision Process (MDP). Then, we present the factorized representation of the large-scale action space, and parametrize the policy by a specialized GNN. Finally, we introduce a customized actor-critic algorithm to train our policy network for deciding the variable subset. 4.1 MDP formulation Most of existing works learn LNS for speci\ufb01c problems [15, 16], which rely on extra domain knowledge and hinder their applicability to other COPs. In this paper, we apply LNS to general IP 2Nevertheless, our method can also work with solvers enhanced by the above methods as repair operators. 3 \fCurrent solution value   Variable to optimize   Reoptimized solution value   \ud835\udc4e11\ud835\udc651 + \ud835\udc4e12\ud835\udc652 + \ud835\udc4e13\ud835\udc653 + \ud835\udc4e14\ud835\udc654 \u2264\ud835\udc4f1 min. \ud835\udc4e21\ud835\udc651 + \ud835\udc4e22\ud835\udc652 + \ud835\udc4e23\ud835\udc653 + \ud835\udc4e24\ud835\udc654 \u2264\ud835\udc4f2 \ud835\udc4e31\ud835\udc651 + \ud835\udc4e32\ud835\udc652 + \ud835\udc4e33\ud835\udc653 + \ud835\udc4e34\ud835\udc654 \u2264\ud835\udc4f3 \ud835\udf0b(\ud835\udc4e\ud835\udc61|\ud835\udc60\ud835\udc61) 4 \u00d7 1 Sample GNN based Policy State \ud835\udc60\ud835\udc61 Environment  Update dynamic features Constraint features \ud835\udc36 Variable features \ud835\udc49 4 \u00d7 \ud835\udc51\u210e 3 \u00d7 \ud835\udc51\ud835\udc50 Destroy operator IP formulation IP formulation Incidence matrix \ud835\udc34 3 \u00d7 4 Static features \ud835\udc651 \ud835\udc652 \ud835\udc653 \ud835\udc654 \ud835\udc651 \ud835\udc652 \ud835\udc653 \ud835\udc654 SOLVER (Repair operator) \ud835\udf071\ud835\udc651 + \ud835\udf072\ud835\udc652 + \ud835\udf073\ud835\udc653 + \ud835\udf074\ud835\udc654 Figure 1: An example of LNS solving an instance with 4 variables and 3 constraints. Given the current state characterized by static and dynamic features, the RL agent (destroy operator) selects a subset of variables from the current solution to be reoptimized. Then this action in\ufb02uences the environment, through which a sub-IP is formulated and solved by the repair operator (i.e., solver). As feedbacks, the new solution updates the state and a reward is provided to the agent. The above process is repeated until reaching the time limit. problems, in which RL is employed to learn a policy that at each step, selects a variable subset from the solution to be reoptimized. We formulate this sequential decision problem as a discrete-time MDP. Speci\ufb01cally, we regard the destroy operator as the agent and the remainder in LNS as the environment. In one episode of solving an IP instance, the MDP can be described as follows: States. A state st \u2208S at each LNS step needs to not only re\ufb02ect the instance information but also the dynamic solving status. For the former, we represent it by static features of variables and constraints, along with the incidence matrix in an instance. The latter is represented by dynamic features, including the current solution xt and dynamic statistics of incumbent solutions up to step t. Speci\ufb01cally, both the current and incumbent solution at t = 0 are de\ufb01ned as the initial solution x0. Actions. At each state st, an action of the agent is to select a variable subset at from all candidate subsets A for reoptimization. Transition. The repair operator (i.e., an IP solver) solves the sub-IP, where only the variables in action at are optimized with the others equaling to their current values in xt. Accordingly, the next state st+1 is deterministically attained by updating st with the new solution: xt+1 = arg min x {\u00b5\u22a4x|Ax \u2264b; x \u22650; x \u2208Zn; xi = xi t, \u2200xi / \u2208at}. (1) Rewards. The reward function is de\ufb01ned as rt = r(st, at) = \u00b5\u22a4(xt \u2212xt+1), which is the change of the objective value. Given the step limit T of the interaction between the agent and environment, the return (i.e., cumulative rewards) from step t of the episode is Rt = PT k=t \u03b3k\u2212trk, with the discount factor \u03b3 \u2208[0, 1]. The goal of RL is to maximize the expected return E[R1] over all episodes, i.e., the expected improvement over initial solutions, by learning a policy. Policy. The stochastic policy \u03c0 represents a conditional probability distribution over all possible variable subsets given a state. Starting from s0, it iteratively picks an action at based on the sate st at each step t, until reaching the step limit T. An example of the proposed LNS framework solving an IP instance is illustrated in Figure 1, in which the GNN based policy network has the potential to process instances of any size via sharing parameters across all variables. 4.2 Action factorization Given the vector of decision variables x in an IP instance, we gather all its elements in a set X = {x1, x2, . . . , xn}. Accordingly, we de\ufb01ne the action space A = {a|a \u2286X}, which contains all possible variable subsets. Thus our RL task is learning a policy to select at \u2208A for reoptimization at each step t of LNS, and the cardinality of at (i.e., |at|) re\ufb02ects the destroy degree on the solution. Apparently, the size of the combinatorial space A is 2n, which grows exponentially with the number 4 \fof variables n. This prevents the application of general RL algorithms on large-scale problems, since they require an explicit representation of all actions and exploration over such huge action space. As a special case under our LNS framework, Song et al. [21] assumes the action space as the subspace of A that merely contains equal-sized variable subsets, that is, Az = {az|az \u2208A, |az| = z}. In doing so, they instead learn classifying variables into groups of equal size, to optimize each iteratively. Despite the good performance on some COPs, such action representation with the \ufb01xed destroy degree makes the LNS search in\ufb02exible and thus limits the class of policies that can be learned. Also, the issue of large action space still exists, since the patterns to group variables could be combinatorial. To involve larger action space and in the meanwhile keep the learning tractable, we factorize the combinatorial action space A into elementary actions on each dimension (i.e., variable). Speci\ufb01cally, we denote ai t \u2208{xi, \u2205} as the elementary action for variable xi at step t. It means that xi is either selected for reoptimization, or not selected and thus \ufb01xed as its value in the current solution. With such representation, any variable subset can also be expressed as at = Sn i=1 ai t. Therefore, our task can be converted into learning policies for binary decisions on each variable. This enables exploration of large action space by traversing binary action spaces, the number of which grows linearly with the number of variables. To this end, we factorize the original policy as below: \u03c0(at|st) = Yn i=1\u03c0i(ai t|st), (2) which expresses \u03c0(at|st), the probability of selecting an action, as the product of probabilities of selecting its elements. Since P ai t\u2208{xi,\u2205}\u03c0i(ai t|st) = 1, \u2200i \u2208{1, . . . , n}, it is easy to verify that the sum of probabilities of all actions in A still equals to 1, i.e., P at\u2208A\u03c0(at|st) = 1. Based on this factorization, we can apply RL algorithms to train policies \u03c0i for each variable xi. In the following subsections, we \ufb01rst parametrize the policy \u03c0 by a GNN \u03c0\u03b8, and then train it by a customized actor-critic algorithm. 4.3 Policy parametrization Policy networks in deep RL algorithms are generally designed to map states to probabilities of selecting each action. In our case, the complexity of directly training such policy networks could exponentially increase with the growing number of variables. Based on Equation (2), an ad hoc way to reduce the complexity is training individual policies for each dimension. However, the IP problems in our study comprise a high volume of variables, and it is unmanageable to train and store this many networks. Learning disjoint policies without coordination could also suffer from convergence problem as shown in [35]. Another possible paradigm could be constructing a semi-shared network, in which all dimensions share a base network and derive outputs from separate sub-networks. Following a similar idea, Tavakoli et al. [26] design a network with branches for deep Q-learning algorithms. However, it is not suitable for IP problems, since the number of sub-networks is only prede\ufb01ned for single problem size, without the generalization ability to different-sized instances. To circumvent the above issues, we design a GNN based policy network to share parameters across all dimensions, and also enable generalization to any problem size [36, 37]. To this end, we \ufb01rst describe the state st by a bipartite graph G = (V, C, A), where V = {v1, \u00b7 \u00b7 \u00b7 , vn} denotes variable nodes with features V \u2208Rn\u00d7dv; C = {c1, \u00b7 \u00b7 \u00b7 , cm} denotes constraint nodes with features C \u2208Rm\u00d7dc; A \u2208Rm\u00d7n denotes the adjacency matrix with aji being the weight (or feature) of the edge between nodes cj and vi, which is practically the incidence matrix A in an IP instance. This state representation is similar to the one in [20], which depicts the status in B&B tree search to learn branching policy. We extend its usage outside the solver to re\ufb02ect the solving process in our LNS framework. Then, we parametrize the policy as \u03c0\u03b8(at|st) by a graph convolutional network (GCN), a GNN variant broadly used in various tasks [38, 39, 40]. The architecture of our design is illustrated in Appendix A.1, with the graph convolution layer expressed as below: C(k+1) = C(k) + \u03c3 \u0010 LN \u0010 AV(k)W (k) v \u0011\u0011 , V(k+1) = V(k) + \u03c3 \u0010 LN \u0010 A\u22a4C(k+1)W (k) c \u0011\u0011 , k = 0, . . . , K (3) where W (k) v , W (k) c \u2208Rdh\u00d7dh are trainable weight matrices in the k-th layer; V(k) = [v(k) 1 \u00b7 \u00b7 \u00b7 v(k) n ]\u22a4 and C(k) = [c(k) 1 \u00b7 \u00b7 \u00b7 c(k) m ]\u22a4are node embeddings for variables and constraints respectively in k-th 5 \flayer; LN and \u03c3(\u00b7) denote layer normalization and Tanh activation function respectively. In particular, we linearly project raw features of variables and constraints into the initial node embeddings V(0) and C(0) with dh dimensions (dh = 128), and keep this dimension through all layers. After K iterations of convolution (K = 2), the embeddings for the two clusters of heterogeneous nodes are advanced as {vK i }n i=1 and {cK j }m j=1. We \ufb01nally process the former by a multi-layer perceptron (MLP) with a single-value output activated by Sigmoid. In this way, the output value can represent the probability of a variable being selected, i.e., \u03c0i(ai t|st) = \u03c0\u03b8(ai t|st) = MLP(vK i ), \u2200i \u2208{1, . . . , n}, such that we can conduct Bernoulli sampling accordingly on each variable and attain the subset. In this paper, we structure the MLP by two hidden layers, which have 256 and 128 dimensions respectively and are activated by Tanh. 4.4 Training algorithm Actor-critic is one of the policy gradient methods developed from REINFORCE [41]. It parametrizes state-value or action-value function as a critic network to estimate the expected return. In this paper, we adopt Q-actor-critic with Q\u03c9(s, a) \u2248Q(s, a) = E[Rt|st = s, at = a], where \u03c9 is the parameter set to be learned. In doing so, Q\u03c9 (i.e., the critic) can be updated through bootstrapping and leveraged in the training of the policy network (i.e., the actor). Speci\ufb01cally, the loss functions for the critic and the actor are de\ufb01ned as follows: L(\u03c9) = ED[(\u03b3Q\u03c9(st+1, at+1) + rt \u2212Q\u03c9(st, at))2], (4) L(\u03b8) = ED[Q\u03c9(st, at) log \u03c0\u03b8(at|st)], (5) where the experience replay buffer D contains transitions (st, at, rt, st+1, at+1), which are collected during solving a batch of instances. \u03b3Q\u03c9(st+1, at+1) + rt is the one-step temporal-difference (TD) target to update the critic. The above Q-actor-critic is not directly applicable to the primitive high-dimensional action spaces in our IP problems. To learn the factorized policies we designed in Section 4.2, one way is to customize it by training an actor and critic on each dimension, and updating the parameters via the loss functions: \u02dc L(\u03c9) = ED[ 1 n Xn i=1(\u03b3Q\u03c9(st+1, ai t+1) + rt \u2212Q\u03c9(st, ai t))2], (6) \u02dc L(\u03b8) = ED[ 1 n Xn i=1Q\u03c9(st, ai t) log(\u03c0\u03b8(ai t|st))], (7) where the parameter-sharing Q\u03c9 is used across all dimensions as the actor \u03c0\u03b8, i.e., Q\u03c9(st, ai t) = Qi(st, ai t), \u2200i \u2208{1, . . . , n}. However, our experiments show that the critic trained by bootstrapping on each elementary action delivers inferior performance, which is similar to the \ufb01nding in [26]. Intuitively, this may stem from excessive action-value regressions with one single neural network. To circumvent this issue, we keep the global TD learning in Equation (4) and adjust elementary policies by the Q value of the state-action pair (st, at), such that: \u02dc L(\u03b8) = ED[Q\u03c9(st, at) Xn i=1 log(\u03c0\u03b8(ai t|st))], (8) where the critic Q\u03c9 is structured in the same manner as \u03c0\u03b8, except that: 1) we add a binary value to the raw features of each variable ai t to indicate whether it is selected; 2) we use MLP to process the graph embedding, which aggregates the embeddings of variables by mean-pooling, to output a real value that represents the Q value. Clipping & masking. To enhance exploration, we clip the probabilities of being selected for each variable in a range [\u03f5, 1 \u2212\u03f5], \u03f5 <0.5. It helps avoid always or never traversing some variables with extreme probabilities. We also do not consider empty or universal sets of variables, which lead to unsuitable sub-IPs. Though the chance to select these two sets are low, we mask them by resampling. Details of the training algorithm are given in Appendix A.2. Besides the policy network we designed in Section 4.3, we also adopt this algorithm to train a MLP based semi-shared network similar to the one in [26], which indicates that the fully-shared one (ours) is more suitable for IP problems. More details are given in Appendix A.3. 6 \f5 Experimental results We perform experiments in this section on four NP-hard benchmark problems: Set Covering (SC), Maximal Independent Set (MIS), Combinatorial Auction (CA) and Maximum Cut (MC), which are widely used in existing works. Our code is available.3 Instance generation. We generate SC instances with 1000 columns and 5000 rows following the procedure in [42]. MIS instances are generated following [43], where we use the Erd\u02dd os-R\u00e9nyi random graphs with 1500 nodes and set the af\ufb01nity number to 4. CA instances with 2000 items and 4000 bids are generated according to arbitrary relationships in [44]. MC instances are generated according to Barabasi-Albert random graph models [45], with average degree 4, and we adopt graphs of 500 nodes. For each problem type, we generate 100, 20, 50 instances for training, validation, and testing. In addition, we double and quadruple the number of variables and generate 50 larger and even larger instances respectively for each problem, to verify the generalization performance. We name the instance groups and display their average sizes in Table 1. Table 1: Average sizes of problem instances. Training Generalization Num. of SC MIS CA MC SC2 MIS2 CA2 MC2 SC4 MIS4 CA4 MC4 Variables 1000 1500 4000 2975 2000 3000 8000 5975 4000 6000 16000 11975 Constraints 5000 5939 2674 4950 5000 11932 5357 9950 5000 23917 10699 19950 Features. To represent an instance, we extract static features of variables and constraints, along with the incidence matrix after presolving by the solver at the root node of B&B. In this way, redundant information could be removed and the extracted features could be more clear and compact in re\ufb02ecting the problem structure. For the dynamic features, we record for each variable its value in the current solution and incumbent, as well as its average value in all incumbents to represent the solving status of LNS. Specially, at the step t = 0, the average incumbent values are naturally attained from the initial solution, i.e., the incumbent at the root node. Note that in practice, we concatenate the static and dynamic features for each variable, and attach them (V) to variable nodes in the bipartite graph. The features of constraints (C) and the incidence matrix (A) are attached to the constraint nodes and edges, respectively. Detailed description of the above features are available in Appendix A.4. Hyperparameters. We use the state-of-the-art open source IP solver SCIP (v6.0.1) [46] as the repair operator, which also serves as a major baseline. We run all experiments on an Intel(R) Xeon(R) E5-2698 v4 2.20GHz CPU. For each problem, we train 200 iterations, during each we randomly draw M =10 instances. We set the training step limit T=50, 50, 70, 100 for SC, MIS, CA and MC respectively. The time limit for repair at each step is 2 seconds, unless stated otherwise. We use \u03f5=0.2 for probability clipping. For the Q-actor-critic algorithm, we set the length of the experience replay TM, the number of updating the network U=4 and the batch size B=TM/U. We set the discount factor \u03b3=0.99 for all problems, and use Adam optimizer with learning rate 1 \u00d7 10\u22124. To show the applicability of our method on other solvers, we have also performed experiments where Gurobi [47] is used as the repair solver, which will be discussed in Section 5.3. 5.1 Comparative analysis Baselines. We compare our method with four baselines: \u2022 SCIP with default settings. \u2022 U-LNS: a LNS version which uniformly samples a subset size, then \ufb01lls it by uniformly sampling variables. We compare with it to show that our method can learn useful subset selection policies. \u2022 R-LNS: a LNS version with hand-crafted rule proposed in [21], which randomly groups variables into disjoint equal-sized subsets and reoptimizes them in order. \u2022 FT-LNS: the best-performing LNS version in [21], which applies forward training, an imitation learning algorithm, to mimic the best demonstrations collected from multiple R-LNS runs. 3https://github.com/WXY1427/Learn-LNS-policy 7 \fTable 2: Comparison with SCIP and LNS baselines. SC MIS CA MC Methods Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% SCIP 567.66 \u00b1 8.76 3.62 -681.02 \u00b1 1.14 0.29 -110181 \u00b1 2.03 2.98 -852.57 \u00b1 1.22 4.37 SCIP\u2217 552.82 \u00b1 8.69 0.91 -681.76 \u00b1 1.06 0.18 -111511 \u00b1 1.85 1.85 -861.10 \u00b1 1.26 3.41 SCIP\u2217\u2217 550.68 \u00b1 8.60 0.53 -682.46 \u00b1 1.02 0.07 -112638 \u00b1 1.68 0.82 -863.63 \u00b1 1.32 3.13 U-LNS 568.60 \u00b1 12.17 5.99 -681.38 \u00b1 0.95 0.23 -103717 \u00b1 1.92 8.67 -869.20 \u00b1 1.53 2.50 R-LNS 560.54 \u00b1 8.07 2.38 -682.20 \u00b1 0.93 0.11 -109550 \u00b1 1.62 3.44 -882.18 \u00b1 1.27 1.05 FT-LNS 564.00 \u00b1 8.03 3.02 -681.82 \u00b1 0.93 0.17 -107370 \u00b1 2.03 5.45 -867.05 \u00b1 1.64 2.75 Ours 551.50 \u00b1 8.59 0.68 -682.52 \u00b1 0.98 0.06 -112666 \u00b1 1.72 0.77 -889.61 \u00b1 1.32 0.27 1 \u2217and \u2217\u2217mean the method run with 500s and 1000s. Table 3: Generalization to large instances. SC2 MIS2 CA2 MC2 Methods Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% SCIP 303.18 \u00b1 8.62 6.80 -1323.90 \u00b1 0.81 3.22 -205542 \u00b1 2.87 7.14 -1691.48 \u00b1 1.18 6.22 SCIP\u2217 298.12 \u00b1 8.08 5.04 -1357.04 \u00b1 1.34 0.80 -214654 \u00b1 1.44 3.02 -1706.45 \u00b1 1.26 5.39 SCIP\u2217\u2217 295.70 \u00b1 7.89 4.21 -1361.98 \u00b1 1.06 0.44 -217271 \u00b1 1.93 1.84 -1714.71 \u00b1 1.02 4.93 U-LNS 303.36 \u00b1 8.24 6.90 -1364.66 \u00b1 0.69 0.24 -197453 \u00b1 1.86 10.79 -1769.00 \u00b1 1.03 1.92 R-LNS 300.84 \u00b1 7.74 6.06 -1339.00 \u00b1 0.81 2.12 -204145 \u00b1 1.57 7.77 -1767.09 \u00b1 1.00 2.03 FT-LNS 303.52 \u00b1 7.86 6.98 -1345.58 \u00b1 0.86 1.63 -212264 \u00b1 1.35 4.10 -1700.58 \u00b1 1.64 5.72 Ours 297.90 \u00b1 8.20 4.98 -1367.78 \u00b1 0.68 0.01 -216006 \u00b1 1.15 2.40 -1803.71 \u00b1 0.92 0.00 SC4 MIS4 CA4 MC4 Methods Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% SCIP 179.88 \u00b1 6.35 6.37 -2652.56 \u00b1 0.58 3.05 -372291 \u00b1 1.22 13.44 -3392.02 \u00b1 0.86 5.45 SCIP\u2217 177.44 \u00b1 6.64 4.91 -2652.56 \u00b1 0.58 3.05 -372291 \u00b1 1.22 13.44 -3392.94 \u00b1 0.82 5.43 SCIP\u2217\u2217 175.38 \u00b1 6.98 3.68 -2673.64 \u00b1 1.54 2.28 -372291 \u00b1 1.22 13.44 -3394.42 \u00b1 0.81 5.39 U-LNS 196.60 \u00b1 10.13 16.25 -2653.42 \u00b1 0.63 3.02 -419973 \u00b1 1.11 1.41 -3522.67 \u00b1 0.80 1.81 R-LNS 188.02 \u00b1 7.13 11.24 -2683.30 \u00b1 0.59 1.93 -427478 \u00b1 0.95 0.61 -3521.90 \u00b1 0.82 1.83 FT-LNS 179.40 \u00b1 8.47 6.04 -2684.94 \u00b1 0.81 1.87 -424052 \u00b1 0.96 1.41 -3525.28 \u00b1 0.83 1.74 Ours 176.84 \u00b1 7.42 4.57 -2735.86 \u00b1 0.50 0.00 -428052 \u00b1 0.12 0.49 -3587.72 \u00b1 0.76 0.00 Following [21], we tune the group number of R-LNS (and FT-LNS since it imitates R-LNS) from 2 to 5, and apply the best one to each problem. To train FT-LNS, we collect 10 demonstrations for each instance, and tune the step limit to 20 for SC, MIS, CA and 50 for MC, which perform the best. Same as our method, all LNS baselines also use SCIP as the repair operator with 2s time limit. To compare solution quality, we use the average objective value and standard deviation over the 50 testing instances as metrics. Also, since all problems are too large to be solved optimally, we measure the primal gap [19] to re\ufb02ect the difference between the solution \u02dc x of a method to the best one x\u2217 found by all methods. We compute |\u00b5\u22a4\u02dc x \u2212\u00b5\u22a4x\u2217|/max{|\u00b5\u22a4\u02dc x|, |\u00b5\u22a4x\u2217|} \u00b7 100% for each instance, then average the gaps for all 50 ones. Below we report the results on testing instances of the same sizes as in training. In this paper, we aim to improve an IP solver from externals to enable more ef\ufb01cient search of highquality solutions in a broad range of COPs. To this end, we compare all methods for time-bounded optimization with the same 200s time limit, and further allow SCIP to run for longer time, i.e., 500s and 1000s. The results are gathered in Table 2. As shown, our method signi\ufb01cantly outperforms all baselines on all problems with the same 200s time limit. It is notable that FT-LNS is inferior to R-LNS which yields demos for its imitation learning. The reason might be that FT-LNS only mimics the random demos of short (training) step limits, and hence lacks the ability of generalizing to longer steps. This limitation might hinder its application since in practice, IP problems are often solved in an anytime manner with \ufb02exible time/step limits. In contrast, our method avoids this myopic issue by RL training. In Appendix A.5, we also show that FT-LNS can outperform R-LNS with the same number of LNS steps as in training. Another key observation from Table 2 is that with longer time limits, SCIP is able to \ufb01nd better solutions than the three LNS baselines on SC, MIS and CA. However, our method still surpasses SCIP (500s) on all problems and SCIP (1000s) on MIS, CA and MC. 8 \fTable 4: Generalization to large instances (500s). SC2 MIS2 CA2 MC2 Methods Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% SCIP\u2217\u2217 295.70 \u00b1 7.89 4.48 -1361.98 \u00b1 1.06 0.56 -217271 \u00b1 1.93 1.59 -1714.71 \u00b1 1.02 5.44 U-LNS 302.94 \u00b1 8.15 7.04 -1368.58 \u00b1 0.72 0.05 -200256 \u00b1 2.08 9.28 -1777.98 \u00b1 1.01 1.95 R-LNS 298.24 \u00b1 7.43 5.39 -1362.04 \u00b1 0.71 0.52 -207937 \u00b1 1.44 5.81 -1776.44 \u00b1 1.02 2.04 FT-LNS 303.34 \u00b1 7.97 7.18 -1345.60 \u00b1 0.83 1.76 -213464 \u00b1 1.21 3.30 -1767.81 \u00b1 1.04 2.51 Ours 295.36 \u00b1 7.81 4.36 -1368.68 \u00b1 0.65 0.04 -218920 \u00b1 2.13 0.85 -1813.02 \u00b1 0.91 0.02 SC4 MIS4 CA4 MC4 Methods Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% SCIP\u2217\u2217 175.38 \u00b1 6.99 5.21 -2673.64 \u00b1 1.54 2.41 -372291 \u00b1 1.22 14.98 -3394.42 \u00b1 0.81 6.04 U-LNS 185.62 \u00b1 8.19 11.26 -2737.24 \u00b1 0.54 0.09 -426480 \u00b1 0.93 2.60 -3556.69 \u00b1 0.80 1.55 R-LNS 172.96 \u00b1 6.43 3.73 -2736.60 \u00b1 0.52 0.11 -431786 \u00b1 1.03 1.39 -3554.98 \u00b1 0.80 1.59 FT-LNS 175.20 \u00b1 6.59 5.11 -2685.30 \u00b1 0.81 1.97 -431234 \u00b1 0.91 1.52 -3526.24 \u00b1 0.79 6.10 Ours 172.38 \u00b1 7.14 3.36 -2738.24 \u00b1 0.50 0.04 -437880 \u00b1 0.72 0.00 -3612.52 \u00b1 0.74 0.00 5.2 Generalization analysis Training deep models that perform well on larger problems is a desirable property for solving IPs, since practical problems are often large-scale. Here we evaluate such generalization performance on instances with the average sizes listed in Table 1. We run our method and baselines on these instances with the same time limits as those in the experiments for comparative analysis. For our method and FT-LNS, we directly apply the policies trained in Section 5.1. All results are displayed in Table 3. It is revealed that with the same 200s time limit, while the LNS baselines only outperform SCIP on speci\ufb01c problems, our LNS policies trained on small instances are consistently superior to all baselines, showing a stronger generalization ability. Also, our method delivers much smaller gaps, e.g., at least 38.35% smaller than that of SCIP (200s), which are more prominent than the results in Section 5.1. It indicates that our policies are more ef\ufb01cient in improving SCIP for larger instances solely by generalization. When SCIP runs with 500s, it surpasses all three LNS baselines on SC2, CA2 and SC4, while our method can still deliver better results on all problems. Compared to SCIP with 1000s, our method is inferior on SC2, CA2 and SC4 but apparently better on the remaining 5 instance groups. We further test all methods with 500s time limit except SCIP, which is allowed to run with 1000s. All results are gathered in Table 4. It is revealed that our method still has clear advantages over others on all problems, and consistently outperforms SCIP with mere 1/2 runtime. We \ufb01nd that LNS baselines can outperform SCIP on some problems, especially the three largest ones, i.e., MIS4, CA4 and MC4. It suggests that as the problem size becomes larger, LNS could be more effective to deliver high-quality solutions by solving successive sub-IPs which have much lower complexity than the original problem. On the other hand, our method outperforms all LNS baselines, showing that it is more ef\ufb01cient in improving the solution. In summary, our LNS policies learned on small instances generalize well to larger ones, with a persistent advantage over other methods. 5.3 Experiments with Gurobi Our LNS framework is generally applicable to any IP solver. Here we evaluate its performance by leveraging Gurobi (v9.0.3) [47] as the repair operator. Gurobi is a commercial solver and offers less interfaces to the internal solving process. Thus, we condense the static features of variables to mere objective coef\ufb01cients, and attain the initial solution by running the solver with 2s time limit. During training, we set the step limit T=50 with 1s time limit for Gurobi at each step. For FT-LNS, we use the same 1s time limit, and tune the group number to 2 in all problems for its best performance. The remaining settings are the same as those with SCIP. During testing, we let all methods run with 100s time limit, roughly twice as much as that for training. To save space, we only show results of two instance groups for each problem in Table 5.4 Despite the shorter runtime, we \ufb01nd that Gurobi (100s) generally attains lower objective values than SCIP (200s), showing a better performance to solve IP 4Specially, we observe that MIS is fairly easy for Gurobi (76 out of 100 instances can be solved optimally with average 40s). Thus, we evaluate this problem with less time limit and show the results in Appendix A.6. 9 \fTable 5: Results with Gurobi. The left part shows the results of inference on the testing set in SC, CA and MC; the right part shows the results of generalization to larger instances in SC2, CA2 and MC2. SC CA MC SC2 CA2 MC2 Methods Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Obj.\u00b1Std.% Gap% Gurobi 554.94 \u00b1 8.34 1.15 -111668 \u00b1 1.96 1.10 -863.91 \u00b1 3.77 3.31 302.52 \u00b1 7.73 2.43 -214271 \u00b1 1.52 3.63 -1652.83 \u00b1 3.63 5.81 U-LNS 562.08 \u00b1 8.16 2.46 -110402 \u00b1 1.67 2.21 -862.59 \u00b1 1.75 3.45 301.48 \u00b1 7.62 2.07 -218986 \u00b1 1.42 1.51 -1733.57 \u00b1 1.22 1.21 R-LNS 563.98 \u00b1 8.29 2.81 -110230 \u00b1 1.56 2.36 -860.22 \u00b1 1.96 3.72 302.86 \u00b1 7.41 2.56 -219462 \u00b1 1.16 1.29 -1723.75 \u00b1 1.32 1.77 FT-LNS 564.14 \u00b1 8.37 2.84 -110041 \u00b1 1.56 2.53 -866.22 \u00b1 1.65 3.03 348.50 \u00b1 9.05 17.99 -206189 \u00b1 1.39 7.26 -1726.07 \u00b1 1.19 1.64 Ours 551.88 \u00b1 8.31 0.59 -111787 \u00b1 2.60 1.00 -888.97 \u00b1 1.55 0.50 297.70 \u00b1 7.40 0.80 -222346 \u00b1 1.35 0.00 -1752.98 \u00b1 1.21 0.11 problems. Hence, the LNS baselines lose their advantages over the solver on several problems, e.g., SC, CA and SC2. In contrast, our method outperforms Gurobi across all problems, showing good performance on instances of both training and generalization sizes. Moreover, our method can be well applied to much larger instances and we provide this evaluation in Appendix A.7. 5.4 Testing on MIPLIB The mixed integer programming library (MIPLIB) [48] contains real-world COPs from various domains. Since the instances in MIPLIB are severely diverse in problem types, structures and sizes, it is not a very suitable testing set to directly apply learning based models and thus seldom used in the related works. We evaluate our method (with Gurobi as the repair solver) on this realistic dataset, in the style of active search on each instance [5, 6], and compare it to SCIP and Gurobi. Results show that: 1) with the same 1000s time limit, our method is superior to both solvers on 24/35 instances and comparable to them on 9/35 instances; 2) our method with 1000s time limit outperforms both solvers with 3600s time limit on 13/35 instances; 3) for an open instance, we \ufb01nd a better solution than the best known one. More details are provided in Appendix A.8. 6"
+                }
+            ],
+            "Yining Ma": [
+                {
+                    "url": "http://arxiv.org/abs/2310.18264v1",
+                    "title": "Learning to Search Feasible and Infeasible Regions of Routing Problems with Flexible Neural k-Opt",
+                    "abstract": "In this paper, we present Neural k-Opt (NeuOpt), a novel learning-to-search\n(L2S) solver for routing problems. It learns to perform flexible k-opt\nexchanges based on a tailored action factorization method and a customized\nrecurrent dual-stream decoder. As a pioneering work to circumvent the pure\nfeasibility masking scheme and enable the autonomous exploration of both\nfeasible and infeasible regions, we then propose the Guided Infeasible Region\nExploration (GIRE) scheme, which supplements the NeuOpt policy network with\nfeasibility-related features and leverages reward shaping to steer\nreinforcement learning more effectively. Additionally, we equip NeuOpt with\nDynamic Data Augmentation (D2A) for more diverse searches during inference.\nExtensive experiments on the Traveling Salesman Problem (TSP) and Capacitated\nVehicle Routing Problem (CVRP) demonstrate that our NeuOpt not only\nsignificantly outstrips existing (masking-based) L2S solvers, but also\nshowcases superiority over the learning-to-construct (L2C) and\nlearning-to-predict (L2P) solvers. Notably, we offer fresh perspectives on how\nneural solvers can handle VRP constraints. Our code is available:\nhttps://github.com/yining043/NeuOpt.",
+                    "authors": "Yining Ma, Zhiguang Cao, Yeow Meng Chee",
+                    "published": "2023-10-27",
+                    "updated": "2023-10-27",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Vehicle Routing Problems (VRPs), prevalent in various real-world applications, present NP-hard combinatorial challenges that necessitate efficient search algorithms [1, 2]. In recent years, significant progress has been made in developing deep (reinforcement) learning-based solvers (e.g., [3\u201311]), which automates the tedious algorithm design with minimal human intervention in a data-driven fashion. Impressively, these neural solvers developed over the past five years have closed the gap to or even surpassed some traditional hand-crafted solvers that have evolved for several decades [5, 12]. In general, neural methods for VRPs can be categorized into learning-to-construct (L2C), learning-tosearch (L2S), and learning-to-predict (L2P) solvers, each offering unique advantages while suffering respective drawbacks. L2C solvers (e.g., [4, 13]) are recognised for their fast solution construction but may struggle to escape local optima. L2P solvers excel at predicting crucial information (e.g., edge heatmaps [14, 15]), thereby simplifying the search especially for large-scale instances, but may lack the generality to efficiently handle VRP constraints beyond the Traveling Salesman Problem (TSP). L2S solvers (e.g., [8, 9]) are designed to learn exploration in the search space directly; however, their search efficiency is still limited and lags behind the state-of-the-art L2C and L2P solvers. In this paper, we delve into the limitations of existing L2S solvers, and aim to unleash their full potential. \u2217Zhiguang Cao is the corresponding author. 37th Conference on Neural Information Processing Systems (NeurIPS 2023). arXiv:2310.18264v1  [cs.LG]  27 Oct 2023 \fOne potential issue of L2S solvers lies in their simplistic action space designs. Current L2S solvers that learn to control k-opt exchanges mostly leverage smaller k values (2-opt [9, 16] or 3-opt [17]), partly because their models struggle to efficiently deal with larger k. To address this issue, we introduce Neural k-Opt (NeuOpt), a flexible L2S solver capable of handling k-opt for any k \u22652. Specifically, it employs a tailored action factorization method that simplifies and decomposes a complex k-opt exchange into a sequence of basis moves (S-move, I-move, and E-move) with the number of I-moves determining the k of a specifically executed k-opt action1. Such design allows kopt exchanges to be easily constructed step-by-step, which more importantly, provides the deep model with the flexibility to explicitly and automatically determine an appropriate k. This further enables varying k values to be combined across different search steps, striking a balance between coarsegrained (larger k) and fine-grained (smaller k) searches. Correspondingly, we design a Recurrent Dual-Stream (RDS) decoder to decode such action factorization, which consists of recurrent networks and two complementary decoding streams for contextual modeling and attention computation, thereby capturing the strong correlations and dependencies between removed and added edges. Besides, existing L2S solvers confine the search space to feasible regions based on feasibility masking. By contrast, we introduce a novel Guided Infeasible Region Exploration (GIRE) scheme that promotes the exploration of both feasible and infeasible regions. GIRE enriches the policy network with additional features that indicate constraint violations in the current solution and the exploration behaviour statistics in the search space. It also includes reward shaping to regulate extreme exploration behaviours and incentivize exploration at the boundaries of feasible and infeasible regions. Our GIRE offers four advantages: 1) it circumvents the non-trivial calculation of ground-truth action masks, particularly beneficial for constrained VRPs or broader action space as in NeuOpt, 2) it fosters searches at the more promising feasibility boundaries, similar to traditional solvers [18\u201321], 3) it bridges (possibly isolated) feasible regions, helping escape local optima and discover shortcuts to better solutions (See Figure 2), and 4) it forces explicit awareness of the VRP constraints, facilitating the deep model to understand the problem landscapes. In this paper, we apply GIRE to the Capacitated Vehicle Routing Problem (CVRP), though we note that it is generic to most VRP constraints. Moreover, our NeuOpt leverages a Dynamic Data Augmentation (D2A) method during inference to enhance the search diversity and escape local optima. Our NeuOpt is trained via the reinforcement learning (RL) algorithm tailored in our previous work [12]. Extensive experiments on classic VRP variants (TSP and CVRP) validate our designs and demonstrate the superiority of NeuOpt and GIRE over existing approaches. Our contributions are four-fold: 1) we present NeuOpt, the first L2S solver that is flexible to handle k-opt with any k \u22652 based on a tailored formulation and a customized RDS decoder, 2) we introduce GIRE, the first scheme that extends beyond feasibility masking, enabling exploration of both feasible and infeasible regions in the search space, thereby bringing multiple benefits and offering fresh perspectives on handling VRP constraints, 3) we propose a simple yet effective D2A inference method for L2S solvers, and 4) we unleash the potential of L2S solvers and allow it to surpass L2C, L2P solvers, as well as the strong LKH-3 solver [20] on CVRP. 2 Literature review We categorize recently developed neural methods for solving vehicle routing problems (VRPs) into learning-to-construct (L2C), learning-to-search (L2S), and learning-to-predict (L2P) solvers. L2C solvers. They learn to construct solutions by iteratively adding nodes to the partial solution. The first modern L2C solver is Ptr-Net [22] based on a Recurrent Neural Network (RNN) and supervised learning (extended to RL in [23] and CVRP in [24]). The Graph Neural Networks (GNN) were then leveraged for graph embedding [25] and faster encoding [26]. Later, the Attention Model (AM) was proposed by Kool et al. [3], inspiring many subsequent works (e.g., [4, 27\u201331]), where we highlight Policy Optimization with Multiple Optima (POMO) [4] which significantly improved AM with diverse rollouts and data augmentations. The L2C solvers can produce high-quality solutions within seconds using greedy rollouts; however, they are prone to get trapped in local optima, even when equipped with post-hoc methods (e.g., sampling [3], beam search [28], etc), or, advanced strategies (e.g., invariant representation [13, 32], learning collaborative policies [33], etc). Recently, the Efficient Active Search (EAS) [5] addressed such issues by updating a small subset of pre-trained 1Broadly, in k-opt, added edges may coincide with removed ones. Thus, 2-opt may be viewed as degenerated 8-opt. To avoid ambiguity, unless specified, we refer to k-opt as an exchange introducing k entirely new edges. 2 \fmodel parameters on each test instance, which could be further boosted if coupled with Simulation Guided Beam Search (SGBS) [34], achieving the current state-of-the-art performance for L2C solvers. L2S solvers. They learn to iteratively refine a solution to a new one, featuring a search process. Early attempts, e.g., NeuRewriter [35] and L2I [36], relied heavily on traditional local search algorithms and long run time. The NLNS solver [8] improved upon them by controlling a ruin-and-repair process that destroys parts of the solution using handcrafted operators and then fixes them using a learned deep model. Besides, the crossover exchanges between solutions were also learned in [37]. Recently, several L2S solvers focused on controlling the k-opt heuristic that is more suitable for VRPs [20, 38]. Wu et al. [39] made an early attempt to guide 2-opt, showing superior performance than the L2C solver AM [3]. Ma et al. [9] improved Wu et al. [39] by replacing vanilla attention with Dual-Aspect Collaborative Attention (DAC-Att) and a cyclic positional encoding method. The DAC-Att was then upgraded to Synthesis Attention (Synth-Att) [12] to reduce computational costs. Besides, Costa et al. [16] proposed an RNN-based policy to govern 2-opt, which was extended to 3-opt in [17] with higher efficiency. However, these neural k-opt solvers are limited by a small and fixed k. Besides, although L2S solvers strive to surpass L2C solvers by directly learning to search, they are still inferior to those state-of-the-art L2C solvers (e.g., POMO [4] and EAS [5]) even when given prolonged run time. L2P solvers. They learn to guide the search by predicting critical information. Joshi et al. [14] proposed using GNN models to predict heatmaps that indicate probabilities of the presence of an edge in the optimal solution, which then uses beam search to solve TSP. It was leveraged for larger-scale TSP instances in [6] based on divide-and-conque, heatmap merging, and Monte Carlo Tree Search. In the GLS solver [40], a similar GNN was used to guide the traditional local search heuristics. More recently, the DIFUSCO solver [15] proposed to replace those GNN models with diffusion models [41]. Compared to L2C or L2S solvers, L2P solvers exhibit better scalability for large instances; however, they are mostly limited to supervised learning and TSP only, due to challenges in preparing training data and the ineffectiveness of heatmaps in handling VRP constraints. Though L2P solver DPDP [42] managed to solve CVRP with dynamic programming, it was outstripped by L2C solver EAS[5]. Recently, L2P solvers also explored predicting a latent continuous space for the underlying discrete solution space, where the latent space is then searched using differential evolution in [43] or gradient optimizer in [7]. Still, they can be either time-consuming or ineffective in tackling VRP constraints. Feasibility satisfaction. Most neural solvers handle VRP constraints using the masking scheme that filters out invalid actions (e.g., [12, 44]). However, few works considered better ways of constraint handling. Although the works [45, 46] attempted to use mask prediction loss to enhance constraint awareness, they overlooked the benefits of the temporary constraint violation applied in many traditional solvers [18\u201321]. Lastly, we note that constraint handling in VRPs largely differs from safe RL tasks that focus on fully avoiding risky actions in uncertain environments [47, 48]. 3 Preliminaries and notations VRP notations. VRP aims to minimize the total travel cost (tour length) while serving a group of customers subject to certain constraints. It is defined on a complete directed graph G ={V, E} where xi \u2208V are nodes (customers) and e(xi \u2192xj)\u2208E are possible edges (route) weighted by the Euclidean distance between xi and xj. In the Traveling Salesman Problem (TSP), the solution is a Hamiltonian cycle that visits each node exactly once and returns to the starting one. In the Capacitated Vehicle Routing Problem (CVRP), a depot x0 is added to V, and each customer node xi(i \u22651) is assigned a demand \u03b4i. A CVRP solution consists of multiple sub-tours, each representing a vehicle departing from the depot, serving a subset of customers, and returning to the depot, where each xi(i \u22651) is visited exactly once and the total demand of a sub-tour must not exceed the vehicle capacity \u2206. For instance, \u03c4 = {x0 \u2192x2 \u2192x1 \u2192x0 \u2192x3 \u2192x0} is a CVRP-3 solution with \u03b4=[5, 5, 9] and \u2206=10. Traditional k-opt heuristic. The k-opt heuristic iteratively refines a given solution by exchanging k existing edges with k (entirely) new ones. The Lin-Kernighan (LK) algorithm [49] narrowed the search with several criteria, where we underscore the sequential exchange criterion. It requires: for each i = 1, . . . , k, the removed edge eout i and added edge ein i must share an endpoint, and so must ein i and eout i+1. This allows a simplified sequential search process, alternating between removing and adding edges. Moreover, the LK algorithm considers scheduling varying k values in a repeated ascending order, so as to escape local optima by varying search neighbourhoods. Inspired by them, our paper proposes a powerful L2S solver that performs flexible k-opt exchanges with automatically chosen k 3 \f\ud835\udc89\ud835\udfd2 Anchor node Node to select GRU \ud835\udc92\ud835\udf41 \ud835\udfce \ud835\udc92\ud835\udf41 \ud835\udfcf \ud835\udc90\ud835\udf41 \ud835\udfcf GRU \ud835\udc92\ud835\udf40 \ud835\udfce \ud835\udc92\ud835\udf40 \ud835\udfcf \ud835\udc89\ud835\udfcf \ud835\udc89\ud835\udfd0 \ud835\udc89\ud835\udfd1 \ud835\udc89\ud835\udfd2 \ud835\udc89\ud835\udfd3 \ud835\udc89\ud835\udfd4 \ud835\udc89\ud835\udfd5 \ud835\udc89\ud835\udfd6 \ud835\udc89\ud835\udfd7 \ud835\udc90\ud835\udf40 \ud835\udfcf \ud835\udc99\ud835\udfcf \ud835\udc99\ud835\udfd0 \ud835\udc99\ud835\udfd1 \ud835\udc99\ud835\udfd2 \ud835\udc99\ud835\udfd3 \ud835\udc99\ud835\udfd4 \ud835\udc99\ud835\udfd5 \ud835\udc99\ud835\udfd6 \ud835\udc99\ud835\udfd7 [(\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd1)] Remove Add   [ ] Input/Output Action: {\ud835\udc7a\ud835\udc99\ud835\udfd0} GRU \ud835\udc92\ud835\udf41 \ud835\udfd0 \ud835\udc90\ud835\udf41 \ud835\udfd0= \ud835\udc89\ud835\udfd0 GRU \ud835\udc92\ud835\udf40 \ud835\udfd0 \ud835\udc89\ud835\udfcf \ud835\udc89\ud835\udfd1 \ud835\udc89\ud835\udfd3 \ud835\udc89\ud835\udfd4 \ud835\udc89\ud835\udfd5 \ud835\udc89\ud835\udfd6 \ud835\udc89\ud835\udfd7 \ud835\udc90\ud835\udf40 \ud835\udfd0= \ud835\udc89\ud835\udfd0 \ud835\udc99\ud835\udfcf \ud835\udc99\ud835\udfd0 \ud835\udc99\ud835\udfd1 \ud835\udc99\ud835\udfd2 \ud835\udc99\ud835\udfd3 \ud835\udc99\ud835\udfd4 \ud835\udc99\ud835\udfd5 \ud835\udc99\ud835\udfd6 \ud835\udc99\ud835\udfd7 (\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd1, (\ud835\udc99\ud835\udfd2\u2192\ud835\udc99\ud835\udfd3)] Remove Add   [(\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd2)] GRU \ud835\udc92\ud835\udf41 \ud835\udfd1 GRU \ud835\udc92\ud835\udf40 \ud835\udfd1 \ud835\udc89\ud835\udfcf \ud835\udc89\ud835\udfd0 \ud835\udc89\ud835\udfd2 \ud835\udc89\ud835\udfd3 \ud835\udc89\ud835\udfd4 \ud835\udc89\ud835\udfd5 \ud835\udc89\ud835\udfd6 \ud835\udc89\ud835\udfd7 \ud835\udc99\ud835\udfcf \ud835\udc99\ud835\udfd0 \ud835\udc99\ud835\udfd1 \ud835\udc99\ud835\udfd2 \ud835\udc99\ud835\udfd3 \ud835\udc99\ud835\udfd5 \ud835\udc99\ud835\udfd6 \ud835\udc99\ud835\udfd7 \ud835\udc99\ud835\udfd4 (\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd1, (\ud835\udc99\ud835\udfd2\u2192\ud835\udc99\ud835\udfd3), (\ud835\udc99\ud835\udfd5\u2192\ud835\udc99\ud835\udfd6)] Remove Add   (\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd2, (\ud835\udc99\ud835\udfd1\u2192\ud835\udc99\ud835\udfd5)] GRU \ud835\udc92\ud835\udf41 \ud835\udfd2 GRU \ud835\udc92\ud835\udf40 \ud835\udfd2 \ud835\udc89\ud835\udfcf \ud835\udc89\ud835\udfd0 \ud835\udc89\ud835\udfd1 \ud835\udc89\ud835\udfd2 \ud835\udc89\ud835\udfd4 \ud835\udc89\ud835\udfd5 \ud835\udc89\ud835\udfd6 \ud835\udc89\ud835\udfd7 \ud835\udc99\ud835\udfcf \ud835\udc99\ud835\udfd0 \ud835\udc99\ud835\udfd1 \ud835\udc99\ud835\udfd2 \ud835\udc99\ud835\udfd3 \ud835\udc99\ud835\udfd6 \ud835\udc99\ud835\udfd7 \ud835\udc99\ud835\udfd5 \ud835\udc99\ud835\udfd4 (\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd1, \ud835\udc99\ud835\udfd2\u2192\ud835\udc99\ud835\udfd3, \ud835\udc99\ud835\udfd5\u2192\ud835\udc99\ud835\udfd6] Remove Add   (\ud835\udc99\ud835\udfd0\u2192\ud835\udc99\ud835\udfd2, (\ud835\udc99\ud835\udfd1\u2192\ud835\udc99\ud835\udfd5), (\ud835\udc99\ud835\udfd3\u2192\ud835\udc99\ud835\udfd6)] Action:{\ud835\udc7a\ud835\udc99\ud835\udfd0, \ud835\udc70\ud835\udc99\ud835\udfd2, \ud835\udc70\ud835\udc99\ud835\udfd5, \ud835\udc6c\ud835\udc99\ud835\udfd6} Last selected node Current source node Action: {\ud835\udc7a\ud835\udc99\ud835\udfd0, \ud835\udc70\ud835\udc99\ud835\udfd2} Removed edge Endpoint \ud835\udf1e[2, 3]=1 \ud835\udf1e[\ud835\udc82, \ud835\udc96] The rank of \ud835\udc65\ud835\udc62 w.r.t anchor \ud835\udc65\ud835\udc4e Move stream \ud835\udf07 Attention Direction change \ud835\udf1e[2, 5]=3 \ud835\udc89\ud835\udfd0 \ud835\udc89\ud835\udfd1 \ud835\udf1e[2, 5]=3 \ud835\udf1e[2, 8]=6 \ud835\udc89\ud835\udfd3 Edge stream \ud835\udf06 Reference Ghost mark Masked attention score Attention score Added edge Decoder \ud835\udf3f= 1 Action:{\ud835\udc7a\ud835\udc99\ud835\udfd0, \ud835\udc70\ud835\udc99\ud835\udfd2, \ud835\udc70\ud835\udc99\ud835\udfd5} Decoder \ud835\udf3f= 2 Decoder \ud835\udf3f= 3 Decoder \ud835\udf3f= 4 \ud835\udc90\ud835\udf40 \ud835\udfd1= \ud835\udc89\ud835\udfd1 \ud835\udc90\ud835\udf41 \ud835\udfd1= \ud835\udc89\ud835\udfd2 \ud835\udc90\ud835\udf40 \ud835\udfd2= \ud835\udc89\ud835\udfd3 \ud835\udc90\ud835\udf41 \ud835\udfd2= \ud835\udc89\ud835\udfd5 Original solution \ud835\udf49 New solution \ud835\udf49\u2032 Figure 1: Illustration of using our RDS decoder to determine a 3-opt exchange on TSP-9 given K = 4 steps (with the E-move chosen at the final decoding step). The upper portion depicts a visual representation of how the dual-stream attentions (move stream \u00b5 and edge stream \u03bb) are computed. The lower portion demonstrates how the inferred basis moves lead to the modification of the current solution. At step \u03ba, RDS computes dual-stream attention from representations of historical decisions q\u03ba \u00b5, q\u03ba \u03bb to node embeddings hi, thereby deciding a basis move \u03a6\u03ba(x\u03ba) by selecting x\u03ba. Ghost marks indicate the same location of a cyclic solution when viewed in a flat perspective as in this figure. based on our action factorization method and the customized decoder. Recently, the LK algorithm has been implemented by Helsgaun [50] in the open-source LKH solver, with additional non-sequential exchanges and an edge candidate set. It was then upgraded to LKH-2 [51], leveraging more general k-opt exchanges, divide-and-conquer strategies, etc. The latest release, LKH-3 [20], further tackled constrained VRPs by penalizing constraint violations, making it a generic and powerful solver that serves as a benchmark for neural methods. Finally, we note that k-opt has been a foundation for various solvers, including the state-of-the-art Hybrid Genetic Search (HGS) solver [21] for CVRP. 4 Neural k-opt (NeuOpt) Designing an effective neural k-opt solver necessitates addressing challenges potentially overlooked in prior works. Firstly, the solver should be generic for any given k\u22652, using a unified formulation and architecture. Secondly, it should coherently parameterize the complex action space while accounting for the strong correlations and dependencies between removed and added edges. Finally, it should dynamically adjust k to balance coarse-grained (larger k) and fine-grained (smaller k) search steps. In light of them, we introduce Neural k-Opt (NeuOpt) in this section. We first present our action factorization method for flexible k-opt exchanges, and then demonstrate our decoder to parameterize such actions, followed by the dynamic data augmentation method for inference. Note that this section focuses on TSP only, and we extend our NeuOpt to handle other VRP constraints in Section 5. 4.1 Formulations We introduce a new factorization method that constructs a k-opt exchange using a combination of three basis moves, namely the starting move, the intermediate move, and the ending move. Concretely, the sequential k-opt can be simplified as performing one S-move, several (possibly none) I-moves, and finally one E-move, where the choice of the k corresponds to determining the number of I-moves. S-move. The starting move removes an edge eout(xa \u2192xb), converting a TSP solution \u03c4 (a Hamiltonian cycle) into an open Hamiltonian path with two endpoints xa and xb. It is executed only at the beginning of action construction. We denote S-move as S(xa) since xb can be uniquely determined if xa is specified. We term the source node xa as anchor node to compute the node rank. Definition 1 (Node rank w.r.t. anchor node) Given an instance G =(V, E) and a solution \u03c4, let xa be the anchor node, as specified in an S-move. The node rank of xu (xu \u2208V) w.r.t. xa, denoted by \u0393[xa, xu] or \u0393[a, u], is defined as the minimal number of edges in \u03c4 needed to reach xu from xa. I-move. The intermediate move adds a new edge, removes an existing edge, and fixes edge directions, transforming the open Hamiltonian path into a new one. Let xi, xj be the endpoints of the Hamiltonian 4 \fpath before the I-move (with \u0393[a, i]<\u0393[a, j]), and let ein(xu \u2192xv) be the introduced new edge. To avoid conflicts between consecutive I-moves, we impose sequential conditions on ein: (1) its source node xu must be the endpoint of the current Hamiltonian path with a lower node rank, i.e., xu =xi, and (2) the node rank of its target node xv, i.e., \u0393[a, v], must be higher than the node ranks of the two current endpoints xi, xj, i.e., \u0393[a, i] < \u0393[a, j] < \u0393[a, v]. The removal of edge eout(xv \u2192xw) followed when xv is chosen, and directions of the edges between xj and xv are reversed to yield a valid Hamiltonian path. Since an I-move can be uniquely determined by xv, we denote it as I(xv). E-move. The ending move adds a new edge connecting the two endpoints of the current Hamiltonian path, converting it into a Hamiltonian cycle, i.e., a new TSP solution \u03c4 \u2032. It is executed only at the end of action construction. Since it is uniquely determined without specifying any node, we denote it as E(xnull). Note that if we relax the condition (2) of I-move to\u0393[a, j]\u2264\u0393[a, v], an E-move can be treated as a general I-move that selects xv =xj, denoted as I\u2032(xj) or E(xj). MDP formulations. The examples of using the above basis moves to factorize 1-opt (void action), 2-opt, 3-opt, and 4-opt are depicted in Appendix A. Next, we present the Markov Decision Process (MDP) formulation M = (S, A, T , R, \u03b3 < 1) for our NeuOpt. At step t, the state st = \b G, \u03c4t, \u03c4 bsf t \t describes the current instance G, the current solution \u03c4t, and the best-so-far solution \u03c4 bsf t found before step t. Given a maximum number of allowed basis moves K(K \u22652), an action consists of K basis moves \u03a6\u03ba(x\u03ba), i.e., at ={\u03a6\u03ba(x\u03ba), \u03ba=1, . . . , K}, where the first move \u03a61(x1) = S(x1) is an S-move, and the rest is either an I-move or an E-move (in the form of general I-move). Note that 1) we permit the I-move as the last move, adding an E-move during state transition if so, and 2) to ensure the action length is always K, we include null actions if E-move early terminates the action. Our state transition rule T is deterministic and it updates st to st+1 based on the above action factorization method. We use reward rt = f(\u03c4 bsf t ) \u2212min \u0002 f(\u03c4t+1), f(\u03c4 bsf t ) \u0003 following [9, 12]. 4.2 Recurrent Dual-Stream (RDS) decoder Our NeuOpt adopts an encoder-decoder-styled policy network. We use the encoder from our previous work [12] but upgrade the linear projection to an MLP for embedding generation (details are presented in Appendix B). Here, we introduce the proposed recurrent dual-stream (RDS) decoder to effectively parameterize the aforementioned k-opt action factorization (see Figure 1 for a concrete example). GRUs for action factorization. An action a={\u03a6\u03ba(x\u03ba), \u03ba=1, . . . , K} is constructed sequentially by K steps, where each decoding step \u03ba specifies a basis move type \u03a6\u03ba and a node x\u03ba to instantiate the move. Such decoding can be further simplified to a node selection process, with move types inferred based on: (1) for \u03ba = 1, it is an S-move; (2) for \u03ba > 1, it is an I-move if \u0393[a, j\u03ba] < \u0393[a, \u03ba] (assuming xi\u03ba, xj\u03ba are the Hamiltonian path endpoints before the \u03ba-th move with \u0393[a, i\u03ba]<\u0393[a, j\u03ba]), otherwise, if x\u03ba =xj\u03ba, it is an E-move that early stops the decoding. Formally, we use factorization: \u03c0\u03b8(a|s) = P\u03b8(\u03a61(x1), \u03a62(x2), . . . , \u03a6K(xK)|s) = K Y \u03ba=1 P \u03ba \u03b8 (\u03a6\u03ba|\u03a61, . . . , \u03a6\u03ba\u22121, s) (1) where P \u03ba \u03b8 is a categorical distribution over N nodes for node selection. Our decoder leverages the Gated Recurrent Units (GRUs) [52] to help parameterize the conditional probabilities P \u03ba \u03b8 . Given node embeddings h\u2208RN\u00d7d from the encoders (hi is a d-dimensional vector), the decoder first computes hidden representations q\u03ba to model the historical move decisions {\u03a61, . . . , \u03a6k\u22121} (the conditions of P \u03ba \u03b8 ) using Eq. (2), where q\u03ba (hidden state of GRU) is derived from q\u03ba\u22121, based on an input o\u03ba. For better contextual modeling, we consider two streams, \u00b5 and \u03bb, which differ from each other by learning independent parameters and taking different inputs o\u03ba at each \u03ba. The q0 \u00b5 = q0 \u03bb = 1 N PN i=1 hi are initialized by the mean pooling of node embeddings (i.e., a graph embedding). q\u03ba \u00b5 = GRU \u0000o\u03ba \u00b5, q\u03ba\u22121 \u00b5 \u0001 , q\u03ba \u03bb = GRU \u0000o\u03ba \u03bb, q\u03ba\u22121 \u03bb \u0001 (2) Dual-stream contextual modeling. We employ a move stream \u00b5 and an edge stream \u03bb during contextual modeling and subsequent attention computation. On the one hand, to provide a relatively overall view, the move stream \u00b5 considers modeling historical move decisions by taking the node embedding of the last selected node x\u03ba\u22121, which is a representation of the past selected move \u03a6\u03ba\u22121(x\u03ba\u22121), as its GRU input, i.e., o\u03ba \u00b5 =h\u03ba\u22121. On the other hand, to offer a relatively detailed view, the edge stream \u03bb focuses on edge proposals in each step \u03ba by taking the node embedding of xi\u03ba, which is stipulated to be the source node of the edge to be introduced in step \u03ba, as its GRU input, 5 \fi.e., o\u03ba \u03bb =hi\u03ba. We intend the two streams to complement each other, coherently capturing complex correlations and dependencies among historical decisions. For the initialization, we use learnable parameters o1 \u00b5 = \u02c6 o, o1 \u03bb = \u02dc o. Based on the attention between q\u00b5, q\u03bb and node embeddings h, the node selection scores suggested by both streams can then be calculated: \u00b5\u03ba = Tanh \u0010 (q\u03ba \u00b5W Query \u00b5 + hW Key \u00b5 ) + (q\u03ba \u00b5W Query\u2032 \u00b5 ) \u2299(hW Key\u2032 \u00b5 ) \u0011 W O \u00b5 , \u03bb\u03ba = Tanh \u0010 (q\u03ba \u03bbW Query \u03bb + hW Key \u03bb ) + (q\u03ba \u03bbW Query\u2032 \u03bb ) \u2299(hW Key\u2032 \u03bb ) \u0011 W O \u03bb , (3) where all matrices W \u2208Rd\u00d7d are trainable parameters and we consider both summation and Hadamard products. Afterwards, the final categorical distribution P \u03ba \u03b8 is induced by: Softmax(C \u00b7 tanh(\u00b5\u03ba+\u03bb\u03ba)) where C =6 and invalid choices that do not satisfy \u0393[a, j\u03ba]\u2264\u0393[a, \u03ba] are masked as \u2212\u221e(an E-move is enforced if all nodes are masked, which happens when the highest-ranked node was chosen in the previous I-move). A node x\u03ba is then sampled from P \u03ba \u03b8 , after which its basis move type \u03a6\u03ba can be inferred. The decoding process continues until it early terminates or reaches the maximum limit K. 4.3 Inference with the dynamic data augmentation (D2A) Our NeuOpt is trained using the RL algorithm tailored in our previous work [12]. During inference, our previous work [12] employed a data augmentation method that applies optimal-solution-invariant transformations (e.g., flipping coordinates) on an instance, G, so as to generate a set of instances {G1, G2, ..., } for the trained model to solve. However, it was performed only once at the beginning of the inference. In this paper, we propose Dynamic Data Augmentation (D2A), an enhancement that generates new augmented instances each time the solver fails to find a better solution (i.e., getting trapped in local optima) within a consecutive maximum of TD2A steps. This allows for more diverse searches. Further details regarding training and inference algorithms are provided in Appendix C. 5 Guided infeasible region exploration (GIRE) Global Optima Trajectory (our GIRE scheme) Trajectory (masking scheme)  Local Optima Feasible (\ud835\udcd5) \ud835\udf50-Feasible (\ud835\udf50\u2212\ud835\udcd5) Infeasible (\ud835\udce4) \ud835\udcd5 \ud835\udce4 \ud835\udf50\u2212\ud835\udcd5 Figure 2: A search space example. As demonstrated in Figure 2, the feasible region is usually fragmented with isolated islands. Masking in neural solvers typically restricts the search to feasible regions, which may lead to inefficient search trajectories or failure to find global optima. Our GIRE explores both feasible and infeasible regions, fostering shortcut discovery, more promising boundary searches, and the identification of possibly isolated regions. We now illustrate our GIRE by handling the capacity constraint in CVRP. Definition 2 (Search space of CVRP): We define the feasible search space F as the set of solutions that satisfy both TSP and the (CVRP-specific) capacity constraint; the infeasible search space U as the set of solutions that satisfy the TSP constraint but not the capacity constraint; and the \u03f5-feasible search space \u03f5-F \u2286U as the set of solutions with a capacity violation percentage not exceeding \u03f5. Feature supplement. GIRE suggests supplementing Violation Indicator (VI) features and Exploration Statistics (ES) features into the policy network to identify specific constraint violations and understand its ongoing exploration behaviour. The VI features flag the specific infeasible portions of the current solution. For CVRP, we use two binary variables indicating if the cumulative demands exceed the capacity before or after visiting a particular node, respectively, which are treated as node features during the node embedding generation. Such an idea could work with most VRP constraints, e.g., indicating the nodes that violate their time windows (TW) for CVRP-TW. The ES features provide the network with ongoing exploration behaviour statistics to guide future exploration behaviour. We define Ht := {(\u03c4t\u2032 \u2192\u03c4t\u2032+ 1)}|t\u22121 t\u2032=t\u2212This as the collection of the most recent This steps of solution transition record (if they exist). The ES features Jt consist of estimated feasibility transition probabilities derived from Ht, including P(\u03c4 \u2208F, \u03c4 \u2032 \u2208U), P(\u03c4 \u2208U, \u03c4 \u2032 \u2208F), P(\u03c4 \u2208F, \u03c4 \u2032 \u2208F), P(\u03c4 \u2208U, \u03c4 \u2032 \u2208U), P(\u03c4 \u2032 \u2208F|\u03c4 \u2208U), P(\u03c4 \u2032 \u2208U|\u03c4 \u2208F), P(\u03c4 \u2032 \u2208F|\u03c4 \u2208F), and P(\u03c4 \u2032 \u2208U|\u03c4 \u2208U), along with a binary indicator w.r.t the feasibility of the current solution \u03c4t. In order to make the policy network decisions dependent on these ES features, we introduce two hypernetworks [53], namely, MLP\u00b5 and MLP\u03bb, which take ES features Jt as inputs and generate the parameters of the last decoder layer, i.e., W O \u00b5 , W O \u03bb of Eq. (3), respectively. We adopt the structure (9 \u00d7 8 \u00d7 d), and share the first hidden layer between W O \u00b5 and W O \u03bb to reduce computational costs. 6 \f0.00 0.25 0.50 0.75 1.00 Probability 0.0 0.5 1.0 Entropy measure Figure 3: Plot of H[P]. Reward shaping. Moreover, GIRE employs reward shaping: rGIRE t = rt + \u03b1 \u00b7 rreg t + \u03b2 \u00b7 rbonus t (4) to guide the reinforcement learning, where rt is the original reward, rreg t regulates extreme exploration behaviours; rbonus t encourages exploration in the \u03f5-feasible regions; and \u03b1 and \u03b2 are reward shaping weights (we use 0.05 for both). Here, the regulation rreg t is determined by an entropy measure H[P] of the estimated conditional transforming probabilities Pt(U|U)= P(\u03c4 \u2032 \u2208U|\u03c4 \u2208U) and Pt(F|F)=P(\u03c4 \u2032 \u2208F|\u03c4 \u2208F): rreg t = \u2212E[rt] \u00d7 [H[Pt(U|U]) + H[Pt(F|F)]] , H[P] = Clip {1 \u2212c1 log2 [c2\u03c0eP(1 \u2212P)] , 0, 1} , c1 = 0.5, c2 = 2.5, (5) where expectation E[rt], that suggests the magnitude of rreg t , is estimated during training; the entropy measure H[P], as shown in Figure 3, imposes larger penalties when P is either too high or too low (indicating extreme exploration behaviour). The bonus rbonus t utilizes a similar reward function as the regular reward rt; however, it only considers an infeasible but \u03f5-feasible solution as a potential new best-so-far solution. More illustrations and discussions on GIRE designs are detailed in Appendix D. 6 Experiments We conduct experiments on TSP and CVRP, with sizes N =20, 50, 100 following the conventions [9, 13, 37]. Training and test instances are uniformly generated following [3]. For NeuOpt, we use K =4; the initial solutions are sequentially constructed in a random fashion for both training and inference. Results were collected using a machine equipped with NVIDIA 2080TI GPU cards and an Intel E5-2680 CPU at 2.40GHz. More hyper-parameter details, discussions, and additional results are available in Appendix E. Our PyTorch code and pre-trained models are publicly available2. 6.1 Comparison studies Setup. In Table 1, we benchmark our NeuOpt (TSP) and NeuOpt-GIRE (CVRP) against a variety of neural solvers, namely, 1) L2P solvers: GCN+BS [14] (TSP only), Att-GCN+MCTS [6] (TSP only), GNN+GLS [40] (TSP only), CVAE-Opt-DE [43], DPDP [42] (state-of-the-art), DIMES [7] (TSP only), DIFUSCO [15] (TSP only, state-of-the-art); 2) L2C solvers: AM+LCP [33], POMO [4], Pointerformer [32] (TSP only), Sym-NCO [13], POMO+EAS [5], POMO+EAS+SGBS [34] (state-ofthe-art); and 3) L2S solvers: Costa et al. [16] (TSP only), Sui et al. [17] (TSP only), Wu et al. [39], NLNS [8] (CVRP only), NCE [37] (CVRP only), DACT [9] (state-of-the-art). To ensure fairness, we test their publicly available pre-trained models on our hardware and test datasets. Those marked with \u2021 are sourced from their original papers due to difficulties in reproducing, among which we find potential issues marked with #. More implementation details are listed in Appendix E. Following the conventions [5, 9, 34], we report the metrics of objective values and optimality gaps averaged on a test dataset with 10k instances, where the total run time is measured under the premise of using one GPU for neural methods and one CPU for traditional ones. The gaps are computed w.r.t. the exact solver Concorde [54] for TSP and the state-of-the-art traditional solver HGS [21] for CVRP. We also include the LKH [20, 51] as baselines. However, we note that it is hard to be absolutely fair when comparing the run time between those CPU-based traditional solvers and GPU-based neural solvers. The baselines are grouped, where the last group comprises variations of our NeuOpt, differentiated by the number of augments (marked as \u2018D2A=\u2019) and the number of inference steps (marked as \u2018T=\u2019). TSP results. Compared to L2P solvers, NeuOpt (D2A=1, T=1k) surpasses GCN+BS, CVAE-OptDE, and GNN+GLS in all problem sizes with shorter run time. With increased T, NeuOpt continues reducing the gaps, and outshines the state-of-the-art DIFUSCO solver with less time at T=10k steps. The NeuOpt (D2A=5, T=1k), with more augmentations, shows lower gaps than NeuOpt (D2A=1, T=5k) in the same solution search count, where it achieves the lowest gap of 0.00% at (D2A=5, T=5k) on all sizes. Despite the longer run time compared to DPDP and Att-GCN+MCTS, their high efficiency is limited to TSP, and our NeuOpt could be potentially boosted by leveraging heatmaps 2https://github.com/yining043/NeuOpt 7 \fTable 1: Performance comparison of NeuOpt with various baselines on TSP and CVRP benchmarks. Method Model Post (PerN =20 N =50 N =100 Type Ins.) Proc. Obj.\u2193 Gap\u2193 Time\u2193 Obj.\u2193 Gap\u2193 Time\u2193 Obj.\u2193 Gap\u2193 Time\u2193 TSP Concorde [54] Exact 3.827 2m 5.696 9m 7.765 43m LKH-2 [51] H 3.827 0.00% 6m 5.696 0.00% 1.3h 7.765 0.00% 5.7h GCN+BS [14]# L2P/SL BS+H 3.827 0.00% 15m 5.698 0.04% 23m 7.869 1.35% 46m Att-GCN+MCTS [6]\u2021,# L2P/SL MCTS (\u22483.830) (\u22480.00%) \u22482m (\u22485.691) (\u22480.01%) \u22488m (\u22487.764) (\u22480.04%) \u224815m GNN+GLS [40] (relocate+2-opt)\u2021 L2P/SL GLS \u22480.00% \u22482.8h \u22480.00% \u22482.8h \u22480.58% \u22482.8h CVAE-Opt-DE [43]\u2021 L2P/UL DE \u22480.00% \u22481.2d \u22480.02% \u22482.5d \u22480.34% \u22481.8d DPDP [42] (100k) L2P/SL DP 7.765 0.00% 1.9h DIMES [7] (T=10)\u2021,# L2P/RL AS+M+M (\u22487.762) (\u22480.01%) DIFUSCO [15] (T=50, S=16)# L2P/SL 2-opt 5.696 0.01% 5.8h 7.766 0.02% 21.7h AM+LCP\u22c6[33] ({1280, 45}) L2C/RL 3.828 0.01% 2.1h 5.699 0.05% 4.9h 7.811 0.60% 10.9h Pointerformer [32] (A=8, T=200) L2C/RL 3.827 0.00% 13m 5.697 0.02% 1.1h 7.773 0.11% 5.6h Sym-NCO [13] (A=8, T=200) L2C/RL 7.771 0.08% 5.6h POMO [4] (A=8, T=200) L2C/RL 3.827 0.00% 13m 5.696 0.00% 1.1h 7.770 0.07% 5.6h POMO+EAS [5] (A=8, T=200) L2C/RL AS 3.827 0.00% 24m 5.696 0.00% 2h 7.769 0.05% 10.9h POMO+EAS+SGBS [34] (short) L2C/RL AS+BS 7.767 0.04% 6.5h POMO+EAS+SGBS [34] (long) L2C/RL AS+BS 7.767 0.03% 1.1d Costa et al. [16] (2-opt, T=2k) L2S/RL 3.827 0.00% 31m 5.703 0.12% 40m 7.824 0.77% 1.1h Sui et al. [17] (3-opt, T=2k)\u2021 L2S/RL \u22483.84 \u22480.00% \u224832m \u22485.70 \u22480.08% \u224848m \u22487.82 \u22480.74% \u22481.3h Wu et al. [39] (2-opt, T=5k) L2S/RL 5.709 0.23% 1.3h 7.884 1.54% 2h DACT [9] (2-opt, A=4, T=10k) L2S/RL 3.827 0.00% 1.5h 5.696 0.00% 4.1h 7.772 0.10% 13.5h NeuOpt (D2A=1, T=1k) L2S/RL 3.827 0.00% 2m 5.697 0.02% 6m 7.790 0.33% 17m NeuOpt (D2A=1, T=5k) L2S/RL 3.827 0.00% 12m 5.696 0.00% 32m 7.768 0.05% 1.4h NeuOpt (D2A=1, T=10k) L2S/RL 3.827 0.00% 23m 5.696 0.00% 1.1h 7.766 0.02% 2.8h NeuOpt (D2A=5, T=1k) L2S/RL 3.827 0.00% 12m 5.696 0.00% 32m 7.767 0.04% 1.4h NeuOpt (D2A=5, T=3k) L2S/RL 3.827 0.00% 35m 5.696 0.00% 1.6h 7.765 0.01% 4.2h NeuOpt (D2A=5, T=5k) L2S/RL 3.827 0.00% 1h 5.696 0.00% 2.7h 7.765 0.00% 7h CVRP HGS [21] H 6.130 10.7h 10.366 1.2d 15.563 2.5d LKH-3 [20] H 6.135 0.08% 17.9h 10.375 0.09% 2.8d 15.647 0.54% 5.7d CVAE-Opt-DE [43]\u2021 L2P/UL DE \u22486.14 \u22482.4d \u224810.40 \u22484.7d \u224815.75 \u224811d DPDP [42] (1000k) L2P/SL DP 15.627 0.41% 1.2d AM+LCP [33] ({2560, 1})\u2021 L2C/RL \u22486.15 \u22480.33% \u224823m \u224810.52 \u22481.48% \u224852m \u224816.00 \u22482.81% \u22482.1h Sym-NCO [13] (A=8, T=200) L2C/RL 15.702 0.89% 7.2h POMO [4] (A=8, T=200) L2C/RL 6.136 0.09% 11m 10.397 0.30% 1.4h 15.672 0.70% 7.2h POMO+EAS [5] (A=8, T=200) L2C/RL AS 6.132 0.04% 38m 10.379 0.13% 3.1h 15.610 0.30% 16h POMO+EAS+SGBS [34] (short) L2C/RL AS+BS 15.587 0.15% 1d POMO+EAS+SGBS [34] (long) L2C/RL AS+BS 15.579 0.10% 4.1d NLNS [8] (Ruin-Repair, T=5k) L2S/RL 6.175 0.73% 48m 10.506 1.35% 1.4h 15.915 2.26% 2.4h NCE [37] (CROSS exchange)\u2021 L2S/SL \u22486.13 \u22480.00% \u224811h \u224810.41 \u22480.42% \u22482.3d \u224815.81 \u22481.59% \u224810.4d Wu et al. [39] (2-opt, T=5k) L2S/RL 10.544 1.72% 4.2h 16.165 3.87% 5h DACT [9] (2-opt, A=6, T=10k) L2S/RL 6.130 0.01% 4h 10.383 0.16% 16h 15.736 1.11% 1.7d NeuOpt-GIRE (D2A=1, T=1k) L2S/RL 6.132 0.03% 4m 10.430 0.61% 12m 15.865 1.94% 28m NeuOpt-GIRE (D2A=1, T=5k) L2S/RL 6.130 0.00% 20m 10.382 0.16% 59m 15.698 0.87% 2.3h NeuOpt-GIRE (D2A=1, T=10k) L2S/RL 6.130 0.00% 41m 10.375 0.08% 2h 15.656 0.60% 4.6h NeuOpt-GIRE (D2A=5, T=6k) L2S/RL 6.130 0.00% 2.1h 10.369 0.03% 5.9h 15.610 0.30% 13.8h NeuOpt-GIRE (D2A=5, T=20k) L2S/RL 6.130 0.00% 6.8h 10.367 0.01% 19.7h 15.586 0.15% 1.9d NeuOpt-GIRE (D2A=5, T=40k) L2S/RL 6.130 0.00% 13.7h 10.367 0.01% 1.6d 15.579 0.10% 3.8d # We found issues with their code, causing an underestimation of their reported gaps by roughly 0.02% (TSP-50) and 0.04% (TSP-100). Thus our reproduced gaps may vary from those in their papers. For not reproduced ones, we use (\u00b7) to indicate the underestimated values sourced directly from their papers. \u2021 The results with \u2018\u2248\u2019 are based on their original papers since their code is either i) not publicly available (e.g., AM+CLP for CVRP), ii) raises (incompatible) errors that are difficult to fix on our hardware (e.g., Att-GCN+MCTS), or iii) requires long computation time to reproduce (e.g., CVAE-Opt-DE). Note: The abbreviations refer to: A Augmentation, D2A Dynamic Data Augmentation, BS Beam Search, H Heuristics, MCTS Monte Carlo Tree Search, DE Differential Evolution, DP Dynamic Programming, GLS Guided Local Search, AS Active Search, AS+M+M AS+MCTS+Meta-Learning. similar to theirs. As for L2C solvers, NeuOpt (D2A=5, T=3k) beats all baselines on TSP-100 with less time, including the state-of-the-art POMO+EAS+SGBS solver. When pitted against L2S solvers which ours also belongs to, NeuOpt is able to at least halve their gaps using a much shorter run time, demonstrating a substantial improvement. Lastly, our NeuOpt (D2A=5, T=5k) is one of the neural solvers (including DPDP) that can solve TSP-100 to near-optimal (i.e., 0.00%) in a reasonable time. CVRP results. Most L2P solvers fail to handle CVRP. The remaining ones, CVAE-Opt-DE and DPDP, are significantly outstripped by our solver (D2A=5, T=6k) with less run time. For L2C solvers, our NeuOpt-GIRE (D2A=1, T=10k) could exceed the previous state-of-the-art POMO solver. Furthermore, with (D2A=5, T=6k) and (D2A=5, T=40k), our method surpasses POMO+EAS and POMO+EAS+SGBS (long) with less run time, respectively, even though they equip with extra posthoc per-instance processing boosters. We note that such post-processing strategies (e.g., per-instance active search) could potentially be integrated into ours for further performance gains. Compared to L2S solvers, due to the advances of our GIRE, even NeuOpt-GIRE (D2A=1, T=5k) could reduce the gaps of those (masking-based) L2S solvers by an order of magnitude in much less time on CVRP-100. Notably, we are the first L2S solver to surpass the strong LKH-3 solver on CVRP. 8 \fTable 2: Comparisons of our proposed RDS decoder and GIRE scheme versus existing designs. Methods Encoder Decoder Feasibility TSP-100 CVRP-20 Type Type Scheme Size(m) Obj.\u2193 Gap\u2193 Time\u2193 Size(m) Obj.\u2193 Gap\u2193 Time\u2193 DACT (2-opt) d=64,FF,DAC-Att DACT masking 0.281 7.933 1.73% 134s 0.281 6.172 0.14% 58s DACT-U (2-opt) d=128,MLP,Synth-Att DACT masking 0.633 7.822 0.32% 171s 0.633 6.171 0.13% 61s DACT-U-GIRE (2-opt) d=128,MLP,Synth-Att DACT GIRE 0.633 7.822 0.32% 171s 0.633 6.168 0.08% 59s NeuOpt (K =2) d=128,MLP,Synth-Att RDS masking 0.683 7.817 0.24% 122s 0.683 6.180 0.27% 60s NeuOpt-GIRE (K =2) d=128,MLP,Synth-Att RDS GIRE 0.683 7.817 0.24% 122s 0.685 6.167 0.05% 53s NeuOpt-GIRE (K =4) d=128,MLP,Synth-Att RDS GIRE 0.683 7.798 0.00% 133s 0.685 6.163 0.00% 60s Table 3: Effects of GRUs, move stream \u00b5, and edge stream \u03bb. Methods TSP-100 CVRP-20 Size(M) Obj.\u2193 Size(M) Obj.\u2193 w/o-GRUs 0.468 7.804 0.470 6.165 w/o-\u00b5 0.617 7.806 0.620 6.165 w/o-\u03bb 0.617 7.799 0.620 6.164 Ours 0.683 7.798 0.685 6.163 Table 4: Effects of dynamic data augmentation (D2A). Inference Type TSP-100 CVRP-100 Gap\u2193 Gap\u2193 w/o-D2A (T=5k) 0.09% 1.00% w-D2A (T=5k) 0.05% 0.87% w/o-D2A (T=10k) 0.04% 0.71% w-D2A (T=10k) 0.02% 0.60% Base +VI +ES VI+ES 6.1625 6.1650 6.1675 6.1700 6.1725 Obj. Value RS No Yes Figure 4: Effects of GIRE designs. Effectiveness analysis. To further reveal the effectiveness of our RDS decoder and GIRE, we conduct a comparison with the state-of-the-art L2S solver, DACT [9]. We upgraded it to DACT-U for fairness, which uses identical embeddings and encoders as our NeuOpt. The evaluation criteria include the model size, the best objective value (averaged over four training runs with T=1k inference steps on 1k validation instances), the gaps w.r.t. the best solver, and the run time. As evidenced in Table 2, our GIRE scheme consistently boosts both NeuOpt and DACT-U on CVRP-20, reducing the mask computation time while marginally increasing model size, which reveals the generality of our GIRE. Moreover, when all are used for 2-opt, NeuOpt (K = 2, for TSP) and NeuOpt-GIRE (K = 2, for CVRP) surpass DACT-U (for TSP) and DACT-U-GIRE (for CVRP), respectively, with less run time. This is further amplified by increasing K to 4 for NeuOpt at the cost of slightly increased run time. 6.2 Ablation studies Ablation on RDS decoder. In Table 3, we remove the key components of our RDS decoder including the GRUs, the move stream \u00b5, and edge stream \u03bb, respectively, and report the model size and the best objective value averaged over four training runs. As depicted, all components largely contribute to effective contextual modeling for parameterizing the k-opt decoder on both TSP-100 and CVRP-20. 0 20 40 60 80 100 Training Epoch 7.75 8.00 8.25 8.50 Obj. Value w/o-S-move Ours (a) Effects of S-move. K=2 K=3 K=4 K=5 K=6 Varying K 7.80 7.83 7.85 7.88 Obj. Value w/o-E-move Ours (b) Effects of E-move. Figure 5: Effects of basis moves. Ablation on D2A inference. In Table 4, we gather the gaps (similar to Table 1) with and without our D2A design when a single data augmentation is leveraged during inference with T=5K and T=10K. The results confirm that our D2A consistently enhances the inference on both TSP-100 and CVRP-100. Ablation on GIRE designs. We examine scenarios, considering the inclusion or exclusion of Violation Indicator (VI), Exploration Statistics (ES), and Reward Shaping (RS), using eight combinations of them during the training of NeuOpt on CVRP-20. We draw the box plots of the best objective values achieved across eight training runs for each scenario in Figure 4. We can conclude that: 1) the supplement of VI consistently enhances training outcomes, irrespective of the presence of RS; 2) the supplement of ES may bring improvement only when RS is present; and 3) our complete GIRE design (VI+ES+RS) yields the lowest objective values and demonstrates superior stability. Ablation on basis moves. In Figure 5 (a), we depict the training curves of NeuOpt on TSP-100 with and without the learnable S-move. When removing the learnable S-move and opting for a random selection of the anchor node xa, a significant degradation in the final performance can be observed, underscoring the necessity of the learnable S-move design. Meanwhile, Figure 5 (b) 9 \fTable 5: Generalization (10 runs) on TSPLIB. AM POMO AMDKD DACT (sol.10k) Ours (sol.10k) -mix -mix (POMO) Avg.\u2193 Best\u2193 Avg.\u2193 Best\u2193 19.59% 0.92% 1.18% 3.29% 1.59% 0.85% 0.50% Table 6: Generalization (10 runs) on CVRPLIB. AM POMO AMDKD DACT (sol.10k) Ours (sol.10k) -mix -mix (POMO) Avg.\u2193 Best\u2193 Avg.\u2193 Best\u2193 15.87% 8.05% 5.77% 5.21% 3.68% 4.80% 3.27% Table 7: Results on N = 200. Methods TSP-200 CVRP-200 Gap.\u2193 Time.\u2193 Gap.\u2193 Time.\u2193 LKH [20, 51] 0.00% 2.3h 1.17% 21.6h Ours (D2A=5,T=10k) 0.04% 4.7h 0.68% 9.6h Ours (D2A=5,T=20k) 0.02% 9.4h 0.48% 19.2h Ours (D2A=5,T=30k) 0.01% 14.1h 0.39% 1.2d Table 8: Influence of K. K T(inf.)\u2193 T(tr.)\u2193 Gap\u2193 2 2m02s 20m 0.30% 3 2m07s 21m 0.15% 4 2m13s 22m 0.05% 5 2m19s 24m 0.01% 6 2m26s 25m 0.00% 0.01 0.05 0.1 6.1625 6.1650 6.1675 Obj. Value Varying Alpha (Beta=0.05) Varying Beta (Alpha=0.05) Figure 6: Influence of \u03b1 and \u03b2. presents the pointplots with confidence intervals that show the performance of our NeuOpt on TSP100 with and without E-move during decoding across varying preset K. When E-move is absent (dotted blue line), the model, downgraded to performing fixed K-opt only, exhibits diminished performance on larger K. Conversely, our NeuOpt (solid orange line) could further benefit from a larger K, due to its flexibility in determining and combining different k across search steps. 6.3 Generalization and scalability studies In Table 5 and Table 6, we further evaluate the generalization capability of our NeuOpt models on more complex instances from TSPLIB [55] and CVRPLIB [56], respectively. Our NeuOpt achieves lower gaps than DACT [9] and L2C solvers (AM [3] and POMO [4]), and even shows superiority over the AMDKD method [11] that is explicitly designed to boost the generalization of POMO through knowledge distillation. Beyond generalization, our NeuOpt also exhibits notable scalability. As shown in Table 7, when trained directly for size 200, NeuOpt finds close-to-optimal solutions and still surpasses the strong LKH-3 solver [20] on CVRP-200. Note that existing L2S solvers, e.g., DACT [9] may struggle with training on such scales. More details and discussions are available in Appendix E. 6.4 Hyper-parameter studies Influence of preset K. In Table 8, we display the performance of NeuOpt on TSP-100, as well as the corresponding inference time (T=1k) and the training time (per epoch) for varying K values. The results highlight trade-offs between better performance and increased computational costs. Influence of GIRE hyper-parameters. Figure 6 depicts the influence of \u03b1 and \u03b2 in Eq. (4) on CVRP-20, where we fix one while varying the other, investigating both extremely smaller (0.01) and larger (0.1) values. The results suggest that more effective reward shaping occurs when the weights are moderate. Please refer to Appendix D for discussions on more GIRE hyper-parameters. 7"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2204.11399v3",
+                    "title": "Efficient Neural Neighborhood Search for Pickup and Delivery Problems",
+                    "abstract": "We present an efficient Neural Neighborhood Search (N2S) approach for pickup\nand delivery problems (PDPs). In specific, we design a powerful Synthesis\nAttention that allows the vanilla self-attention to synthesize various types of\nfeatures regarding a route solution. We also exploit two customized decoders\nthat automatically learn to perform removal and reinsertion of a\npickup-delivery node pair to tackle the precedence constraint. Additionally, a\ndiversity enhancement scheme is leveraged to further ameliorate the\nperformance. Our N2S is generic, and extensive experiments on two canonical PDP\nvariants show that it can produce state-of-the-art results among existing\nneural methods. Moreover, it even outstrips the well-known LKH3 solver on the\nmore constrained PDP variant. Our implementation for N2S is available online.",
+                    "authors": "Yining Ma, Jingwen Li, Zhiguang Cao, Wen Song, Hongliang Guo, Yuejiao Gong, Yeow Meng Chee",
+                    "published": "2022-04-25",
+                    "updated": "2022-10-07",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Ef\ufb01cient neighborhood search functions as the key component of powerful heuristics for pickup and delivery problems (PDPs) [Parragh et al., 2008]. It involves an iterative search process which transforms a solution into another candidate in its current neighborhood, hopefully in an ef\ufb01cient way. Designing the neighborhood and search rules usually determines the ef\ufb01ciency and even the success of a solver. However, they are often problem-speci\ufb01c and manually engineered with a lot of trial and error, which needs to redesign when changes occur in constraints or objectives. These limitations may hinder the applications in the rapidly evolving industries. On the other hand, recent neural methods for vehicle routing problems (VRPs) have emerged as promising alternatives to traditional heuristics (e.g., [Ma et al., 2021b]). They are \u2217Equally contributed. \u2020Zhiguang Cao is the corresponding author. 1Code is available at https://github.com/yining043/PDP-N2S. The published version of this preprint can be found at IJCAI22 (see https://www.ijcai.org/proceedings/2022/662). This preprint \ufb01xes some typos and contains additional discussions and results. usually faster, and more importantly, could automate the design of heuristics for new variants where no hand-crafted rule is available [Kwon et al., 2020]. However, the prevailing neural methods mainly focus on travelling salesman problem (TSP) or capacitated vehicle routing problem (CVRP), where ef\ufb01cient solvers for PDPs are rarely studied. The PDP is ubiquitous in logistics, robotics, meal-delivery services, etc [Parragh et al., 2008], which optimizes the route for pickupdelivery requests and is characterised by the precedence constraint (pickup before delivery). Despite the \ufb01rst attempt in [Li et al., 2021b], which learns a construction method to build a PDP solution (route) in seconds, it leaves a considerable gap to traditional heuristics in solution quality. To reduce the gap, we propose an ef\ufb01cient Neural Neighborhood Search (N2S) approach for PDPs, based on a novel Transformer styled policy network with the encoder-decoder structure. To our knowledge, the most similar existing policy network to ours is the DACT in [Ma et al., 2021b]. Also as an improvement method, DACT learns to encode the current solution and transform it into another one using the 2-opt, insert, or swap decoder. Among them, the 2-opt decoder, which considers reversing a segment of the solution, performed the best for TSP and CVRP. However, it is not suitable for PDPs since the precedence constraint can be easily violated by segment inversion. Though the insert decoder performed well on small-scale PDPs, which considers removing and reinserting a single node, its performance signi\ufb01cantly drops on largerscale or highly constrained PDPs (see Section 5). Conversely, our N2S tackles the precedence constraint more ef\ufb01ciently by allowing a pair of pickup-delivery nodes to be simultaneously operated in the neighborhood search through two customized removal and reinsertion decoders as shown in Figure 1. Another challenge lies in the design of the encoders. It was well revealed in [Ma et al., 2021b] that the vanilla Transformer encoder [Vaswani et al., 2017] failed to correctly encode route solutions since the embeddings of node features (i.e., coordinates) and node positional features (i.e., node positions) involve two different aspects of a route solution which are not directly compatible during encoding. They thus proposed the dual-aspect collaborative attention (DACAtt) to learn dual representations for each feature aspect. In arXiv:2204.11399v3  [cs.LG]  7 Oct 2022 \f1+ 2+ 3+ 132Synth-Att Node-Pair Removal Decoder Node-Pair Reinsertion Decoder Action: { ( 1+, 1) , (0, 3+ ) } \ud835\udc73\u00d7 NFEs PFEs Node Positional Features 0 2+ 3+ 321+ 10 Node Features Compatibility Layer Encoder Problem Instance & Current Solution ( State at time T ) auxiliary  attention scores Enhanced Node Embeddings ( 1+, 1) ( 0, 3+ ) Problem Instance & Next Solution ( State at time T+1 ) decoding state transition encoding 1+ 2+ 3132Input / Output Passing Identify / Reference Update 1+ 2+ 3+ 1320 1+ 2+ 3+ 1320 1+ 2+ 3+ 1320 Figure 1: An example of a PDTSP-7 instance to illustrate our N2S approach. From the left to the right: encoding, decoding and state transition process. The encoders process raw features of the current solution to produce node embeddings, which are then fed into the two decoders to sample an action. In the state transition, the node pair (1+, 1\u2212) are removed and then reinserted after depot (0) and node 3+, respectively. this paper, we propose a simple yet powerful Synthesis Attention (Synth-Att) where the attention scores from various types of node feature embeddings can be synthesized to attain a comprehensive representation. It not only has the potential to encode more aspects than DAC-Att, but also reduces the computation costs while reserving competitive performance. Additionally, we design a diversity enhancement scheme to further ameliorate the performance. The proposed N2S approach was trained through reinforcement learning [Ma et al., 2021b], and we evaluated it on two canonical problems in the PDP family, i.e., the pickup and delivery travelling salesman problem (PDTSP) and its variant with the last-in-\ufb01rstout constraint (PDTSP-LIFO) to verify our design. Experimental results show that our N2S outperforms the state-ofthe-art, and becomes the \ufb01rst neural method to surpass the well-known LKH3 solver [Helsgaun, 2017] when solving the synthesized PDP instances. The contributions of the paper are summarized as follows: 1) we present an ef\ufb01cient N2S approach for PDPs, which learns to perform removal and reinsertion of a pickupdelivery node pair automatically; 2) we propose Synth-Att, which allows the vanilla self-attention mechanism to synthesize node relationships from various feature embeddings in a simple way, and achieves superior expressiveness with much fewer computation costs in comparison to DAC-Att; 3) we explore a diversity enhancement scheme, which further makes our N2S the \ufb01rst neural method with almost no domain knowledge to surpass the LKH3 solver on the synthesized PDP instances (when there are no signi\ufb01cant distribution shift w.r.t training ones). All these advantages highlight the potential of N2S as a powerful neural solver for new PDP variants without much need for manual trial and error. 2 Related Work Neural Methods for VRPs. We classify recent neural methods into construction and improvement ones. The construction methods, e.g., [Nazari et al., 2018; Joshi et al., 2019; Kool et al., 2018], learn a distribution of selecting nodes to autoregressively build solutions from scratch. Despite being fast, they lack abilities to search (near-)optimal solutions, even if armed with sampling (e.g., [Kool et al., 2018]), local search (e.g., [Kim et al., 2021]), or Monte-Carlo tree search (e.g., [Fu et al., 2021]). Among them, POMO [Kwon et al., 2020] which explored diverse rollouts and data augments is recognized as the best construction method. Differently, improvement methods often hinge on a neighborhood search procedure such as node swap in [Chen and Tian, 2019], ruin-and-repair in [Hottung and Tierney, 2020], and 2-opt in [Wu et al., 2021]. [Ma et al., 2021b] extended the Transformer styled model of [Wu et al., 2021] to Dual-Aspect Collaborative Transformer (DACT), and achieved the stateof-the-art performance, which was also competitive to the hybrid neural methods, e.g., the ones combined with differential evolution [Hottung et al., 2021] and dynamic programming [Kool et al., 2021]. Despite the success of the above methods for CVRP or TSP, they are not veri\ufb01ed on the precedence constrained PDPs. Though [Li et al., 2021b] made the \ufb01rst attempt to learn a construction solver for PDPs by introducing the heterogeneous attention to [Kool et al., 2018], the resulting solution qualities are still far from the optimality. Neighbourhood Search for PDPs. Various heuristics based on neighborhood search have been proposed for PDPs. For PDTSP, [Savelsbergh, 1990] studied the k-interchange neighborhood. A ruin-and-repair neighborhood was later proposed in [Renaud et al., 2000], and further extended in [Renaud et al., 2002] with multiple perturbation methods. Aside from PDTSP, other PDP variants were also studied, normally solved by designing new problem-speci\ufb01c neighborhoods [Parragh et al., 2008], e.g., [Veenstra et al., 2017] proposed \ufb01ve neighborhoods to tackle the PDTSP with handling costs. Among them, the PDTSP-LIFO attracts much attention due to its largely constrained search space. To solve it, [Carrabs et al., 2007] introduced additional neighborhoods such as double-bridge and shake. In [Li et al., 2011], neighborhoods with tree structures were further proposed. Different from the above PDP solvers, the well-known LKH3 solver [Helsgaun, 2017] combines neighborhood restriction strategies to achieve more effective local search, which could solve various VRPs with superior performance. Recently, LKH3 was extended to tackle several PDP variants including PDTSP \f0 1+ 2+ 23+ 310 1+ 1+ 2+ 1+ 1+ 3+ 1+ 0 1+ 2+ 13+ 320 1+ 1+ 2+ (a) \u03b41 (feasible to both PDPs) 0 1+ 2+ 23+ 310 1+ 1+ 2+ 1+ 1+ 3+ 1+ 0 1+ 2+ 13+ 320 1+ 1+ 2+ (b) \u03b42 (only feasible to PDTSP) Figure 2: Two PDP solutions. (a) all goods are on top of the stack at the delivery node; (b) goods from 1+ is blocked by 2+ at 1\u2212. and PDTSP-LIFO, and delivered comparable performance to [Renaud et al., 2002] and [Li et al., 2011] on PDTSP and PDTSP-LIFO, respectively. Also given the open-sourced nature2, we use LKH3 as the benchmark heuristic in this paper. 3 Problem Formulation 3.1 Notations We de\ufb01ne the studied PDPs over a graph G = (V, E), where nodes in V = P \u222aD \u222a0 represent locations and edges in E represent trips between locations. With n oneto-one pickup-delivery requests, an PDP instance contains |V | = 2n + 1 different locations, where node 0 is depot, set P = {1+, 2+, ..., n+} contains pickup nodes, and set D = {1\u2212, 2\u2212, ..., n\u2212} contains delivery nodes3. Each pickup node i+ has a number of goods to be transported to its delivery node i\u2212. The objective is to \ufb01nd the shortest Hamiltonian cycle to ful\ufb01ll all requests. In this paper, we consider two representative PDP variants, i.e., PDTSP and PDTSP-LIFO. The solution \u03b4 is de\ufb01ned as a cyclic sequence (x0, ..., x2n+1), where x0 and x2n+1 are the depot, and the rest is a permutation of nodes in P \u222aD. For PDTSP, such permutation is under the precedence constraint that requires each pickup i+ to be visited before its delivery i\u2212. For PDTSP-LIFO, the last-in-\ufb01rst-out constraint is further imposed which requires loading and unloading to be executed in the corresponding order. This implies that unloading at a delivery node is allowed if and only if the goods is at the top of the stack. In Figure 2, we present two example solutions with n = 3 and |V | = 7. The two solutions are both feasible to PDTSP, however, solution \u03b42 in Figure 2(b) is infeasible to PDTSP-LIFO as the goods from 1+ is NOT at the top of the stack when it needs to be delivered at 1\u2212. 3.2 MDP Formulations We de\ufb01ne the process of solving PDPs by our N2S as a Markov Decision Process M=(S, A, T , R, \u03b3) as follows. State S. At time t, the state is de\ufb01ned to include: 1) features of the current solution \u03b4t, 2) action history, and 3) objective value of the best incumbent solution, i.e., st = {{l(x)}x\u2208V , {pt(x)}x\u2208V , H(t, K), f(\u03b4\u2217 t )} , (1) where \u03b4t is described from two aspects following [Ma et al., 2021b]: l(x) contains 2-dim coordinates of node x (i.e., node features) and pt(x) indicates the index position of x in \u03b4t (i.e., node positional feature); H(t, K) stores the most recent K actions at time t if any; and f(\u00b7) denotes the objective function and \u03b4\u2217 t = arg min\u03b4t\u2032\u2208{\u03b40,...,\u03b4t} f(\u03b4t\u2032). 2http://webhotel4.ruc.dk/\u223ckeld/research/LKH-3/ 3We also refer x0 to the depot, {x1, ..., xn} to the pickup nodes, and {xn+1, ..., x2n} to the delivery nodes from here on. Action A. With action at ={(i+, i\u2212), (j, k)} where j, k \u2208 V \\{i+, i\u2212}, the agent removes node pair (i+, i\u2212), and then reinserts node i+ and i\u2212after node j and k, respectively. State Transition T . We utilize a deterministic transition rule to perform at. An example is illustrated in Figure 1. Reward R. The reward function is de\ufb01ned as rt =f(\u03b4* t)\u2212 min \u0002 f(\u03b4t+1), f(\u03b4* t) \u0003 which is the immediate reduced cost w.r.t. f(\u03b4* t). The N2S agent aims to maximize the expected total reduced cost w.r.t. \u03b40 with a discount factor \u03b3 < 1. 4 Methodology 4.1 Encoder and Synth-Att Given state s = {{l(x)}x\u2208V , {p(x)}x\u2208V , H(t, K), f(\u03b4\u2217)}, the N2S encoder takes {l(x)}x\u2208V and {p(x)}x\u2208V as inputs4 to learn embeddings for representing the current solution. Following DACT, we \ufb01rst project these raw features into two sets of embeddings, i.e., node feature embeddings (NFEs) {hi}|V | i=0 and positional feature embeddings (PFEs) {gi}|V | i=0. Different from [Ma et al., 2021b], we treat NFEs as the primary set of embeddings whereas PFEs as auxiliary ones. NFEs. We de\ufb01ne hi as the linear projection of its node features l(xi) for any xi \u2208V with output dimension dh = 128. PFEs. By extending the absolute positional encoding in the vanilla Transformer [Vaswani et al., 2017], the cyclic positional encoding (CPE) was proposed in [Ma et al., 2021b], which enables Transformer to encode cyclic sequences (as our PDP solutions) more accurately. The PFE gi with output dimension dg = 128 are initialized by CPE as follows, gi(d) = \uf8f1 \uf8f2 \uf8f3 sin(\u03c9d\u00b7 \f \f \f(z(i) mod 4\u03c0 \u03c9d ) \u22122\u03c0 \u03c9d \f \f \f), if d is even cos(\u03c9d\u00b7 \f \f \f(z(i) mod 4\u03c0 \u03c9d ) \u22122\u03c0 \u03c9d \f \f \f), if d is odd (2) where superscript d of gi(d) refers to the d-th dimension of gi, and the scalar z(i) as well as the angular frequency \u03c9d = 2\u03c0 Td are de\ufb01ned accorting to Eq. (3) and Eq. (4), respectively. z(i) = i |V | 2\u03c0 \u03c9d \u0018 |V | 2\u03c0/\u03c9d \u0019 , (3) Td = ( 3\u230ad/3\u230b+1 dg (|V |\u2212|V | 1 \u230adg/2\u230b)+|V | 1 \u230adg/2\u230b, ifd<\u230adg 2 \u230b |V |. otherwise (4) According to [Ma et al., 2021b], directly fusing the two sets of embeddings (i.e., hi+gi) may cause undesired noises to the vanilla self-attention. As shown in Figure 3(a), they thus proposed the DAC-Att in ways that each embedding set independently computes attention scores and shares the other aspect to learn dual-aspect representations. Different from it, we propose a simple and generic mechanism by incorporating a multilayer perception (MLP). As shown in Figure 3(b), besides the original self-attention scores (the orange squares), multiple auxiliary attention scores learned from other feature embeddings (the blue squares) are leveraged and fed into an 4H(t, K) is input to decoder and f(\u03b4\u2217) is input to critic network (introduced later). Here we omit time step t for better readability. \fMatMul Scale Element-wise MLP SoftMax MatMul Qh Kh Vh MatMul Scale SoftMax Qh Kh Vh Vhref Concat & Linear MatMul Scale SoftMax MatMul Kg Qg Vgref Vg MatMul MatMul MatMul Concat & Linear (a) The DAC-Att [Ma et al., 2021b] MatMul Scale Element-wise MLP SoftMax MatMul Qh Kh Vh MatMul Scale SoftMax Qh Kh Vh Vhref Concat & Linear MatMul Scale SoftMax MatMul Kg Qg Vgref Vg MatMul MatMul MatMul Concat & Linear (b) Our Synth-Att Figure 3: Comparison of attention modules for VRPs. The blue and orange squares are used to represent self-attention score matrices. element-wise MLP, which allows it to synthesize heterogeneous attention relationships into comprehensive ones. We call it Synthesis Attention (Synth-Att). It is able to not only leverage more attention scores from various feature embeddings, but also achieve competitive performance to DAC-Att with less computation costs. Below, we present more details. Auxiliary Attention Scores. In our N2S, PFEs are used to generate multi-head auxiliary attention scores as follows, \u03b1aux i,j,m = 1 \u221adk \u0000giW Qaux m \u0001 \u0000gjW Kaux m \u0001T , (5) where W Qaux m \u2208Rdg\u00d7dq, W Kaux m \u2208Rdg\u00d7dk are trainable matrices for each head m. We set m = 4 and dq = dk = dg/m. Syn-Att. The Syn-Att is de\ufb01ned as follows, \u02dc hi = Syn-Att(W Q, W K, W V , W O, MLP). (6) In speci\ufb01c, it \ufb01rst computes the multi-head self-attention scores \u03b1self i,j,m for NFEs based on the trainable matrices W Q m \u2208 Rdh\u00d7dq and W K m \u2208Rdh\u00d7dk for head m using Eq. (7) as \u03b1self i,j,m = 1 \u221adk \u0000hiW Q m \u0001 \u0000hjW K m \u0001T . (7) Thereafter, the attention scores \u03b1aux and \u03b1self are fed into a three-layer MLP with structure (2m \u00d7 2m \u00d7 m) to compute the synthesized multi-head attention scores as follows, \u03b1Synth i,j,1 , ..., \u03b1Synth i,j,m =MLP \u0000\u03b1self i,j,1, ..., \u03b1self i,j,m, \u03b1aux i,j,1, ..., \u03b1aux i,j,m \u0001 . (8) The scores are then normalized to \u02dc \u03b1i,j,m through Softmax, which are further used to calculate the attention values for each head as Eq. (9). Finally, the outputs are given by Eq. (10) with trainable matrix W O \u2208Rmdv\u00d7dh (dv = dh/m). headi,m = |V | X j=1 \u02dc \u03b1i,j,m \u0000hjW V m \u0001 , (9) \u02dc hi = Concat [headi,1, ..., headi,m] W O. (10) N2S Encoder. We stack L (L=3) encoders, each of which is the same as the Transformer encoder, except that the vanilla multi-head self-attention is replaced with our multihead Synth-Att and we use the same instance normalization layer as [Ma et al., 2021b]. Note that the auxiliary attention scores \u03b1aux i,j,m are only computed once and shared among all stacked encoders to reduce the computation costs. 4.2 Decoder The N2S decoder \ufb01rst adopts the max-pooling layer in [Wu et al., 2021] to aggregate the global representation of all embeddings into each individual one as follows, \u02c6 hi = \u02dc h(L) i W Local h + max h {\u02dc h(L) i }|V | i=1 i W Global h . (11) Node-Pair Removal Decoder. Given the enhanced embeddings {\u02c6 hi}|V | i=1 and the set H(t, K), the removal decoder outputs a categorical distribution over n requests for removal action. In speci\ufb01c, it \ufb01rst computes a score \u03bbi for each xi \u2208V indicating the closeness between node xi and its neighbors as \u03bbi = (\u02c6 hpred(xi)W Q \u03bb )(\u02c6 hiW K \u03bb )T + (\u02c6 hiW Q \u03bb )(\u02c6 hsucc(xi)W K \u03bb )T \u2212(\u02c6 hpred(xi)W Q \u03bb )(\u02c6 hsucc(xi)W K \u03bb )T , (12) where pred(xi) and succ(xi) refer to the predecessor and the successor nodes of xi, respectively, and W Q \u03bb \u2208Rdh\u00d7dh, W K \u03bb \u2208Rdh\u00d7dh. We use multi-head technique to obtain \u03bbi,1 to \u03bbi,m. Then the decoder aggregates the scores for each pickup-delivery pair (i+, i\u2212) based on a three-layer MLP\u03bb, \u02dc \u039b(i+,i\u2212) = MLP\u03bb(\u03bbi+,1, ..., \u03bbi+,m, \u03bbi\u2212,1, ..., \u03bbi\u2212,m, c(i), 1last(1)=i, 1last(2)=i, 1last(3)=i), (13) where the MLP structure is (2m+4, 32, 32, 1), scalar c(i) counts the frequency of request (i+, i\u2212) being selected for removal in the past K steps, and 1last(t)=i\u2032 is a binary variable indicating whether request i\u2032 was selected at the t-th last step. An activation layer \u02c6 \u039b = C \u00b7 Tanh(\u02dc \u039b) is then applied (C = 6), followed by Softmax to normalize the distribution which is then used to sample a node pair (i+, i\u2212) as removal action. Node-Pair Reinsertion Decoder. Given a request (i+, i\u2212) for removal, the reinsertion decoder outputs the joint distribution that reinserts the two nodes back to the solution. We \ufb01rst de\ufb01ne two scores \u00b5p(x\u03b1, x\u03b2) and \u00b5s(x\u03b1, x\u03b2) for a node x\u03b1 indicating the degree of preference of accepting a node x\u03b2 as its new predecessor and successor nodes, respectively, \u00b5p [x\u03b1, x\u03b2] = (\u02c6 h\u03b1W Qp \u00b5 )(\u02c6 h\u03b2W Kp \u00b5 )T , \u00b5s [x\u03b1, x\u03b2] = (\u02c6 h\u03b1W Qs \u00b5 )(\u02c6 h\u03b2W Ks \u00b5 )T , (14) where W Qp \u00b5 , W Qs \u00b5 \u2208Rdh\u00d7dh, and W Kp \u00b5 , W Ks \u00b5 \u2208Rdh\u00d7dh. Again, we use multiple heads. Based on the scores, the decoder predicts the distribution of reinserting node i+ after node j, and reinserting node i\u2212after node k using MLP\u00b5, \u02dc \u00b5[j, k] = MLP\u00b5(\u00b5p 1[i+, succ(j)], ..., \u00b5p m[i+, succ(j)], \u00b5p 1[i\u2212, succ(k)], ..., \u00b5p m[i\u2212, succ(k)], \u00b5s 1[i+, j], ..., \u00b5s m[i+, j], \u00b5s 1[i\u2212, k], ..., \u00b5s m[i\u2212, k]), (15) where the MLP structure is (4m, 32, 32, 1). Note that here pred(\u00b7) and succ(\u00b7) should be considered in the new solution where nodes i+, i\u2212have already been removed. Afterwards, \u02c6 \u00b5 = C \u00b7Tanh(\u02dc \u00b5) is applied and infeasible choices are masked as \u2212\u221ebefore normalizing by Softmax. Finally, a node pair (j, k), as the reinsertion action, is sampled according to the resulting distribution to indicate the positions of reinserting the node-pair (i\u2212, i+) back to the solution. \fAlgorithm 1 n-step PPO with CL strategy Input: policy \u03c0\u03b8, critic v\u03c6, PPO clipping threshold \u03b5, learning rate \u03b7\u03b8, \u03b7\u03c6, learning rate decay \u03b2, epochs E, batches B, mini-batch \u03ba, training steps Ttrain, CL scalar \u03c1CL 1: for e = 1 to E do 2: for b = 1 to B do 3: Generate training data Db on the \ufb02y; 4: Initialize random solutions {\u03b4i} to Db; 5: Improve {\u03b4i} to {\u03b4\u2032 i} via \u03c0\u03b8 for T = e/\u03c1CL steps; 6: Set initial state s0 based on {\u03b4\u2032 i} and Eq. (1); t\u21900; 7: while t < Ttrain do 8: Get {(st\u2032,at\u2032,rt\u2032)}t+n t\u2032=t where at\u2032 \u223c\u03c0\u03b8(at\u2032|st\u2032); 9: t \u2190t + n, \u03c0old \u2190\u03c0\u03b8, vold \u2190v\u03c6; 10: for k = 1 to \u03ba do 11: \u02c6 Rt+1 = v\u03c6(st+1); 12: for t\u2032 \u2208{t, t \u22121, ..., t \u2212n} do 13: \u02c6 Rt\u2032 \u2190rt\u2032 + \u03b3 \u02c6 Rt\u2032+1; 14: \u02c6 At\u2032 \u2190\u02c6 Rt\u2032 \u2212v\u03c6(st\u2032); 15: end for 16: Compute RL loss JRL(\u03b8) using Eq. (18) and clipped critic loss LBL(\u03c6) using Eq. (19); 17: \u03b8 \u2190\u03b8 + \u03b7\u03b8\u2207JRL(\u03b8); 18: \u03c6 \u2190\u03c6 \u2212\u03b7\u03c6\u2207LBL(\u03c6); 19: end for 20: end while 21: end for 22: \u03b7\u03b8 \u2190\u03b2\u03b7\u03b8, \u03b7\u03c6 \u2190\u03b2\u03b7\u03c6; 23: end for 4.3 Training Algorithm As presented in Algorithm 1, the training algorithm for N2S is the same as [Ma et al., 2021b], namely the proximal policy optimization with a curriculum learning strategy. It learns a policy \u03c0\u03b8 (our N2S) with the help of a critic v\u03c6 as follows. Critic Network. Given the embeddings {\u02dc h(L) i }|V | i=0 from policy network \u03c0\u03b8, the critic network \ufb01rst enhances them by a vanilla multi-head attention layer (with m = 4 heads) to get {yi}|V | i=0. The enhanced embeddings are then fed into a meanpooling layer in [Wu et al., 2021] to aggregate the global representation of all embeddings into each individual one as \u02c6 yi = yiW Local v + mean h {yi}|V | i=1 i W Global v , (16) where we use trainable matrices W Local v , W Global v \u2208Rdh\u00d7 dh 2 . Lastly, the state value v(st) is output by a four-layer MLP in Eq. (17) with structure (129, 128, 64, 1). Here, f(\u03b4\u2217) is added as an additional input to MLPv. v(st) = MLPv \u0010 max h {\u02c6 yi}|V | i=1 i , mean h {\u02c6 yi}|V | i=1 i , f(\u03b4\u2217) \u0011 (17) We train N2S for E epochs and B batches per epoch, where the training dataset Db is randomly generated on the \ufb02y with the uniform distribution (line 3). Following [Ma et al., 2021b], we introduce a similar CL strategy (in lines 4 to 6) to further ameliorate the training. In speci\ufb01c, it improves the randomly generated solutions {\u03b4i} to {\u03b4\u2032 i} by running the current policy \u03c0\u03b8 for T = e/\u03c1CL steps. The improved solutions Algorithm 2 N2S-A Inference Input: Instance I with size |V |, policy \u03c0\u03b8, maximum step T 1: for i = 1, ..., \u230a1 2|V |\u230bdo 2: Ii \u2190I; 3: Ai \u2190RandomShuf\ufb02e([\ufb02ip-x-y, 1-x, 1-y, rotate]); 4: for each augment method j \u2208Ai do 5: \u03f1j \u2190RandomCon\ufb01g(j); 6: Ii \u2190perform augment j on Ii with con\ufb01g \u03f1j; 7: end for 8: end for 9: Solve all instances Ii in parallel with \u03c0\u03b8 for T steps; 10: return the best solution found among all Ii {\u03b4\u2032 i} with higher quality are then used to initialize the \ufb01rst state s0. In such a way, the hardness level of neighborhood search is gradually increased per epoch following the idea of curriculum learning. The remaining algorithm follows the original design of the n-step PPO, where we present the reinforcement learning loss function in Eq. (18), and the baseline loss function of critic in Eq. (19), respectively. Here, we clip the estimated value \u02c6 v(st) around the previous value estimates using vclip \u03c6 (st\u2032) = clip [v\u03c6(st\u2032), vold(st\u2032) \u2212\u03b5, vold(st\u2032) + \u03b5]. JRL(\u03b8) = 1 n|Db| X Db t+n X t\u2032=t min \u0012 \u03c0\u03b8(at\u2032|st\u2032) \u03c0old(at\u2032|st\u2032) \u02c6 At\u2032, clip \u0014 \u03c0\u03b8(at\u2032|st\u2032) \u03c0old(at\u2032|st\u2032), 1 \u2212\u03b5, 1 + \u03b5 \u0015 \u02c6 At\u2032 \u0013 , (18) LBL(\u03c6) = 1 n|Db| X Db t+n X t\u2032=t max \u0010\f \f \fv\u03c6(st\u2032) \u2212\u02c6 Rt\u2032 \f \f \f , \f \f \fvclip \u03c6 (st\u2032) \u2212\u02c6 Rt\u2032 \f \f \f \u00112 . (19) 4.4 Diversity Enhancement To be more resistant to local minima, we further equip our N2S with an augmentation-based inference scheme, which leads to N2S-A in Algorithm 2. Such idea was originally explored in [Kwon et al., 2020] for a neural construction method, and we extend it to an improvement one. The rationale is that an instance I can be transformed into different ones for searching while reserving the same optimal solution, e.g., rotating all locations of nodes by \u03c0/2 radian. For an instance of size |V |, our N2S-A performs \u230a1 2|V |\u230baugments, each of which is generated by sequentially applying four preset invariant transformation operations with different orders and different con\ufb01gurations as listed in Table 1. Note that although the mentioned transformations are conducted on instances de\ufb01ned in the Euclidean space, we believe that such an idea has favourable potential to be also exploited in nonEuclidean space, as long as there are proper invariant transformations for coordinates in the target space. Meanwhile, we use K = |V | for training and K\u2032 = \u230a1 2|V |\u230bfor inference. This is because we found that when a speci\ufb01c K in H(t, K) is used for training, a smaller K during inference can improve the diversity of the solutions, thus better performance. \fTransformations Formulations Con\ufb01gurations \ufb02ip-x-y (x\u2032, y\u2032) = (y, x) perform or skip 1-x (x\u2032, y\u2032) = (1 \u2212x, y) perform or skip 1-y (x\u2032, y\u2032) = (x, 1 \u2212y) perform or skip rotate \u0012 x\u2032 y\u2032 \u0013 = \u0012 x cos \u03d1 \u2212y sin \u03d1 x sin \u03d1 + y cos \u03d1 \u0013 \u03d1 \u2208{0, \u03c0/2, \u03c0, 3\u03c0/2} Table 1: Descriptions of the four invariant transformations. 5 Evaluation We design experiments to answer the following questions: 1. How good is the proposed N2S approach against the baselines, including the state-of-the-art neural methods and the strong LKH3 solver? (see Table 3 and Table 4) 2. Can Synth-Att reduce computation costs while achieving competitive performance to DAC-Att? (see Table 5) 3. How crucial are the proposed learnable node-pair removal and node-pair reinsertion decoders for achieving an ef\ufb01cient search? (see Table 6) 4. Can our N2S generalize well to benchmark instances that are different from training ones? (see Table 7) 5.1 Setup We evaluate N2S on PDTSP and PDTSP-LIFO with three sizes |V | = 21, 51, 101 following the conventions in [Kwon et al., 2020; Ma et al., 2021b], where the node coordinates of instances are randomly and uniformly generated in the unit square [0, 1]\u00d7[0, 1]. For our N2S, the initial solution \u03b40 is sequentially constructed in a random fashion. Our experiments were conducted on a server equipped with 8 RTX 2080 Ti GPU cards and Intel E5-2680 CPU @ 2.4GHz. The training time of our N2S varies with problem sizes, i.e., around 1 day for |V | = 21, 3 days for |V | = 51, and 7 days for |V | = 101, all of which are shorter than the baselines in Table 2. Hyper-parameters. Our N2S is trained with E = 200 epochs and B = 20 batches per epoch using batch size 600. We set n=5, Ttrain =250 for the n-step PPO with \u03ba=3 minibatch updates and a clip threshold \u03f5=0.1. Adam optimizer is used with learning rate \u03b7\u03b8 =8\u00d710\u22125 for \u03c0\u03b8, and \u03b7\u03c6 =2\u00d710\u22125 for v\u03c6 (decayed \u03b2 = 0.985 per epoch). The discount factor \u03b3 is set to 0.999 for both PDPs. We clip the gradient norm to be within 0.05, 0.15, 0.35, and set the curriculum learning \u03c1CL to 2, 1.5, 1 for the three problem sizes, respectively. 5.2 Comparison Evaluation We compare our N2S with the state-of-the-art (SOTA) neural methods and the highly-optimized LKH3 solver. Regarding the former, we consider the SOTA improvement method DACT [Ma et al., 2021b]5 and the SOTA construction method Heter-AM [Li et al., 2021b] (specially designed for PDPs). To make a fair comparison with our N2S-A (with the diversity enhancement), we upgrade Heter-AM to HeterPOMO, also given the known superiority of POMO to AM. In speci\ufb01c, we reserve the policy network in Heter-AM as the backbone while adopting the diverse rollouts and the data augmentation techniques in POMO [Kwon et al., 2020] to 5We use the insert decoder which is the best for PDPs. Method Code Training Time Hyper-parameters Heter-AM online6 \u223c20 days train 800 epochs as per the original setting. Heter-POMO online7 \u223c14 days train 2,000 epochs as per the original setting. DACT online8 \u223c10 days \u03beCL = 0.25, 1, 4 for sizes |v| = 21, 51, 101, respectively; n=5, Ttrain =250 (same as ours); train 200 epochs (same as ours). Table 2: Training details of the adopted neural baselines. leverage the advantages of them for the best performance. Each neural baseline is trained using the respective implementation code that is publicly available. For the upgraded Heter-POMO, we adapt and combine the model architecture from the original Heter-AM and the original POMO. The links to their original implementations, approximate training time for the size |V | = 101 , and the used hyper-parameters are presented in Table 2. For other hyper-parameters, we follow the recommendation in their papers. Regarding the latter, LKH3 is a strong heuristic (as reviewed in Section 2) which is widely used as a baseline to benchmark neural methods in recent studies (e.g., [Ma et al., 2021b; Hottung et al., 2021; Li et al., 2021c; Wu et al., 2021]). We report its results with two settings of iterations, i.e., LKH (5k) and LKH (10k). All baselines are evaluated on a test dataset with 2,000 instances, and we report the metrics of averaged objective values, averaged gaps to LKH3 (10K) and the total solving time. Note that it is hard to perform an absolutely fair time comparison between running Python codes on GPUs (neural methods) and running ANSI C codes on CPUs (LKH solver). Thus we follow the guidelines in [Accorsi et al., 2022] to perform the facilitate comparison that lets each method make full use of the best settings on our machine. In particular, we report the time of LKH3 when running in parallel with 16 CPU cores and the time of each neural method when all 8 GPU cards are available (but do not need to be fully used). Results on PDTSP. Table 3 shows the results on PDTSP. In the \ufb01rst group, we compare N2S with Heter-AM (greedy and sampling), and DACT. Compared to Heter-AM (5k), our N2S with only 1k steps attains lower gaps with less time for all sizes. Although DACT offers the best gap on PDTSP-21, its performance drops signi\ufb01cantly as the problem size increases, partly because its decoder is less ef\ufb01cient than our node-pair removal and reinsertion ones when tackling largerscale problems. Instead, our N2S achieves higher performance and consistently dominates DACT in terms of both the gaps and the time on PDTSP-51 and PDTSP-101. In the second group, our augmented N2S-A is compared to the upgraded Heter-POMO method with three variants9. It is shown 6https://github.com/Demon0312/Heterogeneous-AttentionsPDP-DRL 7https://github.com/yd-kwon/POMO 8https://github.com/yining043/VRP-DACT 9Heter-POMO (gr.), Heter-POMO-A (gr.), Heter-POMO-A (3k) refer to: greedily generate |V | solution; greedily generate |V | solutions with 8 augments; sample |V |\u00d73k solutions with 8 augments. \fMethods PDTSP-21 PDTSP-51 PDTSP-101 Obj. Value Gap to LKH Total Time Obj. Value Gap to LKH Total Time Obj. Value Gap to LKH Total Time LKH (5k) 4.563 0.00% 3m 6.866 0.06% 10m 9.443 0.16% 49m LKH (10k) 4.563 0.00% 5m 6.862 0.00% 19m 9.428 0.00% 98m Heter-AM (gr.) 4.655 2.02% (0s) 7.333 6.86% (1s) 10.348 9.76% (2s) Heter-AM (5k) 4.578 0.33% (33s) 7.108 3.58% (1.5m) 10.051 6.61% (5m) DACT (1k) 4.572 0.20% (18s) 7.245 5.57% (29s) 10.551 11.91% (51s) DACT (2k) 4.566 0.07% (37s) 7.118 3.72% (1m) 10.312 9.38% (1.5m) DACT (3k) 4.564 0.03% (1m) 7.057 2.83% (1.5m) 10.195 8.13% (2.5m) N2S (1k) 4.573 0.21% (21s) 7.103 3.51% (31s) 10.030 6.38% (1m) N2S (2k) 4.567 0.09% (42s) 7.053 2.77% (1m) 9.905 5.06% (2m) N2S (3k) 4.565 0.05% (1m) 7.027 2.40% (1.5m) 9.846 4.44% (3m) Heter-POMO (gr.) 4.634 1.56% (0s) 7.168 4.45% (1s) 10.060 6.70% (2s) Heter-POMO-A (gr.) 4.584 0.46% (1s) 6.995 1.93% (5s) 9.681 2.68% (11s) Heter-POMO-A (3k) 4.564 0.03% (7m) 6.916 0.77% (32m) 9.567 1.47% (135m) N2S-A (1k) 4.563 0.01% (1m) 6.865 0.03% (8m) 9.475 0.50% (40m) N2S-A (2k) 4.563 0.00% (2m) 6.860 -0.03% (16m) 9.427 -0.01% (80m) N2S-A (3k) 4.563 0.00% (3m) 6.860 -0.04% (24m) 9.409 -0.20% (121m) Table 3: Comparison results for solving PDTSP instances of sizes |V | = 21, 51, and 101. Methods PDTSP-LIFO-21 PDTSP-LIFO-51 PDTSP-LIFO-101 Obj. Value Gap to LKH Total Time Obj. Value Gap to LKH Total Time Obj. Value Gap to LKH Total Time LKH (5k) 5.539 0.00% 1m 10.218 0.17% 8m 17.115 0.34% 33m LKH (10k) 5.539 0.00% 3m 10.200 0.00% 16m 17.057 0.00% 67m Heter-POMO (gr.) 5.636 1.75% (0s) 10.540 3.33% (1s) 17.583 3.08% (2s) Heter-POMO-A (gr.) 5.567 0.51% (1s) 10.353 1.50% (5s) 17.276 1.29% (10s) Heter-POMO-A (4k) 5.545 0.12% (10m) 10.209 0.09% (48m) 16.890 -0.98% (180m) N2S-A (1k) 5.539 0.00% (3m) 10.144 -0.55% (13m) 16.976 -0.47% (54m) N2S-A (2k) 5.539 0.00% (6m) 10.137 -0.62% (27m) 16.859 -1.16% (107m) N2S-A (3k) 5.539 0.00% (9m) 10.135 -0.64% (40m) 16.806 -1.47% (161m) Table 4: Comparison results for solving PDTSP-LIFO instances of sizes |V | = 21, 51, and 101. that even with only 1k steps, our N2S-A attains a signi\ufb01cantly smaller gap than all three Heter-POMO variants, by almost an order of magnitude. Although Heter-POMO-A (gr.) tends to be competitive with fast speed, the gap is hard to be further reduced by increasing inference time if we refer to HeterPOMO-A (3k). Moreover, our N2S-A (2k) keeps abreast of, or even slightly exceeds the strong LKH3 solver, achieving gaps of -0.03% and -0.01% with less time on PDTSP-51 and PDTSP-101, respectively. Those gaps are further reduced to -0.04% and -0.20% with more steps, i.e., 3k. Results on PDTSP-LIFO. In Table 4, we report the results on PDTSP-LIFO. Due to the more constrained search space, DACT failed to work well (see Table 6). Therefore, we mainly compare our N2S-A with Heter-POMO (the best neural baseline in Table 3) and the LKH3 solver. As exhibited, the advantages of neural methods over LKH3 have been further enhanced on this harder problem, where our approach consistently outperforms Heter-POMO for all sizes. Compared to LKH3, our N2S-A with 3k steps presents superior performance again, and attains gaps of -0.64% and -1.47% on PDTSP-LIFO-51 and PDTSP-LIFO-101, respectively. 5.3 Ablation Evaluation Effects of Different Encoding Methods. We replace the proposed Synth-Att in our N2S encoder with vanilla-Att and DAC-Att, respectively. Using only one GPU card, we report Att. in Encoders Dim. # Param.(M) Time(s) Gap(%) Vanilla-Att 64 0.18 (1.00\u00d7) 239 (1.00\u00d7) 4.66 DAC-Att 0.31 (1.72\u00d7) 285 (1.19\u00d7) 2.89 Synth-Att 0.19 (1.06\u00d7) 255 (1.07\u00d7) 2.88 Vanilla-Att 128 0.72 (1.00\u00d7) 322 (1.00\u00d7) 3.92 DAC-Att 1.25 (1.73\u00d7) 400 (1.24\u00d7) 2.43 Synth-Att 0.76 (1.06\u00d7) 340 (1.06\u00d7) 2.40 Table 5: Effects of different encoding methods. the number of model parameters, time, and gaps for solving 2,000 PDTSP-51 instances with 3k steps in Table 5. As exhibited, Synth-Att attains slightly smaller gaps than DAC-Att with much fewer computation costs, and considerably lower gaps than vanilla-Att with slight extra computation costs. Effects of Different Decoding Methods. To highlight the desirability of our two decoders, we replace the trainable decoders with hand-crafted ones (i.e., random and the \u03f5-greedy with \u03f5 = 0.1) while ensuring feasibility. We gather the results on PDTSP-51 and PDTSP-LIFO-51 with 3k steps in Table 6 where mark \u2018\u0013\u2019 means our proposed decoder is used (the one without augments) and mark \u2018\u0017\u2019 means the decoder in the parentheses is used instead. As revealed, the methods of retaining at least one trainable decoder always attain much lower gaps than the ones equipped with only hand-crafted decoders. The method with both trainable decoders (i.e., N2S) \f(a) N201P3 (b) N201P6 (c) N201P7 Figure 4: Example instances from [Renaud et al., 2002]. (a) brd14051 101 (b) pr1002 101 (c) fnl4461 101 Figure 5: Example instances from [Carrabs et al., 2007]. Removal Decoder Reinsertion Decoder Gap(%) on PDTSP Gap(%) on PDTSP-LIFO \u0017(random) \u0017(random) 210.82 112.37 \u0017(random) \u0017(\u03f5-greedy) 18.03 17.87 \u0017(\u03f5-greedy) \u0017(random) 86.31 47.57 \u0017(\u03f5-greedy) \u0017(\u03f5-greedy) 15.62 12.64 \u0013 \u0017(random) 43.12 17.75 \u0013 \u0017(\u03f5-greedy) 3.54 3.88 \u0017(random) \u0013 6.38 4.38 \u0017(\u03f5-greedy) \u0013 8.42 6.28 \u0017(DACT) \u0017(DACT) 2.83 11.73 \u0013 \u0013 2.40 0.64 Table 6: Effects of different decoding methods. achieves the best performance. We also notice that DACT fail to solve PDTSP-LIFO well. This might be because its decoder considers removing and reinserting only one node instead of a pickup-delivery node pair in each action, which is a serious limitation for more constrained PDPs. For example, on PDTSP-LIFO, over 92% of the action space need to be masked for DACT, which leads to extremely low ef\ufb01ciency. 5.4 Generalization Evaluation We further evaluate our N2S on benchmark instances, including all the ones from [Renaud et al., 2002] for PDTSP and the ones with size |V | \u2264201 from [Carrabs et al., 2007] for PDTSP-LIFO, which are largely different from our training ones, e.g., different sizes (i.e., 200 nodes) as shown in Figure 4 and different node distributions as shown in Figure 5. In Table 7, we report the best and the average gaps (with 10 independent runs) achieved by N2S-A and neural baseline Heter-POMO-A w.r.t. optimal solutions for PDTSP, or heuristic baseline B1 [Carrabs et al., 2007] and B2 [Li et al., 2011] for PDTSP-LIFO. It can be seen that our N2S signi\ufb01cantly outstrips Heter-POMO in all cases. Without re-training or leveraging other techniques, this is fairly hard to achieve because machine learning often suffers from a mediocre outof-distribution zero-shot generalization [Zhou et al., 2021; Li et al., 2021a]. The results also imply slight inferiority of our N2S to the LKH3 solver, given that LHK3 reports similar performance to the B2 baseline on those instances [Helsgaun, 2017]. Accordingly, we will focus on improving the out-ofdistribution generalization for our N2S in future. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2110.02544v3",
+                    "title": "Learning to Iteratively Solve Routing Problems with Dual-Aspect Collaborative Transformer",
+                    "abstract": "Recently, Transformer has become a prevailing deep architecture for solving\nvehicle routing problems (VRPs). However, it is less effective in learning\nimprovement models for VRP because its positional encoding (PE) method is not\nsuitable in representing VRP solutions. This paper presents a novel Dual-Aspect\nCollaborative Transformer (DACT) to learn embeddings for the node and\npositional features separately, instead of fusing them together as done in\nexisting ones, so as to avoid potential noises and incompatible correlations.\nMoreover, the positional features are embedded through a novel cyclic\npositional encoding (CPE) method to allow Transformer to effectively capture\nthe circularity and symmetry of VRP solutions (i.e., cyclic sequences). We\ntrain DACT using Proximal Policy Optimization and design a curriculum learning\nstrategy for better sample efficiency. We apply DACT to solve the traveling\nsalesman problem (TSP) and capacitated vehicle routing problem (CVRP). Results\nshow that our DACT outperforms existing Transformer based improvement models,\nand exhibits much better generalization performance across different problem\nsizes on synthetic and benchmark instances, respectively.",
+                    "authors": "Yining Ma, Jingwen Li, Zhiguang Cao, Wen Song, Le Zhang, Zhenghua Chen, Jing Tang",
+                    "published": "2021-10-06",
+                    "updated": "2022-12-01",
+                    "primary_cat": "cs.LG",
+                    "cats": [
+                        "cs.LG",
+                        "cs.AI"
+                    ],
+                    "main_content": "Introduction Vehicle Routing problems (VRPs), such as the Traveling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) which consider \ufb01nding the optimal route for a single or \ufb02eet of vehicles to serve a set of customers, have ubiquitous real-world applications [1, 2]. Despite being intensively studied in the Operations Research (OR) community, VRPs still remain challenging due to their NP-hard nature [3]. Recent studies on learning neural heuristics are gathering attention as promising extensions to traditional hand-crafted ones (e.g., [4\u201314]), where reinforcement learning (RL) [15] is usually exploited to train a deep neural network as an ef\ufb01cient solver without hand-crafted rules. A salient motivation is that deep neural networks may learn better heuristics by identifying useful patterns in an end-to-end and data-driven fashion. Solutions to VRPs, i.e., routes, are sequences of nodes (customer and depot locations). Naturally, deep models for Natural Language Processing (NLP), which deal with sequence data as well, are ideal \u2217Zhiguang Cao and Wen Song are the corresponding authors. 35th Conference on Neural Information Processing Systems (NeurIPS 2021). arXiv:2110.02544v3  [cs.LG]  1 Dec 2022 \f1 2 3 4 Enhanced NFE Enhanced PFE 1 2 3 4 Aspect 1 Aspect 2 0      2      3      1      4      0 0      2      1 3 4      0 \ud835\udc82\ud835\udc95 0 0 Aspect 1 Aspect 2 0      2      3      1      4      0 DAC Encoder DAC Decoder \u00d7\ud835\udc73 Decoder Enhanced Node Embedding Encoder (\ud835\udc4e) (\ud835\udc4f) \ud835\udc82\ud835\udc95 1 2 3 4 0 1 2 3 4 0 \u00d7\ud835\udc73 Node Embedding PE Vector + Node Feature Embedding (NFE) Positional Feature Embedding (PFE) (a) 1 2 3 4 Enhanced NFE Enhanced PFE 1 2 3 4 Aspect 1 Aspect 2 0      2      3      1      4      0 0      2      1 3 4      0 \ud835\udc82\ud835\udc95 0 0 Aspect 1 Aspect 2 0      2      3      1      4      0 DAC Encoder DAC Decoder \u00d7\ud835\udc73 Decoder Enhanced Node Embedding Encoder (\ud835\udc4e) (\ud835\udc4f) \ud835\udc82\ud835\udc95 1 2 3 4 0 1 2 3 4 0 \u00d7\ud835\udc73 Node Embedding PE Vector + Node Feature Embedding (NFE) Positional Feature Embedding (PFE) (b) Figure 1: Transformer frameworks for VRPs. (a) Wu et al. [11] (the original one); (b) DACT (ours). choices for encoding VRP solutions. Given its remarkable performance in NLP tasks, Transformer [16] is standing at the forefront in the learning based methods for VRPs (e.g., [5, 7, 8, 11\u201313, 17]). The original Transformer encodes a sentence, i.e., a sequence of words, into a uni\ufb01ed set of embeddings by injecting word positional information into its word embeddings through positional encoding (PE). When it comes to VRPs, while is not required in construction models, positional information is critical for deep models that learn improvement heuristics since the input are solutions to be improved. Although some success has been achieved, learning improvement heuristics for VRPs based on the original Transformer encoder is yet lacking from our perspective. Firstly, directly applying addition operation on PE vectors and the embeddings in absolute PE method (i.e., Figure 1(a)) could limit the representation of the model [18], as the mixed correlations2 existing in the self-attention can bring unreasonable noises and random biases to the encoder (details in Appendix A). Secondly, existing PE methods tend to fuse the node and positional information into one uni\ufb01ed representation. NLP tasks such as translation may bene\ufb01t from this owing to the deterministic and instructive nature of the positional information. However, such design may not be optimal for routing tasks because the positional information therein can be non-deterministic and sometimes even random. This may cause disharmony or disturbance in the encoder and may thus deteriorate the performance. Finally, most VRPs seek the shortest loop of the nodes, making their solutions to be cyclic sequences. However, existing PE methods are only designated to encode linear sequences3, which may fail to identify such circular input. As will be shown in our experiments, this could severely damage the generalization performance, since the cyclic feature of VRP solutions is not correctly re\ufb02ected by the encoder. In this paper, we address the above issues and contribute to the line of using RL to learn neural improvement heuristics for VRPs. We introduce the Dual-Aspect Collaborative Transformer (DACT), where we revisit the solution representations and propose to learn separated groups of embeddings for the node and positional features of a VRP solution as shown in Figure 1(b). Our DACT follows the encoder-decoder structure. In the encoder, each set of embeddings encodes the solution mainly from its own aspect, and at the same time exploits a cross-aspect referential attention mechanism for better perceiving the consistence and differentiation with respect to the other aspect. The decoder then collects action distribution proposals from the two aspects and synthesizes them to output the \ufb01nal one. Meanwhile, we design a novel cyclic positional encoding (CPE) method to capture the circularity and symmetry of VRP solutions, which allows Transformer to encode cyclic inputs, and also boost the generalization performance for solving VRPs. As the last contribution, we design a simple yet effective curriculum learning strategy to improve the sample ef\ufb01ciency. This further leads to faster and more stable convergence of RL training. Extensive experiments show that our DACT can outperform existing Transformer based improvement models with fewer parameters, and also generalizes well across different sizes of synthetic and benchmark instances, respectively. 2The term correlation refers to the dot product between Query and Key in the self-attention module. The term mixed correlation refers to the case where Query and Key are projected from different types of embeddings. 3The relative PE method seems to help, however, it is found to be even worse than the absolute PE method for VRPs in Wu et al. [11], partly due to the disharmony issue caused by learning the uni\ufb01ed representation. 2 \f2 Related work 2.1 Positional encoding (PE) in Transformer. The original Transformer adopted the absolute PE method to describe the absolute position of elements in the sequence [16], especially for NLP. As formulated in Eq. (1), each generated positional embedding pi \u2208Rd is added together with the i-th word embedding xi in the \ufb01rst layer of the encoder, \u03b1Abs i,j = 1 \u221a d ((xi + pi)W Q)((xj + pj)W K)T . (1) The relative PE method was further proposed in Shaw et al. [19] to better capture the relative order information. On the basis of absolute PE, it introduces an inductive bias to the attention as follows, \u03b1Rel i,j = 1 \u221a d ((xi + pi)W Q)((xj + pj)W K + aj\u2212i)T , (2) where aj\u2212i \u2208Rd is learnable parameters for encoding the relative position j\u2212i. To avoid the mixed and noisy correlations between word semantics and positional information in the above two PEs, the Transformer with United Positional Encoding (TUPE) [18] was proposed for NLP which utilizes separated projection metrics Wx and Wp for each information as follows, \u03b1TUPE i,j = 1 \u221a 2d (xiW Q x )(xjW K x )T + 1 \u221a 2d (piW Q p )(pjW K p )T + bj\u2212i. (3) However, as mentioned previously, existing PE methods are less effective for VRPs since they simply fuse the node and positional information into one uni\ufb01ed set of embeddings during or after the calculation of the attention correlation \u03b1i,j. Meanwhile, they are also unable to properly encode and handle cyclic input sequences as in VRP solutions. 2.2 Deep models for VRP. Various deep architectures such as Recurrent Neural Network (RNN), Graph Neural Network (GNN), and Transformer have been employed in solving VRPs. RNN based models. As the pioneering work of neural VRP solvers, Pointer Network adopted RNN and supervised learning to solve TSP [20] (extended to RL in Bello et al. [21] and CVRP in Nazari et al. [22]). While the models in [20\u201323] learn construction heuristics, NeuRewriter [4] learns improvement heuristic for CVRP using LSTM to encode the positional information of a solution. In Hottung et al. [13], the conditional variational autoencoder was adopted to learn a continuous and latent search space for VRP, where high-quality solutions were taken as input and encoded by RNNs. However, recurrence structures in RNN are less ef\ufb01cient in both representation and computation [5]. GNN based models. In Dai et al. [24], GNN was combined with Q-learning for solving TSP. Based on supervised learning, Joshi et al. [6] used GNN to learn heatmaps that prescribe the probability of each edge appearing in the optimal TSP tour. This idea was extended in Fu et al. [25] with additional components such as graph sampling and heatmap merging to enable generalization to larger TSP instances. These models often require post-processing to construct feasible solutions from heatmaps (e.g., beam search [6], Monte-Carlo tree search [25], and dynamic programming [26]). Transformer based models. The Attention Model (AM) by Kool et al. [5] was recognized as the \ufb01rst success of Transformer based models for VRPs. Based on AM, Xin et al. [7] proposed a MultiDecoder AM that learns multiple diverse policies for better performance. In Kwon et al. [8], the RL algorithm of AM was improved which leaded to a new solver, i.e., POMO (Policy Optimization with Multiple Optima), and achieved the state-of-the-art performance. However, POMO is still lacking in generalization. Besides these construction models, Transformer was also explored to learn improvement heuristics. Hottung and Tierney [27] learned \ufb01rst neural large neighborhood search algorithm for VRPs. Lu et al. [12] proposed the L2I model that learns to select local search operators from a pool of traditional ones. Both methods used a Transformer-style encoder, but the positional information is captured in the node features (information of previous and next nodes) instead of using PE methods. Though L2I was shown to outperform LKH3 [28], it is limited to CVRP and the required time is considerably long. Wu et al. [11] proposed a Transformer model which learns to pick node pair in each step to perform a pairwise local operator (e.g., 2-opt). However, it suffers from the inaccurate representation of positional information given the original Transformer encoder. 3 \f\ud835\udc65! \ud835\udc65\" \ud835\udc65# \u2026 \ud835\udc54! \ud835\udc54\" \ud835\udc54# \u2026 Linear Projection \ud835\udc54! (#$!)\ud835\udc54& (#$!) \ud835\udc54' (#$!) \u2026 \ud835\udc54\" (%) \ud835\udc54# (%) \u2026 \ud835\udc54! (%) Dual-Aspect Collaborative Attention Add & Norm Add & Norm Feed Forward Feed Forward Add & Norm Add & Norm (using CPE) NFEs PFEs DAC Encoder Max Pooling Max Pooling # \u210e! # \u210e\" # \u210e# \u2026 % \ud835\udc54! % \ud835\udc54\" % \ud835\udc54# \u2026 Multi-head Compatibility Multi-head Compatibility Feed Forward Mask & Softmax DAC Decoder L\u00d7 \u2026 \u2026 Linear Linear Linear Linear Matmul & Scale Softmax Matmul Matmul \ud835\udc4a \" # Concat & Linear Projection Linear Linear Linear Linear Matmul & Scale Softmax Matmul Matmul Concat & Linear Projection Multi-Head DAC-Att Cross-Aspect Referential Attention \u210e! (#$!)\u210e& (#$!) \u210e' (#$!) \u2026 \u210e\" (%) \u210e# (%) \u2026 \u210e! (%) \u210e! (#$!)\u210e& (#$!) \u210e' (#$!) \ud835\udc54! (#$!)\ud835\udc54& (#$!) \ud835\udc54' (#$!) \ud835\udc4a \" % \ud835\udc4a \" & \ud835\udc4a \" #'() \ud835\udc4a * #'() \ud835\udc4a * & \ud835\udc4a * % \ud835\udc4a * # \ud835\udc4c !! \" \ud835\udc4c !# \" \ud835\udc4c #! \" \ud835\udc4c ## \" \ud835\udc4c !! $ \ud835\udc4c !# $ \ud835\udc4c #! $ \ud835\udc4c ## $ \u2026 \ud835\udc43 !! \ud835\udc43 !# \ud835\udc43 #! \ud835\udc43 ## \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 \u2026 NFEs PFEs \ud835\udc5d! \ud835\udc5d\" \ud835\udc5d# \u2026 Figure 3: Architecture of our policy network, dual-aspect collaborative Transformer (DACT). 3 Problem formulation We de\ufb01ne a VRP instance as a group of N nodes to visit, where the node feature xi of node i contains 2-dim coordinates and other problem-speci\ufb01c features (e.g., customer demand). A solution \u03b4 consists of a sequence of nodes visited in order where we denote pi to be the position (indices) of node i in the solution which is deemed as the positional feature of node i. The objective is to minimize the total travel distance D(\u03b4) under certain problem-speci\ufb01c constraints. 0 2 3 1 4 0 0 1 3 2 4 0 2-\ud835\udc5c\ud835\udc5d\ud835\udc61 0 2 3 1 4 0 0 3 1 2 4 0 \ud835\udc56\ud835\udc5b\ud835\udc60\ud835\udc52\ud835\udc5f\ud835\udc61 0 2 3 1 4 0 0 1 3 2 4 0 \ud835\udc60\ud835\udc64\ud835\udc4e\ud835\udc5d Figure 2: Illustration examples of three pairwise operators for routing problems when node pair (i = 2, j = 1) is speci\ufb01ed for operating. From left to right: 2-opt, insert, and swap. Starting with an initial yet complete solution, our neural RL policy tries to improve the solution iteratively. At each step, the policy automatically selects a pair of nodes and locally adjusts the solution using a preset pairwise operator such as 2-opt, insert, or swap. As illustrated in Figure 2, given a node pair (i, j), the 2-opt operator adjusts a solution by reversing the segment between node i and node j; the insert operator adjusts a solution by placing node i after node j; and the swap operator adjusts a solution by exchanging the position of node i and node j. Such operation is repeated until reaching the step limit T and we model it in the form of Markov Decision Process (MDP) as follows. State. For an instance with N nodes, a state describes current solution \u03b4t using its node and positional features of each node, i.e., st = \u03a8(\u03b4t) = {xt 1, ..., xt N, pt 1, ..., pt N}. Action. The action at = (i, j) speci\ufb01es a node pair (i,j) for the pairwise operator. Reward. The reward function is de\ufb01ned as, rt = D(\u03b4* t)\u2212min \u0002 D(\u03b4t+1), D(\u03b4* t) \u0003 where \u03b4* t is the best incumbent solution found until time t. It refers to the immediate reduced cost at each step with respects to the best incumbent solution, which ensures the cumulative reward equal to the total reduced cost over the initial solution. Hence the reward rt >0 if and only if a better solution is found. Policy. The policy \u03c0\u03b8 is parameterized by the proposed DACT model with parameters \u03b8. At each time step, the action (i, j) is obtained by sampling the stochastic policy for both training and inference. Transition. The next state st+1 is originated from st by performing the preset pairwise operator on the given node pair (action). Our state transient is deterministic, in the sense that it always accepts the next solution as the next state (infeasible solutions will be masked), regardless of its objective value. With such simple rule, the RL agent is expected to automatically learn how to combine multiple steps of simple local movements to achieve better solutions, even if some of them may worsen the current solution. Note that the step limit T can be any user-speci\ufb01ed value according to the allowed time budget. Hence, our MDP can have in\ufb01nite horizon and we consider the reward discount factor \u03b3 <1. Besides, we stipulate that if Tr (a hyper-parameter) subsequent steps do not yield a better solution, we will reinitialize the search by assigning the best solution found thus far as the next solution to move on. 4 \f0 16 32 48 64 PE 0 5 10 15 20 0 16 32 48 64 CPE (Ours) 0 5 10 15 20 0 5 10 15 PE 0 5 10 15 20 0 5 10 15 CPE (Ours) 0 5 10 15 20 4 2 0 2 4 PE 3 2 1 0 1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4 2 0 2 4 CPE (Ours) 4 2 0 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (a) 0 16 32 48 64 PE 0 5 10 15 20 0 16 32 48 64 CPE (Ours) 0 5 10 15 20 0 5 10 15 20 PE 0 5 10 15 20 0 5 10 15 20 CPE (Ours) 0 5 10 15 20 4 2 0 2 4 PE 3 2 1 0 1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4 2 0 2 4 CPE (Ours) 4 2 0 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (b) 0 16 32 48 64 PE 0 5 10 15 20 0 16 32 48 64 CPE (Ours) 0 5 10 15 20 0 5 10 15 PE 0 5 10 15 20 0 5 10 15 CPE (Ours) 0 5 10 15 20 4 2 0 2 4 PE 3 2 1 0 1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4 2 0 2 4 CPE (Ours) 4 2 0 2 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 (c) Figure 5: Comparison of our CPE method with absolute PE method on a TSP instance with 20 nodes. (a) the embedding vectors, (b) the correlations (dot products) between every two embeddings, and (c) the top two principal components after PCA (principal component analysis) projection. 4 Dual-aspect collaborative Transformer model We now present the details of our Dual-Aspect Collaborative Transformer (DACT). The concrete architecture of DACT is presented in Figure 3, where we take the TSP with N nodes as an illustration example. Our DACT leverages separate aspects of embeddings to encode a VRP solution. In the DAC encoder, the self-attention correlations are computed individually for each aspect, and a cross-aspect referential attention mechanism is proposed to enable one aspect to effectively exploit attention correlations from the other aspect as optional references. The DAC decoder then collects action distribution proposals from both aspects and synthesize them to the \ufb01nal one. 4.1 Dual-aspect solution representation Speci\ufb01cally, we propose to learn two sets of embeddings, i.e., the node feature embeddings (NFEs) for node representation and the positional feature embeddings (PFEs) for positional representation. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 \ud835\udc41 1 \ud835\udc51=1 \ud835\udc51=2 \ud835\udc51=3 \ud835\udc51=4 1 0 0 1 Base symmetry pattern: (Gray code) (Ours) Node index: Digits Figure 4: An example of cyclic Gray code where 4 digits are used to encode N =16 nodes. The top left shows the base symmetry pattern \u20181001\u2019 in Gray code, and the top right plots its representation in our method. NFEs. Following [5, 11], the NFE hi of node i is initialized as the linear projection of its node feature xi with output dimension4 dim = 64. PFEs. The PFE gi of the positional feature pi is initialized as a real-valued vector (dim = 64) by applying our cyclic positional encoding (CPE), which is designed based on cyclic Gray codes [29]. As illustrated in Figure 4, the cyclic Gray codes present a cyclic property (\u20181110\u2019 in the last column is adjacent to \u20181111\u2019 in the \ufb01rst column) and an adjacency similarity property (any codes in adjacent columns only differ in one digit), both of which are desirable for cyclic sequences. To preserve these properties in designing our CPE, we follow two observed patterns: 1) each numerical digit contains a periodic cycle with re\ufb02ectional symmetry, e.g., the \u201810|01\u2019 in the lowest digit; and 2) the higher the numerical digit, the longer the period. Accordingly, we create similar patterns based on the sinusoidal functions in Eq. (4), where a periodic function with period 4\u03c0 \u03c9d (induced by modulus) is used to generate one base symmetry pattern (the top right in Figure 4), \u2212 \u2192 gi (d) := \uf8f1 \uf8f2 \uf8f3 sin(\u03c9d\u00b7 \f \f \f(z(i) mod 4\u03c0 \u03c9d ) \u22122\u03c0 \u03c9d \f \f \f), if d is even cos(\u03c9d\u00b7 \f \f \f(z(i) mod 4\u03c0 \u03c9d ) \u22122\u03c0 \u03c9d \f \f \f), if d is odd (4) where z(i) = i N 2\u03c0 \u03c9d l N 2\u03c0/\u03c9d m is to make N nodes linearly spaced in the generated pattern; the angular frequency \u03c9d is decreasing along the dimension to make the wavelength longer within the range [N 1 \u230adim/2\u230b, N] (see Appendix B for details). In Figure 5, we visualize the comparison between the absolute PE and our CPE for encoding a TSP instance of 20 nodes. Figure 5(a) demonstrates that our real-valued base symmetry pattern has a longer cyclic period as the digit grows. Figure 5(b) indicates that our method (blue) is able to correctly re\ufb02ect the adjacency between the head and tail of the cyclic 4Different from Wu et al. [11], we reduce the dimension of the embeddings from 128 to 64. 5 \fsequence whereas the PE method (red) fails to do so. Figure 5(c) veri\ufb01es that our CPE vectors are well distributed in space with desired cyclic and adjacency similarity properties. 4.2 The encoder The encoder consists of L = 3 stacked DAC encoders. In each DAC encoder, we retain relatively independent encoding stream for NFEs and PFEs as in Eq. (5) and Eq. (6), respectively, each of which consists of a shared Dual-Aspect Collaborative Attention (DAC-Att) sub-layer and an independent feed-forward network (FFN) sub-layer. DAC-Att takes both sets of embeddings as input and then outputs their respective enhanced embeddings, i.e., NFEs {\u02dc h}N i=1 and PFEs {\u02dc g}N i=1. Each sub-layer is followed by skip connection [30] and layer normalization [31] as same as the original Transformer. h(l) i =LN \u0010 h\u2032 i + FFN(l) h (h\u2032 i) \u0011 , h\u2032 i =LN \u0010 h(l \u2212 1) i + \u02dc h(l) i \u0011 , (5) g(l) i =LN \u0010 g\u2032 i + FFN(l) g (g\u2032 i) \u0011 , g\u2032 i =LN \u0010 g(l \u2212 1) i + \u02dc g(l) i \u0011 . (6) DAC-Att. The DAC-Att sub-layer enhances each set of embedding from its own aspect, while leveraging attention correlations from the other aspect to achieve the synergy. Given the two sets of embeddings5, {hi}N i=1 and {gi}N i=1, we \ufb01rst compute the self-attention correlation from both aspects, \u03b1h i,j = 1 \u221adk \u0010 hiW Q h \u0011 \u0000hjW K h \u0001T , \u03b1g i,j = 1 \u221adk \u0000giW Q g \u0001 \u0000gjW K g \u0001T , (7) where independent matrices W Q h , W K h , W Q g and W K g \u2208Rdim\u00d7dk are used to calculate queries and keys. The obtained correlations are further normalized to \u02dc \u03b1h i,j and \u02dc \u03b1g i,j via Softmax. Note that the correlations are computed from their own aspect, which eliminates possible noises and conduces to correctly describe the incompatible node pair relationships in different aspects of VRP solutions. We then exploit a cross-aspect referential attention mechanism, which allows computed correlations to be shared between each other, as additional references for both contradistinction and collaboration, outh i =Concat \uf8ee \uf8f0 N X j=1 \u02dc \u03b1h i,j \u0000hjW V h \u0001 , N X j=1 \u02dc \u03b1g i,j \u0010 hjW Vref h \u0011 \uf8f9 \uf8fb, (8) outg i =Concat \uf8ee \uf8f0 N X j=1 \u02dc \u03b1g i,j \u0000gjW V g \u0001 , N X j=1 \u02dc \u03b1h i,j \u0000gjW Vref g \u0001 \uf8f9 \uf8fb, (9) where W V h , W V g \u2208Rdim\u00d7dv are trainable parameter matrices for formulating values in each aspect; and W Vref h , W Vref g \u2208Rdim\u00d7dv are parameter matrices for each aspect to generate referential values. We \ufb01nally use the multi-head attention to get NFEs \u02dc hi and PFEs \u02dc gi as follows, \u02dc hi, \u02dc gi = DAC-Att \u0000W Q, W K, W V , W Vref, W O\u0001 , \u02dc hi = Concat h headh i,1, ..., headh i,m i W O h , \u02dc gi = Concat \u0002 headg i,1, ..., headg i,m \u0003 W O g , (10) where headh i,k = outh i,k, headg i,k = outg i,k, and W O h , W O g \u2208R2mdv\u00d7dim are trainable parameter matrices. In our model, we adopt m = 4 and dk = dv = 16. FFN. Our FFN sub-layer has only one hidden layer with 64 hidden unites and adopts the ReLU activation function. The parameters of FFNh and FFNg are different for each group of embeddings. 4.3 The decoder In the DAC decoder, the two sets of embeddings {h(L) i }N i=1 and {g(L) i }N i=1 are \ufb01rst passed through a Max-pooling sub-layer and a multi-head compatibility (MHC) sub-layer to independently generate 5We omit the encoder index l for better readability. 6 \fdiversi\ufb01ed node-pair selection proposals from their own aspect, which are then aggregated through a feed-forward aggregation (FFA) sub-layer for output. Max-pooling. For each set of embeddings, we adopt the max-pooling sub-layer in Wu et al. [11] to aggregate the global representation of all N embeddings into each respective one6. MHC. The compatibility sub-layer computes the attention correlations for each embedding pair, where the obtained correlations with size N \u00d7 N will be deemed as a proposal distribution for node pair selection. Our correlations are computed based on multiple heads for diversity. And we calculate separated attention score matrices Y h k , Y g k \u2208RN\u00d7N (of head k) from the two aspects independently. Accordingly, the action distribution proposals would be different due to their aspect-speci\ufb01c focus and cognitions of the current solution, which will provide the subsequent FFA layer with a rich pool of proposals and allow our model to be more \ufb02exible and robust. FFA. Once all proposals from two aspects are collected, a FFN with four layers (dimensions are 2m, 32, 32 and 1, respectively) and ReLU activation is used to aggregate them, \u02dc Yi,j = FFA \u0000Y g i,j,1, ..., Y g i,j,m, Y h i,j,1, ...Y h i,j,m \u0001 , (11) where m = 4 is the number of heads; and the output \u02dc Yi,j is a scalar indicating the likelihood of selecting node pair (i, j) as an action. Afterwards, we apply \u02c6 Yij = C \u00b7 Tanh( \u02dc Yi,j) with C = 6 to control the entropy, and mask 7 the infeasible node pairs (i\u2032, j\u2032) as \u02c6 Yi\u2032j\u2032 =\u2212\u221e. Lastly, the likelihoods are normalized using Softmax function to obtain the \ufb01nal action distribution Pi,j. 4.4 Reinforcement learning algorithm We adopt the proximal policy optimization [32] with n-step return estimation for training (details are given in Appendix C), and design a curriculum learning (CL) strategy for better sample ef\ufb01ciency. Curriculum learning strategy. The strategy in Wu et al. [11] sets a maximum of Ttrain steps for training and estimates future returns by bootstrapping [33]. However, due to the concern of training cost, Ttrain is usually much smaller than actual T for inference (e.g., 200 v.s. 10k), which may leave the agent a poor chance of observing high-quality solutions (states) during training. Consequently, it may cause high variance for bootstrapping because the value function is mostly \ufb01tted on low-quality solutions and may render it less knowledgeable in estimating long-term future returns accurately. In this paper, we tackle this issue by a simple yet ef\ufb01cient strategy which gradually prescribes higher-quality solutions as the initial states for training. In doing so, 1) it increases the probability for the agent to observe better solutions and thus reduce the variance of the value function; 2) it increases the dif\ufb01culty of the learning task (higher-quality solutions are harder to improve) in a gradual manner and achieves better sample ef\ufb01ciency [34]. In practice, those higher-quality solutions can be easily achieved by improving the randomly generated ones using the current policy for a few Tinit steps, where Tinit could be slightly increased as the epoch grows. 5 Experiments We evaluate our DACT model on two representative routing problems, i.e., TSP and CVRP [5, 8, 11]. For each problem, we abide by existing conventions to randomly generate instances on the \ufb02y for three sizes, i.e., N = 20, 50 and 100. Initial experiments with three operators including 2-opt, swap and insert show that 2-opt performs best for both TSP and CVRP (with insert better than swap), hence we report results of our method based on 2-opt. Following [4, 11, 27] we use randomly generated initial solutions for training and the solutions generated by the greedy algorithm for inference. Since each problem has its own constraints and node features, we adjust the input, feasibility masks, and problem-dependent hyperparameters for each problem, the details of which are provided in Appendix D and E. The DACT models and baselines are tested on a server equipped with RTX 2080 TI GPU cards and Intel Xeon E5-2680 CPU @ 2.40GHz. Our code in PyTorch are available here8. 6E.g., for \u02c6 h, \u02c6 hi =h(L) i W Local h +max h {h(L) i }N i=1 i W Global h , where W Local h ,W Global h \u2208R64\u00d764 are parameters. 7Besides, we also mask all the diagonal elements since they are not meaningful to the pair-wise operators, and the node pair selected at the last step to forbid possible dead loops [11]. 8https://github.com/yining043/VRP-DACT 7 \fTable 1: Comparison with various baselines on TSP and CVRP. Method N=20 N=50 N=100 Obj. Gap Time\u2217 Obj. Gap Time\u2217 Obj. Gap Time\u2217 TSP Concorde [35] 3.83 5.70 7.76 LKH [36] 3.83 0.00% (6m) 5.70 0.00% (1.5h) 7.76 0.00% (6h) OR-Tools [37] 3.86\u2021 0.94%\u2021 (1m)\u2021 5.85\u2021 2.87%\u2021 (5m)\u2021 8.06\u2021 3.86%\u2021 (23m)\u2021 Neural-2-Opt [23] (T=3k) 3.83 0.00% (47m) 5.70 0.10% (1h) 7.81 0.61% (1.5h) Neural-2-Opt [23] (T=6k) 3.83 0.00% (1.5h) 5.70 0.06% (2h) 7.79 0.39% (3.5h) Wu et al. [11] (T=1k) 3.83\u2021 0.03%\u2021 (12m)\u2021 5.74 0.82% (16m) 8.02 3.31% (24m) Wu et al. [11] (T=5k) 3.83\u2021 0.00%\u2021 (1h)\u2021 5.71 0.24% (1.5h) 7.88 1.54% (2h) DACT (T=1k) 3.83 0.04% (2m)[40s] 5.70 0.14% (7m)[2m] 7.89 1.62% (20m)[5m] DACT (T=5k) 3.83 0.00% (10m)[3m] 5.70 0.02% (32m)[8m] 7.81 0.61% (1.5h)[25m] DACT (T=10k) 3.83 0.00% (21m)[7m] 5.70 0.01% (1h)[17m] 7.79 0.37% (3.5h)[50m] DACT\u00d74 augment 3.83 0.00% (1.5h)[28m] 5.70 0.00% (4h)[1h] 7.77 0.09% (14h)[3.5h] GCN-BS [6] 3.84\u2021 0.01%\u2021 (12m)\u2021 5.70\u2021 0.01%\u2021 (18m)\u2021 7.87\u2021 1.39%\u2021 (40m)\u2021 AM-sampling [5] 3.84\u2021 0.08%\u2021 (5m)\u2021 5.73\u2021 0.52%\u2021 (24m)\u2021 7.94\u2021 2.26%\u2021 (1h)\u2021 MDAM-BS [7] 3.84\u2021 0.00%\u2021 (3m)\u2021 5.70\u2021 0.03%\u2021 (14m)\u2021 7.79\u2021 0.38%\u2021 (44m)\u2021 POMO [8] 3.83\u2021 0.04%\u2021 (1s)\u2021 5.70\u2021 0.21%\u2021 (2s)\u2021 7.80 0.46% (12s) POMO\u00d78 augment [8] 3.83\u2021 0.00%\u2021 (3s)\u2021 5.69\u2021 0.03%\u2021 (16s)\u2021 7.78 0.19% (2m) CVAE-Opt-DE [13] \u22480.00% (\u22481d) \u22480.02% (\u22482d) \u22480.34% (\u22486d) DPDP [26] (100k) 7.77\u2021 0.00%\u2021 (3h)\u2021 CVRP LKH [28] 6.14 0.00% (18h) 10.38 0.00% (3d) 15.65 0.00% (6d) OR-Tools [37] 6.46\u2021 5.68%\u2021 (2m)\u2021 11.27\u2021 8.61%\u2021 (13m)\u2021 17.12\u2021 9.54%\u2021 (46m)\u2021 NeuRewriter [4] 6.15\u2021 (22m)\u2021 10.51\u2021 (35m)\u2021 16.10\u2021 (1h)\u2021 NLNS [27] (T=1k) 6.19 0.85% (9m)# 10.56 1.82% (14m)# 16.09 2.84% (31m)# NLNS [27] (T=10k) 6.17 0.60% (1.5h)# 10.49 1.12% (2.5h)# 15.88 1.49% (5h)# Wu et al. [11] (T=1k) 6.16\u2021 0.90%\u2021 (23m)\u2021 10.68 2.89% (50m) 16.39 4.76% (1h) Wu et al. [11] (T=5k) 6.12\u2021 0.39%\u2021 (2h)\u2021 10.54 1.63% (4h) 16.17 3.31% (5h) DACT (T=1k) 6.14 0.04% (4m)[1m] 10.55 1.73% (16m)[3m] 16.20 3.52% (41m)[5m] DACT (T=5k) 6.13 -0.07% (20m)[6m] 10.46 0.80% (1.5h)[13m] 15.95 1.92% (3.5h)[26m] DACT (T=10k) 6.13 -0.07% (40m)[12m] 10.43 0.58% (2.5h)[25m] 15.88 1.51% (7h)[52m] DACT\u00d76 augment 6.13 -0.08% (4h)[1h] 10.38 0.08% (16h)[2.5h] 15.74 0.57% (41h)[5h] AM-sampling [5] 6.25\u2021 1.87%\u2021 (6m)\u2021 10.62\u2021 2.40%\u2021 (28m)\u2021 16.23\u2021 3.72%\u2021 (2h\u2021) MDAM-BS [7] 6.14\u2021 0.18%\u2021 (5m)\u2021 10.48\u2021 0.98%\u2021 (15m)\u2021 15.99\u2021 2.23%\u2021 (1h)\u2021 POMO [8] 6.17\u2021 0.82%\u2021 (1s)\u2021 10.49\u2021 1.14%\u2021 (4s)\u2021 15.84 1.21% (21s) POMO\u00d78 augment [8] 6.14\u2021 0.21%\u2021 (5s)\u2021 10.42\u2021 0.45%\u2021 (26s)\u2021 15.75 0.69% (2m) CVAE-Opt-DE [13] \u22486.14 (\u22482d) \u224810.40 (\u22485d) \u224815.75 (\u224811d) DPDP [26] (100k) 15.69\u2021 0.31%\u2021 (6h)\u2021 \u2217We report the total time using one 2080 TI GPU or one CPU in (\u00b7), and the total time using multiple GPUs (4 for TSP, 8 for CVRP) in [\u00b7]. \u2021 The results (i.e., obj. values, gaps, and time) are adopted from its original paper or from Wu et al. [11]. # The run time of NLNS is based on one GPU and ten CPUs (in parallel) following its default setting. 5.1 Comparison studies In Table 1, we compare our DACT with, (1) learning based improvement methods, including Wu et al. [11], Neural-2-Opt [23] (TSP only), NeuRewriter [4] (CVRP only), NLNS [27] (CVRP only), (2) learning based construction methods, including AM-sampling [5], GCN-BS [6] (TSP only), MDAM-BS [7], POMO [8], (3) conventional optimization algorithms equipped with learning based component(s), including DPDP [26], CVAE-Opt-DE [13], and (4) strong conventional solvers including Concorde [35], LKH [28, 36], and OR-Tools [37]. We do not include L2I [12] as a baseline since it requires a prohibitively longer inference time than others9. All results are averaged over 10,000 randomly generated instances except for CVAE-Opt-DE (with \u2248) which only infers 1,000 instances, and we report the metrics of objective values, (optimality) gaps and run time. We test strong baselines POMO (TSP100, CVRP100), Neural-2-Opt, Wu et al. (except for size 20), and NLNS on our machine based on the pre-trained models. Other results (with \u2021) are adopted from their original papers or from Wu et al. [11]. For TSP, Concorde is adopted to get the optimal solutions; CVRP is harder to be solved optimally, and the gaps are based on objective values of LKH3. We report the total time using one GPU (for neural solvers) or one CPU (for conventional solvers) in (\u00b7), and the total time using multiple GPUs (for DACT) in [\u00b7]; for NLNS (with #), we use one GPU and ten CPUs (in parallel) following its default setting. Note that the run time (for CPU solvers, and those with \u2021 and #) may not be directly comparable with ours and hard to compare due to various factors (e.g., GPU v.s. CPU, Python v.s. C, different devices) and thus we focus more on gaps. Pertaining to TSP, our DACT with inference steps 1,000 (T=1k) and 5,000 (T=5k) signi\ufb01cantly outperforms the traditional solver OR-Tools and the baseline Wu et al. [11] which directly adopted 9L2I needs 167 days for 10,000 CVRP100 instances, as estimated from 24min/instance in its original paper. 8 \fTable 2: Generalization performance. (a) DACT v.s. baselines on benchmark datasets (up to 200 customers, see Appendix E.4 for detailed results and discussion); (b) PE v.s. CPE on different sizes. Method TSPLIB CVRPLIB OR-Tools [37] 3.34% 8.06% POMO [8] 10.06% 6.10% Wu et al. [11] 4.17% 5.20% DACT-Avg 1.90% 3.85% DACT-Best 1.01% 2.97% (a) Method N=20 N=100 Obj. Gap Obj. Gap DACT-PE (T=5k) 3.84 0.21% 8.38 7.93% DACT-CPE (T=5k) 3.83 0.10% 7.99 2.98% Wu et al. [11] (T=5k) 3.91 2.14% 9.03 16.37% OR-Tools [37] 3.83 0.00% 8.06 3.87% (b) Table 3: Dual v.s. single aspect representation Steps Method # Params N=50 N=100 T=1k SA-T 0.37M 0.35% (2m) 3.49% (4m) DACT 0.29M 0.14% (2m) 1.62% (5m) T=5k SA-T 0.37M 0.05% (7m) 1.55% (22m) DACT 0.29M 0.02% (8m) 0.61% (25m) the original Transformer encoder. The DACT (T=5k) also deliveries lower gaps than construction methods including AM-sampling and GCN-BS on TSP100. With larger steps T=10k, our DACT further boosts the solution qualities and outperforms other construction methods including MDAMBS (with beam search), and POMO (the current state-of-the-art). Besides, our DACT consistently outperforms improvement method Neural-2-Opt with shorter or similar run time. To further reduce the gaps, we also leverage the data augmentation technique in POMO (which considers \ufb02ipping node coordinates without changing the optimal solution) to solve same instances multiple times in different ways. Although the inference time increases (we run data augmentation in serial), our DACT with 4 augments not only outstrips POMO with 8 augments but also achieves the lowest objective values and gaps among all purely learning based models. In particular, we achieve the gap of 0.09% on TSP100, which is superior to most of the recent neural solvers. Pertaining to CVRP, our DACT with T=5k achieves better objective values than that of NeuRewriter. It also consistently outperforms the important baseline Wu et al. [11] in all cases. Compared with the NLNS solver that learned to control the large neighborhood search heuristics, the gaps achieved by DACT (T=10k) are smaller on CVRP20 and CVRP50, and highly comparable on CVRP100. With T=10k and 6 augments, our DACT exhibits even better performance than the highly specialized heuristic solver LKH on CVRP20 and delivers the smallest gap of 0.57% on CVRP100 against other neural solvers including POMO with 8 augments. Compared with hybrid solver CVAE-Opt-DE, despite that it is averaged over 1,000 instances and integrated with search algorithm like differential evolution, our objective values are still lower on both TSP and CVRP. Nevertheless, our DACT is inferior to a more advanced hybrid solver DPDP which leverages learnt heat-maps and traditional dynamic programming to search solutions. In terms of the inference time, our DACT is highly competitive against most of the neural solvers except for POMO which learns a construction model by sampling diverse trajectories. However, when it comes to the generalization performance on benchmark datasets, i.e., TSPLIB [38] and CVRPLIB [39] in Table 2(a), DACT produces signi\ufb01cantly lower average gaps than the POMO with 8 augments, which indicates that our DACT is more advantageous in practice despite its longer inference time. On the other hand, it is possible to adopt a similar diverse rollout strategy for DACT to \ufb01nd better solutions earlier, or explore other model compression techniques such as the knowledge distillation [40] to learn a lighter DACT model for faster inference. Since our focus is to ameliorate Transformer for neural improvement solvers, we will investigate these possibilities in the future. 5.2 Ablation studies Dual-aspect representation. In Table 3, we evaluate the effectiveness of our dual-aspect representation against the single-aspect one (SA-T) on TSP50 and TSP100, where SA-T mainly follows the Transformer in Wu et al. [11] but equipped with the CPE, multi-head attentions and CL strategy for fair comparison. We observe that our DACT with fewer parameters consistently outperforms SA-T, which veri\ufb01es the effectiveness of the dual-aspect representation. 9 \f0 5 10 15 20 TSP20 Policy tested on TSP20 (PE) 0 5 10 15 20 0 10 20 30 40 50 TSP20 Policy tested on TSP50 (PE) 0 10 20 30 40 50 0 5 10 15 20 TSP20 Policy tested on TSP20 (CPE) 0 5 10 15 20 0 10 20 30 40 50 TSP20 Policy tested on TSP50 (CPE) 0 10 20 30 40 50 (a) 0 5 10 15 20 TSP20 Policy tested on TSP20 (PE) 0 5 10 15 20 0 10 20 30 40 50 TSP20 Policy tested on TSP50 (PE) 0 10 20 30 40 50 0 5 10 15 20 TSP20 Policy tested on TSP20 (CPE) 0 5 10 15 20 0 10 20 30 40 50 TSP20 Policy tested on TSP50 (CPE) 0 10 20 30 40 50 (b) Figure 6: Visualization of the (multi-head) attention scores for the encoder when a trained model is used to solve instances with a larger size. (a) using PE method; (b) using CPE method (ours). 0 25 50 75 100 Epoch 6.2 6.3 6.4 6.5 6.6 Objective Value PPO PPO+CL Figure 7: Training curves of PPO with and without CL on CVRP20 (random seeds 1-5). Cyclic positional encoding. Here we show that CPE signi\ufb01cantly improves the generalization performance across different problem sizes. In Table 2(b), we record the results of our DACT with PE and CPE, and Wu et al. [11], when the model trained on TSP50 is directly used to solve instances from TSP20 and TSP100 with T=5k. We see that even with PE, our DACT outperforms Wu et al. [11]. Further equipped with CPE, DACT outstrips DACT-PE and OR-Tools on TSP100. We continue to compare the two DACT variants by visualizing their attention scores. As depicted in Figure 6(a), although the absolute PE is designed for linear sequences, it did attempt to capture the circularity of VRP solutions (as highlighted in the green boxes) after training. However, the ability to perceive such properties signi\ufb01cantly drops when generalizing over different problem size, which instead engenders random attention scores when generalizing to larger size (see right side of Figure 6(a)). In contrast, our DACT with CPE is able to capture the circularity as depicted in Figure 6(b), which veri\ufb01es the effectiveness of CPE in representing cyclic sequences (i.e., VRP solutions). Curriculum learning (CL) strategy. In Figure 7, we plot training curves of PPO algorithm with and without our CL strategy for solving CVRP, where the results are averaged over 5 independent runs with 90% con\ufb01dence intervals. It shows that our CL strategy signi\ufb01cantly improves the sample ef\ufb01ciency while reducing the variance of training, which aligns with our analysis in Section 4.4. 6"
+                }
+            ],
+            "Yeow Meng Chee": [
+                {
+                    "url": "http://arxiv.org/abs/2403.01454v1",
+                    "title": "Maximum Length RLL Sequences in de Bruijn Graph",
+                    "abstract": "A timing and synchronization system based on a de Bruijn sequence has been\nproposed and studied recently for a channel associated with quantum\ncommunication that requires reliable synchronization. To avoid a long period of\nno-pulse in such a system on-off pulses are used to simulate a zero and on-on\npulses are used to simulate a one. However, these sequences have high\nredundancy. To reduce the redundancy, run-length limited sequences in the de\nBruijn graph are proposed for the same purpose. The maximum length of such\nsequences in the de Bruijn graph is studied and an efficient algorithm to\nconstruct a large set of these sequences is presented. A maximum length\nsequence for which the position of each window can be computed efficiently is\nconstructed. Finally, an enumeration of the number of such sequences is given\nand some generalizations are discussed.",
+                    "authors": "Yeow Meng Chee, Tuvi Etzion, Tien Long Nguyen, Duy Hoang Ta, Vinh Duc Tran, Van Khu Vu",
+                    "published": "2024-03-03",
+                    "updated": "2024-03-03",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "Introduction In a free-space quantum key distribution, one of the goals is to synchronize the transmitted and received signals accurately. There are some e\ufb00orts in designing e\ufb03cient and reliable timing and synchronization systems [6, 18]. In [26] a de Bruijn sequence-based timing and synchronization system is introduced using a beacon with an on-o\ufb00model. In this model, a \u2217Parts of this work have been presented at the IEEE International Symposium on Information Theory, Espoo, Finland, June-July 2022 and the IEEE International Symposium on Information Theory, Taipei, Taiwan, June 2023. 1 \fsequence of beacon pulses is used to represent a binary de Bruijn sequence. Hence, once a sub-string of beacon pulses is received, its position is also determined uniquely. Furthermore, to consider the timing jitter performance, a long period of no-pulses is forbidden. Assume one pulse slot is used to represent a binary bit. If on-pulse is one and o\ufb00-pulse is zero, then a long run of zeros in the sequence, which is a long period of no-pulses, would impact the timing jitter. In [26], two pulse slots are suggested to represent a single bit, on-on is a one and on-o\ufb00is a zero so that two consecutive no-pulses are avoided. However, this scheme requires 2N pulse slots to represent a sequence of length N in the order n de Bruijn graph and it is required to receive a sub-sequence of 2n pulse slots to locate its position, i.e., if the sequence is long of about 2N = 2n+1 bits (representing a sequence of length N in the de Bruijn graph), about 2 log N pulse slots are required to locate its position instead of log N pulse slots if any sequence of n-pulses was permitted (all logarithms in this paper are in base 2). A second disadvantage in the proposed scheme is that the sequences of consecutive zeros were constrained to length one, while in reality, a few consecutive no-pulses are permitted subject to a constraint that their length will not be larger than a certain threshold s. Therefore, there is a target to use less redundant pulse slots to achieve both goals, to synchronize accurately, and to avoid long periods of no-pulses. For this purpose run-length limited (RLL) de Bruijn sequences are proposed [3, 4], where one pulse is represented by one binary bit. An on-pulse is represented by a one and an o\ufb00-pulse is represented by a zero. This scheme is more general and more e\ufb03cient with lower redundancy and higher rate than the ones in the previous work. The scheme combines two concepts, RLL sequences and sequences in the de Bruijn graph which will be de\ufb01ned now. RLL sequences are binary sequences in which there is an upper bound on the number of consecutive zeros in a sequence. An (n, s)-word is a binary word of length n in which the longest run of consecutive zeros is of length at most s. An (n, s)-sequence is a sequence whose windows of length n are distinct (n, s)-words. The family of sequences in which there are no runs of more than s consecutive zeros was extensively studied due to many applications that require such sequences [16]. The de Bruijn graph of order n, Gn, was de\ufb01ned \ufb01rst by Nicolaas Govert de Bruijn [1] and in parallel by Good [14]. The graph is a directed graph with 2n vertices which are represented by the set of all binary words of length n. The edges of Gn are represented by the 2n+1 binary words of length n + 1. There is a directed edge (x0, x1, . . . , xn\u22121, xn) from the vertex (x0, x1, . . . , xn\u22121) to the vertex (x1, . . . , xn\u22121, xn). A span n de Bruijn sequence is a cyclic binary sequence in which each binary n-tuple is contained exactly once in a window of length n. A walk (tour) in the graph is a sequence of directed edges such that the end-vertex of one edge is the start-vertex of the next edge. In a tour, the \ufb01rst vertex is also the last one. Any binary sequence can be represented by a walk in Gn, where any n consecutive symbols represent a vertex and any n + 1 consecutive symbols represent an edge. A cyclic sequence can be represented by a tour. The consecutive n symbols and n + 1 symbols represent a walk with its consecutive vertices and consecutive edges, respectively. A path (cycle) in the graph is a walk (tour) with no repeated vertices. In a cycle, each vertex can be considered the \ufb01rst vertex of the cycle. A trail (circuit) in the graph is a walk (tour) with no repeated edges. A span n de Bruijn sequence can be represented by an Eulerian circuit in Gn\u22121, i.e., a circuit which traverses each edge exactly once. In a circuit, the \ufb01rst vertex is also the last one. It can be also represented by a Hamiltonian cycle in 2 \fGn, i.e., a cycle that visit each vertex of Gn exactly once. Similarly, any path in Gn can be represented by a sequence with no repeated n-tuples. This sequence can be represented also by a trail in Gn\u22121. One structure that will be used in our exposition and is extensively studied in the literature associated with the de Bruijn graph is a necklace. A necklace of order n is a cycle in Gn whose length is a divisor of n. If the cycle is of length n, then the necklace is of full-order. If the necklace is of length which is a divisor of n that is smaller than n, then the necklace is degenerated. Each vertex in Gn is represented by a word of length n and in the necklace, we have all cyclic shifts of such a word. The necklaces of order n are the equivalence classes of the relation de\ufb01ned on words of length n, where two words are related if one is a cyclic shift of the other. If the necklace has length d which is a divisor of n, then it contains d words of length n, but the necklace can be represented by a word of length d. In Gn\u22121 such a necklace of order n is also a cycle of length which is a divisor of n with no repeated edges. The necklaces in Gn\u22121 is represented by the edges whose representation are words of length n. The runs of zeros in a necklace are considered to be a cyclic run since the necklace is a cycle. Let (n, s)-necklace be a necklace of order n that does not have a (cyclic) run of more than s consecutive zeros. Such a necklace contains only (n, s)-words. Example 1. Assume n = 6 and consider the (6, 2)-necklace with the vertex (001011). This necklace is of full-order and it contains the six words of length 6 in the cycle (001011) \u2192(010110) \u2192(101100) \u2192(011001) \u2192(110010) \u2192(100101) \u2192(001011) in G6. These six words are cyclic shifts of each other and their necklace is represented for example by [001011]. In G5 this necklace is represented by the cycle (00101) \u2192(01011) \u2192(10110) \u2192(01100) \u2192(11001) \u2192(10010) \u2192(00101) where the representation of the edges in G5 is the same as the representation of the vertices in G6 for these two cycles in G5 and G6, respectively. The degenerated necklace [011] of order 6 contains the three vertices in G6 which form the cycle of length 3 whose vertices and edge are given by (011011) \u2192(110110) \u2192(101101) \u2192(011011) . In G5 this cycle is (01101) \u2192(11011) \u2192(10110) \u2192(01101) . \u25a0 In this work, we will consider a combination of sequences in the de Bruijn graph that have a run-length constraint on the number of consecutive zeros. We will be interested in the maximum length cyclic and acyclic (n, s)-sequences. By their de\ufb01nition, if s is small, these sequences can be used in free-space quantum key distribution with \ufb02exibility in the parameters. 3 \fLet Gn(s) be the subgraph of Gn induced by all the vertices of Gn whose representations are (n, s)-words. Our goal is to \ufb01nd the maximum length path and maximum length cycle in Gn(s). The de\ufb01nition of Gn(s) implies that the edges of this graph also do not have a run of zeros whose length is greater than s. This property will be important in Gn\u22121(s). An upper bound on the maximum length of such a sequence will use a trail in Gn\u22121(s) (which is a circuit if the sequence is cyclic) and a lower bound on the maximum length will use a path in Gn(s) (which is a cycle if the sequence is cyclic). This is unique to our paper as usually it is not required to use di\ufb00erent orders of the graph for similar tasks. Note, that a cyclic sequence of length N with no repeated n-tuples has N binary digits, N vertices, and N edges in Gn and in Gn(s). An acyclic sequence with no repeated n-tuples has N + n \u22121 binary digits, N vertices, and N \u22121 edges in Gn and Gn(s). Henceforth, unless stated otherwise, all the (n, s)-sequences that will be considered will be cyclic. Example 2. The following cycle in G6(2) \u2192(100100) \u2192(001001) \u2192(010011) \u2192(100111) \u2192(001110) \u2192(011101) \u2192(111010) \u2192(110100) \u2192(101001) \u2192(010010) \u2192(100101) \u2192(001011) \u2192(010111) \u2192(101110) \u2192(011100) \u2192(111001) \u2192(110010) form the (6, 2)-sequence 10010011101001011 of length 17. The sequence contains two words (001110) and (011100) which are (6, 2)-words not in a (6, 2)-necklace since their necklace contains the words (000111), (100011), (110001), and (111000), that are not (6, 2)-words. \u25a0 Example 2 indicates that there could be long (n, s)-sequences which contain (n, s)-words that are not contained in (n, s)-necklaces. This implies that it is not straightforward to know the maximum length of an (n, s)-sequence. The rest of the paper is organized as follows. In Section 2 we consider an upper bound on the maximum length of (n, s)-sequences which are cycles in Gn. We prove that this bound is equal to the number of words in the (n, s)-necklaces. The trail with the maximum length can have another s edges (and s vertices) from Gn\u22121(s). The proof will be based on the necklaces obtained by the edges of Gn\u22121(s). In Section 3 it will be proved that the upper bound which was derived in Section 2 can be attained by many (n, s)-sequences. This is proved by considering Hamiltonian cycles in a subgraph of Gn(s) that contains only the vertices of (n, s)-necklaces. By adding s vertices (and s edges) of Gn\u22121(s) from necklaces which contain a cyclic run with s + 1 zeros a longer acyclic (n, s)-sequence is obtained. An e\ufb03cient algorithm to construct these sequences will be presented. In Section 4 enumeration of the number of (n, s)-necklaces is done and a formula for the length of a maximum length (n, s)-sequence is derived. In Section 5 a maximum length (n, s)-sequence from which the position of each n-tuple can be e\ufb03ciently decoded is constructed. Conclusion, several possible generalizations, and future research are discussed in Section 6. 4 \f2 The Maximum Length of an (n, s)-Sequence In this section, we will present an upper bound on the length of (n, s)-sequences. Let C be a circuit of maximum length in Gn\u22121 which does not contain an edge with s + 1 consecutive zeros in its representation. This circuit is a cycle in Gn with no vertex with s+1 consecutive zeros in its representation and hence it represents an (n, s)-sequence. To \ufb01nd an upper bound on the length of C we will try to remove the edges from Gn\u22121(s) which are not contained in C. These edges are associated with vertices whose in-degree is not equal to their out-degree. The following lemmas are immediate consequences from the binary representation of the vertices, their in-edges, and out-edges. Lemma 2.1. The vertices with in-degree one in Gn\u22121(s) are all the vertices that start in their binary representation with exactly s zeros followed by a one. Lemma 2.2. The vertices with out-degree one in Gn\u22121(s) are all the vertices that end in their binary representation with a one followed by exactly s zeros. The following lemmas are simple observations from the de\ufb01nitions. Lemma 2.3. All the words in a necklace of order n with at least two disjoint runs of s + 1 or more zeros have at least one run with s + 1 consecutive zeros. Corollary 2.4. All the words of each necklace of order n with at least two disjoint runs of s + 1 zeros are not contained in the circuit C. Lemma 2.5. If a necklace of order n has a word with a run of at least 2s + 1 zeros, then each word on this necklace has at least one run with more than s zeros. Corollary 2.6. All the words of each necklace of order n with a run of at least 2s + 1 zeros are not contained in the circuit C. Henceforth, a string which starts and ends with a one and has no run of more than s zeros will be called an s-ones string. Lemma 2.7. In Gn\u22121 a necklace of order n that contains a unique run with more than s zeros and the length of this run is s + k, 1 \u2264k \u2264s, contains n \u22121 \u2212s + k edges with a run of more than s zeros. Proof. Consider the edge 0k\u22121X0s+1, 1 \u2264k \u2264s, where X is an s-ones string of length n \u2212s \u2212k. This is the \ufb01rst edge in Gn\u22121 that has a run of s + k zeros, associated with a necklace of order n, to be considered. The last edge with a run of more than s zeros on this necklace to be considered is 0s+1X0k\u22121. Now, in the path from the edge 0k\u22121X0s+1 to the edge 0s+kX in this necklace there are 1 + k \u22121 + n \u2212s \u2212k = n \u2212s edges including the \ufb01rst edge and the last edge. All these edges have a run of zeros with a length greater than s. From the edge 0s+kX to the edge 0s+1X0k\u22121 in the necklace there are exactly k \u22121 edges including the last edge, excluding the \ufb01rst edge. Again, all these edges have a run of zeros with a length greater than s. Hence, in a necklace with a unique long run of more than s zeros whose length is exactly s + k, the total number of edges with a run of at least s + 1 zeros is n \u2212s + k \u22121. \u25a0 5 \fLemma 2.8. Any edge in Gn\u22121(s) of the form e = 0sX0k, where 1 \u2264k \u2264s, and X is an s-ones string of length n \u2212s \u2212k, yields a path, which starts at the vertex (0sX0k\u22121), whose length is at least s \u2212k + 1, its edges do not have a run of more than s zeros, and these edges are not contained in C. Proof. The in-vertex of the edge e = (v, u) has the form v = (0sX0k\u22121). By Lemma 2.1, the in-degree of the vertex v is one and hence any circuit of Gn\u22121(s) can contain at most one of its two out-edges. Remove one of its out-edges v \u2192v1 that is not contained in C from Gn\u22121(s). Now, v1 has in-degree one and hence C contains at most one of its out-edges. Remove the edge which is not contained in C from Gn\u22121(s). The process continues until the out-degree of the vertex that is reached is already one. The smallest number of edges removed by this process is s\u2212k+1 since the shortest path from v to such a vertex (0k\u22121X0s) is of length s \u2212k + 1 (by Lemma 2.2 the vertex (0k\u22121X0s) has out-degree one.). \u25a0 For a vertex of the form v = (0sX0k\u22121), where 1 \u2264k \u2264s and X is a s-ones string, in Gn\u22121(s), let P(v) denote the sub-path of length at least s \u2212k + 1 in Gn\u22121(s), that starts in the vertex v and ends at a vertex v\u2032 whose out-degree one (as described in the proof of Lemma 2.8) and whose edges are not contained in C. The sub-path P(v) will be called the deleted path of v. Corollary 2.9. After all the deleted paths induced by the proof of Lemma 2.8 are removed from Gn\u22121(s) the graph remains only with vertices whose in-degree equals to their out-degree. Proof. This is an immediate consequence of the proof of Lemma 2.8. It should be noted that for a vertex v with two out-edges v \u2192v1 and v \u2192v2, one vertex (either v1 or v2) ends with a zero and hence the edge e in Lemma 2.8 is taken without loss of generality as e = 0sX0k, where 1 \u2264k \u2264s. \u25a0 Corollary 2.10. There exists exactly one deleted path P(v) of length s \u2212k + 1 from the vertex v = (0sX0k\u22121), where 1 \u2264k \u2264s and X is a s-ones string. Remark: Note that in this section we are considering trails and circuits in Gn\u22121(s), but some of these trails, e.g., in the proof of Lemma 2.8 are paths and they are mentioned as such. Lemma 2.11. In Gn\u22121(s), the \ufb01rst s \u2212i edges in a deleted path P(v) of v = (0sX10i) are disjoint from the \ufb01rst s \u2212j edges in a deleted path P(u) of u = (0sX20j), where X1 and X2 are two distinct s-ones strings and i, j \u22650. Proof. The \ufb01rst s \u2212i edges in a deleted path that starts in the vertex (0sX10i) are exactly the edges either in the path 0sX10s or in a path of the form 0sX10i+k1Y, where 0 \u2264k < s \u2212i, and the length of Y is s \u2212i \u22121 \u2212k. 6 \fSimilarly, the \ufb01rst s \u2212j edges in a deleted path that starts in the vertex (0sX20j) are exactly the edges either in the path 0sX20s or in a path of the form 0sX20j+k1Z, where 0 \u2264k < s \u2212j, and the length of Z is s \u2212j \u22121 \u2212k. The binary representation of any vertex from the \ufb01rst s vertices in these four paths starts in one of the zeros of the \ufb01rst run of s zeros. Hence, a common vertex for these two paths implies that X1 = X2, a contradiction. Thus, the claim of the lemma follows. \u25a0 Corollary 2.12. The length of the circuit C (an (n, s)-sequence) in Gn\u22121(s) is at most the number of edges in all the (n, s)-necklaces. If a circuit of such length exists it contains exactly all the words of the (n, s)-necklaces. Proof. Let B be a necklace of order n with a run of more than s consecutive zeros. By Lemma 2.5 all the words in a necklace with a run of at least 2s + 1 zeros have at least one run with more than s zeros and hence all the vertices of these necklaces are not contained in C. If the longest run of consecutive zeros in B is s + k, 1 \u2264k \u2264s, then by Lemma 2.7 there are n \u22121 \u2212s + k edges in the necklace with a run of more than s consecutive zeros. Such a necklace has an edge e in Gn\u22121(s) of the form e = 0sX0k, where 1 \u2264k \u2264s and X is an s-ones string, whose \ufb01rst vertex is v = (0sX0k\u22121) and by Lemma 2.8 it yields a deleted path P(v) whose length is at least s \u2212k + 1. Together, B (n \u22121 \u2212s + k forbidden edges) and the associated deleted path P(v) (at least s\u2212k +1 forbidden edges) have at least (n \u22121 \u2212s + k) + (s \u2212k + 1) = n edges which are not contained in C. By Lemma 2.11 all the deleted paths of di\ufb00erent vertices are disjoint and hence the number of edges of Gn\u22121(s) which are not contained in C is at least the number of edges in the necklaces of order n which contain edges with a run of more than s consecutive zeros. Thus, the length of the circuit C in Gn\u22121(s) is at most the number of edges in all the (n, s)-necklaces in Gn\u22121(s). For the fact that such a circuit cannot contain any (n, s)-word not from an (n, s)-necklace we have to add the consequence of Corollary 2.10. \u25a0 What about the length of the maximum length trail P in Gn\u22121(s)? All the arguments we have used so far hold also for a trail with one exception. The trail can contain two vertices v and u, where the in-degree of v is one and its out-degree is two; the in-degree of u is two and its out-degree is one. The \ufb01rst vertex in the trail is v and the last vertex in the trail is u. We can apply the process in the proof of Lemma 2.8 on all the vertices in Gn\u22121(s) except for v. After all the deleted paths are removed from Gn\u22121(s) if we continue and apply the process in the proof of Lemma 2.8 on v we will obtain a deleted path P(v) which starts in v and ends at u. This deleted path can be added to the maximum length (n, s)-sequence to obtain a maximum length acyclic (n, s)-sequences. In other words, we have two vertices v1 and v2 for which we have the edges v \u2192v1 and v \u2192v2 in Gn\u22121(s) and the in-degree of v is one. Such vertices have the form v = (0sX), v1 = (0s\u22121X0), and v2 = (0s\u22121X1). Note, that v is a vertex contained in the (n, s)-necklace [10sX] and v is also a vertex contained in the 7 \fnecklace [0s+1X] which is not an (n, s)-necklace, but s words of the necklace are (n, s)-words. The maximum length circuit in Gn\u22121 cannot contain both edges v \u2192v1 and v \u2192v2 since the in-degree of v is one. But, a trail can contain both edges if one of them will be the \ufb01rst edge in the trail. In other words, if both edges are on the trail P, then since they have the same unique predecessor in Gn\u22121(s), it follows that one of them must be the \ufb01rst edge in P. It can be only at the beginning of a sub-path of P which was excluded by Lemma 2.8. The length of this sub-path in Lemma 2.8 is s and its structure is as follows: (0sX) \u2192(0s\u22121X0) \u2192\u00b7 \u00b7 \u00b7 \u2192(0X0s\u22121) \u2192(X0s) , where all the s edges are (n, s)-words that are contained in the necklace [0s+1X] which is not an (n, s)-necklace. Thus, we have the following conclusion. Theorem 2.13. The length of a trail P of maximum length in Gn\u22121(s) is at most the number of edges in all the (n, s)-necklaces plus s (unless the (n, s)-necklaces contain all the (n, s)words in which case the bound on the length of a trail is the same as the one on the length of a circuit). Example 3. If n = 3 and s = 2, then an (3, 2)-sequence of maximum length is 0010111 and an acyclic (3, 2)-sequence of maximum length is 001011100 . Both sequences have seven (3, 2)-words and they contain all the (3, 2)-necklaces. If n = 5 and s = 2, then an (5, 2)-sequence of maximum length has length 21 and one such sequence is. 001010011101011011111 . An acyclic (5, 2)-sequence has length 27 (4 additional bits for the acyclic representation and 2 additional bits from words which are not contained in (5, 2)-necklaces) as follows 001100101001110101101111100 . \u25a0 A maximum length acyclic (n, s)-sequence (trail) in Gn\u22121(s) can contain s words which are not contained in a maximum length (n, s)-sequence. It was explained how to add them to a maximum length (n, s)-sequence. The maximum length trail can be constructed from an (n, s)-sequence in which one deleted path P(v) was taken from a necklace and it was added to obtain a maximum length (n, s)-sequence. It should be noted that the sequence starts and ends with the two possible shortest deleted paths. This is demonstrated in the following example. Example 4. The sequence 00110101111101100101 8 \fis a (5, 2)-sequence of length 20 which contains two (5, 2)-words (00110) and (01100) which are not contained in (5, 2)-necklaces. This sequence falls short by one from the upper bound of a maximum length (5, 2)-sequence. The deleted path which was removed from G4(2) is (0011) \u2192(0111) \u2192(1110) \u2192(1100) and its words of length 5 are contained in the (5, 2)-necklace [00111]. The sequence 001101011111011001010011100 of length 27 is an acyclic (5, 2)-sequence which contains 23 (5, 2)-words, two of which are not contained in an (5, 2)-necklace. This sequence attains the upper bound of a maximum length acyclic (5, 2)-sequence. In this sequence the deleted path was added at the end of the sequence and hence all the words of the (5, 2)-necklaces are contained in this trail. \u25a0 3 Construction of Sequences of Maximum Length In this section, we will concentrate on constructing (n, s)-sequences of maximum length. We will show that there exist (n, s)-sequences which attain the upper bound of Corollary 2.12 and hence these sequences are maximum length (n, s)-sequences. We will design a method and an e\ufb03cient algorithm to generate a large class of maximum length (n, s)-sequences. The algorithms which will be designed are implemented for contructing the next bit given the last n constructed bits of the sequence. We start by describing the general method, continue with an algorithm to construct a very large class of maximum length (n, s)-sequences, but with a large amount of storage and the complexity for computing the next bit of the sequence is not linear if n is large. This algorithm is quite e\ufb03cient when n is not large. We \ufb01nish with an algorithm that requires a more reasonable storage size, but with a slightly smaller number of generated sequences, but the best possible for the amount of storage that is used. The complexity for \ufb01nding the next bit of the sequence is linear in n. The following method and its algorithm is a modi\ufb01cation of the method and the algorithms used frequently to generate de Bruijn sequences, such as the one in [9]. While the upper bound on the length of a maximum length (n, s)-sequences was based on an analysis of trails in Gn\u22121(s), the (n, s)-sequences which will be generated are cycles in Gn(s). The basic principle in the method is taking all the (n, s)-necklaces and merging them into one cycle. For a vertex X = (x1, x2, . . . , xn\u22121, xn) in Gn, its companion X\u2032 is de\ufb01ned by X\u2032 = (x1, x2, . . . , xn\u22121, \u00af xn), where \u00af x is the binary complement of x. Two vertex-disjoint cycles in Gn, C1 and C2, are joined together into one cycle, if they contain a pair of companion states X on C1 and X\u2032 on C2. If Y \u2192X is an edge on C1 and Z \u2192X\u2032 is an edge on C2, then keeping all the edges of C1 and C2 except for these two edges and adding the edges Y \u2192X\u2032 and Z \u2192X, will merge the two cycles into one cycle. The merging of all the cycles is done as follows. The (n, s)-necklaces are ordered by their weight from the one with the largest weight to the one with the smallest weight. We start with the necklace which contains only ones and continue to the necklace with a unique zero and start with step 2. At each step, we have a main cycle composed of the (n, s)-necklaces that were merged so far and also the remaining (n, s)-necklaces. At 9 \fstep i, i \u22652, we choose on each (n, s)-necklace X with weight n \u2212i (this is the number of ones in each word of the necklace) a vertex whose representation ends with a zero, e.g., X0. Its companion X1 on a necklace X \u2032 has weight n \u2212i + 1 and it is readily veri\ufb01ed that X \u2032 does not have a run of more than s zeros, i.e., X \u2032 in also an (n, s)-necklace. Hence, X1 is on the main cycle, so we can merge the necklace X into the main cycles and continue. This procedure ends when all the (n, s)-necklaces are merged into the main cycle. To apply this procedure, we have to choose on each (n, s)-necklace whose weight is at most n \u22122 a vertex whose binary representation ends with a zero. For example, this vertex can be the one which is the smallest one as a binary number among all the vertices whose binary representation ends with a zero. The companion of this vertex has a weight greater by one and hence its cycle was merged before and hence each necklace can be merged in its time to the main cycle. This procedure will result in a cycle of Gn which contains all the vertices of all the (n, s)-necklaces. If we want to form more than one cycle we have to choose two vertices that end in a zero, e.g., one vertex ending in a zero with the smallest binary representation and a second one ending in a zero with the largest binary value. Some necklaces have only one word ending in a zero. On such a necklace X the unique word which ends with a zero has the form (1d\u221210)n/d which implies that the length d of the necklace is a divisor of n. On these necklaces, we cannot have two choices. Fortunately, the number of such necklaces is less than the number of divisors of n and we can ignore them. We should use a binary vector whose length is the number of (n, s)-necklaces. This vector indicates which one of the two vertices should be taken to perform the merging of the cycles. If K is the number of (n, s)-necklaces of weight at most n \u22122 in which there are at least two words ending in a zero, then the algorithm can be used to generate 2K distinct maximum length (n, s)-sequences. There is one disadvantage in this method \u2013 K is exponential in n and hence the algorithm will require an exponential area of storage. To overcome this disadvantage we can choose two such vertices only for some of the (n, s)-necklaces, for example, all the (n, s)-necklaces of some given weights. Algorithm to merge (n, s)-necklaces: Choose two integers 1 \u2264k1 \u2264k2 \u2264n \u22122. For each k, k1 \u2264k \u2264k2, use an order of the (n, s)-necklaces (with at least two words ending in a zero) whose words have weight k, using any given e\ufb03cient algorithm. For each such k choose a binary vector V (k) whose length is the number of such (n, s)-necklaces with weight k. Using the order of these necklaces, the ith entry of V (k) (associated with the ith necklace in this order) is set to 0 if we merge the i-necklace through the word of the smallest value and is set to 1 if we merge through the word with the largest value. Initially, \u03b20 = (1, 1, . . ., 1) = 1n, \u03b21 = (1, 1, . . . , 1, 0) = 1n\u221210 and w(\u03b21) = n \u22121. Given \u03b2i = (bi, bi+1, . . . , bi+n\u22121) and its weight w(\u03b2i), proceed to produce the next bit which implies the next n-tuple \u03b2i+1 = (bi+1, . . . , bi+n\u22121, bi+n) as follows. (A1) Let \u03b2\u2217 i = (bi+1, . . . , bi+n\u22121, 0) and w(\u03b2\u2217 i ) = w(\u03b2i) \u2212bi be the weight of \u03b2\u2217 i . (A2) If the necklace which contains \u03b2\u2217 i contains a run of more than s zeros, then go to (A5). If the necklace contains exactly one word ending in a zero, then go to (A6). (A3) If k1 \u2264w(\u03b2\u2217 i ) \u2264k2, then let d be the value of the entry in V (w(\u03b2\u2217 i )) associated with the necklace that contains \u03b2\u2217 i . 10 \f(A4) Examine the cyclic shifts of \u03b2\u2217 i for the existence of a shift X which end with a zero whose value is the smallest one if d = 0 (or if w(\u03b2\u2217 i ) is not between k1 and k2) or whose value is the largest one if d = 1. If \u03b2\u2217 i = X, then go to (A6). (A5) Set bi+n := bi, w(\u03b2i+1) := w(\u03b2i), and stop. (A6) Set bi+n := \u00af bi, w(\u03b2i+1) := w(\u03b2i) \u2212bi + \u00af bi. The correctness of the algorithm and the proof of the next theorem is done similarly to the related proof in [9]. Theorem 3.1. (a) For every choice of k1 and k2, algorithm merge (n, s)-necklaces produces the next bit of a maximum length (n, s)-sequence. (b) For a given choice of k1 and k2, there are 2K distinct choices of vertices for merging the (n, s)-necklaces, where K is the number of (n, s)-necklaces with weight between k1 and k2 having at least two words which end in a zero. Thus, algorithm merge (n, s)-necklaces can be used to produce 2K distinct maximum length (n, s)-sequences. (c) The working space that algorithm merge (n, s)-necklaces requires to produce the next bit of a maximum length (n, s)-sequence is 3n + K bits and the work required to produce the next bit is at most n cyclic shifts and about the same number of n-bits comparisons. Other algorithms for generating de Bruijn sequences can be also adapted to merge all the (n, s)-necklaces for constructing (n, s)-sequences. The order of the necklaces can be done by some known algorithms as the one by Kociumaka, Radoszewski, and Rytter [19]. But, if n is large, then the known algorithms for ordering the necklaces, including the one in [19] are not linear and it will increase the time complexity of our overall algorithm. If n is small, then we can use this method as the complexity will not be an issue. To overcome the complexity problem, assuming that n is large enough, we suggest choosing in most cases the word that ends in a zero with the maximal value for merging the necklace into the main cycle, but in some cases to use other words as follows. Choose a large integer k which depends on the number of maximum length (n, s)sequences that we want to generate. On most (n, s)-necklaces the word the word ending in a zero with maximum value will be used to merge the necklace to the main cycle. On k necklaces it will be another word. Each such a word is de\ufb01ned by \u03b8(n, s, k) free parameters and hence 2k\u00b7\u03b8(n,s,k) maximum length (n, s)-sequences will be generated by the algorithm. The value of \u03b8(n, s, k) will be increases as n increases, or s increases, or k increases. Let m = \u2308log k\u2309and consider the ordered set V = {V (i)}k\u22121 i=0 of k distinct (n, s)-words in k distinct (n, s)-necklaces, constructed as follows: (c1) The \ufb01rst (s + 1)\u2308m s \u2309+ 1 bits of V (i) contain ones at positions (s + 1)j, 0 \u2264j \u2264\u2308m s \u2309 (this guarantee that in these position there will not be a run of more than s consecutive zeros). In the other s\u2308m s \u2309positions (between the positions of the ones) V (i) forms the binary representation of i. These (s + 1)\u2308m s \u2309+ 1 bits are followed by a single zero. 11 \f(c2) The last (s + 1)\u2308m s \u2309+ 4 bits of V (i) start and end with single zero and between these two zeros there are (s + 1)\u2308m s \u2309+ 2 ones. We want that this run of (s + 1)\u2308m s \u2309+ 2 ones will be the unique longest run of ones in the word. (c3) For each j \u22651, in position \u0000(s + 1)\u2308m s \u2309+ 2 \u0001 j, where \u0000(s + 1)\u2308m s \u2309+ 2 \u0001 j \u2264n \u2212(s + 1)\u2308m s \u2309\u22123, there is a zero. This guarantee that the requirement for the longest run of ones in (c2) is satis\ufb01ed. (c4) Between the \ufb01rst (s + 1)\u2308m s \u2309+ 1 bits and the last (s + 1)\u2308m s \u2309+ 4 bits each s + 1 bits we have a one (at the end of these s + 1 positions). If there is a collision between these ones and the zeros of (c3), then at the place of the collision we will have the one after s bits and not after s + 1 bits and after this one the zero (to avoid such a collision). This guarantees that there will not be a run of more than s zeros in this section of the word. (c5) The rest of the positions which were not speci\ufb01ed. These are free positions in which there are arbitrary assignments of zeros and ones. We \ufb01rst note that each word of V is an (n, s)-word from an (n, s)-necklace since the word starts with a one (by (c1)) and ends with an isolated zero (by (c2)) and in each s + 1 consecutive positions there is at least one one by (c1), (c2), (c3), and (c4). Note further that each V (i) has a unique run of exactly (s+1) \u0000\u2308m s \u2309 \u0001 +2 ones which ends just one position before the last position . All the other runs of ones in each V (i) are shorter. These properties are guaranteed by (c2) and (c3). This implies that two such words from V cannot be a shift of the other and also all the words are of full-order. In the \ufb01rst (s+1) \u0000\u2308m s \u2309 \u0001 +1 bits there is the binary representation of i (by (c1)) with separations of ones to satisfy the constraint of no more than s consecutive zeros (again by (c1)). This will enable to \ufb01nd if the word that we consider is from the set V . There are many free parameters whose number is increased as s get larger and/or if we choose larger k. The algorithm will have similar steps to algorithm merge (n, s)-necklaces. The algorithm will \ufb01rst try to see if the necklace of \u03b2\u2217 i is a word from V in which case the necklace is joined to the main cycle only if \u03b2\u2217 i is the associated word of V . If \u03b2\u2217 i is not a word in the set V , then it examines whether \u03b2\u2217 i is word ending in a zero with the maximum value in its necklace. Note, that since the largest run of ones is at the end of the word followed by a unique zero, it follows that no V (i) can be the word of maximum value end in a zero in its necklace (this longest run of ones starts the word of the maximum value in the necklace. The procedure itself is very similar, but slightly more complicated, than the one described in [9]. The construction of the maximum length (n, s)-sequences is based on merging all the (n, s)-necklaces. There are (n, s)-sequences that contain (n, s)-words that are not contained in (n, s)-necklaces. But, these sequences cannot be of maximum length and this can be proved as a consequence from the proofs of Lemmas 2.8 and 2.12. Finally, we consider a maximum length acyclic (n, s)-sequence. This is done by adding a short path with (n, s)-words, which are not contained in (n, s)-necklaces, to the maximum length (n, s)-sequence C. Consider an (n, s)-word (10sX), for some s-ones string X, whose length is n \u22121 \u2212s, on an (n, s)-necklace with the edge (10sX) \u2192(0sX1) of the associated necklace on the cycle C. We start the maximum length path with the vertex (10sX) and 12 \fcontinue with the edge 10sX1 to the vertex (0sX1) as on the cycle C until we reach the vertex (10sX) again. So far the path is the same as the cycle C. Now, we continue with the s edges, (10sX) \u2192(0sX0) \u2192\u00b7 \u00b7 \u00b7 \u2192(0X0s) , to obtain an acyclic (n, s)-sequence of maximum length, i.e., s edges and s vertices were added to the maximum length (n, s)-sequence C and the outcome is an acyclic (n, s)-sequence which attain the bound of Theorem 2.13. Note, that all the added vertices are on a necklace which is not an (n, s)-necklace, but the added vertices are represented by the (n, s)-words which are contained in the necklace. Note, further that the added sub-path is to the end of the cycle C, while in the upper bound it was considered in its beginning, but this is equivalent. 4 Enumeration of the Number of Sequences In this section, we will consider enumeration associated with the number of (n, s)-words, (n, s)-necklaces, and (n, s)-sequences related to the bounds and the construction that were presented in the previous sections. For this purpose, we de\ufb01ne two important concepts, the rate and the redundancy of the set of sequences discussed in this paper. The rate of an (n, s)-sequence with maximum length N will be de\ufb01ned as R(n, s) = lim sup n\u2192\u221e log N n . The redundancy of an (n, s)-sequence S of length N will be de\ufb01ned as red(S) = n \u2212log N . The algorithm which uses a de Bruijn sequence of length 2n and each one is simulated by \u201911\u2019 and each zero by \u201910\u2019 considers a sequence of length 2n+1 with windows of length 2n for each required n-tuple. Hence, it has a high redundancy of 2n \u2212log 2n+1 = n \u22121. Our (n, s)-sequences reduce this redundancy quite dramatically. What is the exact length of a maximum length (n, s)-sequence? The computation of this size can be done similarly to other computations associated with constrained codes [15, 21], and asymptotic computations are done in the same way. Some of the computations can be done more accurately. For example, we will present the computations for s = 1. Let gn be the number of (n, 1)-words. Lemma 4.1. The number of (n, 1)-words is gn = \u0010 1+ \u221a 5 2 \u0011n+2 \u2212 \u0010 1\u2212 \u221a 5 2 \u0011n+2 \u221a 5 . Proof. It is well-known and easily computed that gn = gn\u22121 +gn\u22122, where g1 = 2 and g2 = 3. The solution for this recurrence is gn = \u03d5n+2\u2212\u03c8n+2 \u221a 5 , where \u03d5 = 1+ \u221a 5 2 and \u03c8 = 1\u2212 \u221a 5 2 . \u25a0 Lemma 4.2. The number of (n, 1)-words in the (n, 1)-necklaces is gn \u2212gn\u22124. 13 \fProof. A necklace that is not an (n, 1)-necklace contains (n, 1)-words if it starts with 01 and ends with 10. In between we have an (n \u22124, 1)-word and hence by de\ufb01nition there are gn\u22124 such words. Thus, the number of (n, 1)-words in the (n, 1)-necklaces is gn \u2212gn\u22124. \u25a0 Now we can combine Corollary 2.12 and the constructions in Section 3 with Lemma 4.2 to obtain the following conclusion. Corollary 4.3. The length of a maximum length (n, 1)-sequence is gn \u2212gn\u22124. Corollary 4.4. The length of a maximum length acyclic (n, 1)-sequence is gn \u2212gn\u22124 + 1. Corollary 4.5. The rate of maximum length (n, 1) sequences is 0.6942 and their redundancy is 0.3058n. Lemma 4.1 can be simply generalized to (n, s)-words as follows. Lemma 4.6. The number hn,s of (n, s)-words satisfy the following recursive formula: hn,s = s+1 X i=1 hn\u2212i,s , where hi,s = 2i for 1 \u2264i \u2264s and hs+1,s = 2s+1 \u22121. Proof. Each (n, s)-word can start in i zeros, 0 \u2264i \u2264s, followed by a one and after that there is an (n \u2212i \u22121, s)-word. This implies the claim in the lemma. \u25a0 It was proved in [2, 20, 22] that hn,s can be expressed as hn,s = \u0016 \u03bbk+1(\u03bb \u22121) (s + 2)\u03bb \u22122(s + 1) + 0.5 \u0017 , where \u03bb is the unique positive real root of the equation xs+1 \u2212 s X i=0 xi = 0 . (1) Lemma 4.2 can be generalized as follows. Lemma 4.7. The number of (n, s)-words in the (n, s)-necklaces is hn,s \u2212 s X i=1 i \u00b7 hn\u22122s\u22123+i,s . Proof. A necklace which is not an (n, s)-necklace contains (n, s)-words if such a word starts with 0i1 and ends with 10j, where 1 \u2264i, j \u2264s and s < i + j. In between these two substrings we have an (n \u22122 \u2212i \u2212j, s)-word and hence by de\ufb01nition, there are hn\u22122\u22122s,s such words if i = j = s. If i = s \u22121, j = s or i = s, j = s \u22121, and s > 1, then there are 2hn\u22121\u22122s,s such words (for both together). If i + j = 2s \u22122 and s > 2, then i = s \u22122 or i = s \u22121 or i = s and hence there are 3hn\u22122s,s such words. This analysis is continued for each s+1 \u2264i+j \u22642s, where 1 \u2264i, j \u2264s. Finally, if i+j = s+1, then 1 \u2264i \u2264s and hence there are s \u00b7 hn\u2212s+3,s such words. Thus, the number of (n, s)-words in the (n, s)-necklaces is hn,s \u2212Ps i=1 i \u00b7 hn\u22122s\u22123+i,s. \u25a0 14 \fCorollary 4.8. The length of a maximum length (n, s)-sequence is hn,s \u2212Ps i=1 i \u00b7 hn\u22122s\u22123+i. Corollary 4.9. The length of a maximum length acyclic (n, s)-sequence is hn,s \u2212Ps i=1 i \u00b7 hn\u22122s\u22123+i + s. Corollary 4.10. The rate of maximum length (n, s) sequences is log |\u03bb| and their redundancy is n \u2212|\u03bb| n, where \u03bb is the root of Eq. ( 1). We are also interested in the number of generated cycles which is 2K, where K is the number of (n, s)-necklaces whose weights are between k1 and k2. This is associated with the algorithm to merge (n, s)-necklaces. When s = 1 this number is not di\ufb03cult to compute. Lemma 4.11. The number of (n, 1)-words with weight k is \u0000k+1 n\u2212k \u0001 if k + 1 \u2265n \u2212k. Proof. An (n, 1)-word with weight k has k ones and the n \u2212k zeros are isolated between the ones (including at the beginning or the end) and hence there are \u0000k+1 n\u2212k \u0001 such words. \u25a0 Lemma 4.12. The number of words in the (n, 1)-necklaces with words of weight k is \u0012 k n \u2212k \u0013 + \u0012 k \u22121 n \u2212k \u22121 \u0013 if k + 1 \u2265n \u2212k. Proof. A word in an (n, 1)-necklace is an (n, 1)-word that does not start and end in a zero. By Lemma 4.11 the total number of (n, 1)-words with weight k is \u0000k+1 n\u2212k \u0001 . An (n, 1)-word which starts with a zero and ends with a zero, starts with 01 and ends with 10. In between we have an (n \u22124, 1)-word with weight k \u22122. By Lemma 4.11, there are \u0000 k\u22121 n\u2212k\u22122 \u0001 such words and hence the number of words in the (n, 1)-necklaces with words of weight k is \u0012k + 1 n \u2212k \u0013 \u2212 \u0012 k \u22121 n \u2212k \u22122 \u0013 = \u0012 k n \u2212k \u0013 + \u0012 k \u22121 n \u2212k \u22121 \u0013 . \u25a0 It should be noted that on one hand the set V de\ufb01ned in Section 3 is less e\ufb00ective when s = 1, but based on Lemmas 4.11 and 4.12 an e\ufb03cient algorithm to construct a large set of maximum length (n, 1)-sequences can be designed. This algorithm will be based on algorithm merge (n, 1)-necklaces and an e\ufb03cient algorithm to enumerate the associated binomial coe\ufb03cient, e.g., the enumerative encoding of Cover [5]. 5 Sequences with E\ufb03cient Positioning Decoding In this section, we present a simple algorithm for generating an (n, s)-sequence of maximum length. The advantage of this algorithm is that the position of any given n-tuple can be decoded e\ufb03ciently. The disadvantage of such an algorithm for generating an (n, s)-sequence is that it is possible to generate with it only one such (n, s)-sequence. Moreover, the time complexity to generate the next bit is larger. The algorithm is based on an idea of concatenating necklaces that was presented by Fredricksen and Maiorana [12] which improved on 15 \fa previous idea of Fredricksen and Kessler [10] that generated a de Bruijn sequence based on a partition of n into smaller positive integers. The algorithm was improved later by Fredricksen and Kessler [11] and it has a few variants. It is natural to call this algorithm, the FKM algorithm for Fredricksen, Kessler, and Maiorana. The variant of the algorithm that we consider uses representatives of the necklaces which are called Lyndon words. For a given necklace of order n, its Lyndon word of order n is the word of the least value in the necklace, whereas for a full-order necklace of length n, a word of length n is taken, while for a degenerated necklace of length d < n which divides n, a word of length d is taken. The Lyndon words are now ordered lexicographically from the smallest to the largest one and concatenated together in this order. The outcome is a de Bruijn sequence of length 2n and therefore, it is called the lexicographically least composition sequence. In the original papers [11, 12] the word of maximum value in base 2 was taken from a necklace and the necklaces were ordered from the maximum value to the minimum value in this representation. Hence, the generation of the associated de Bruijn sequence is going down to the ordering of the necklaces lexicographically based on their Lyndon words. But, to generate an (n, s)sequence of maximum length we have to use only the (n, s)-necklaces and therefore we have to make a small modi\ufb01cation to the original sequence and also to the original algorithm. As mentioned, the original sequences generated in [11, 12] considered the necklaces ordered by the words with the largest binary value from the largest one to the smallest one, but later algorithms with the same technique to the order based on Lyndon words, e.g. [24, 25]. The algorithm was analyzed by Ruskey, Savage, and Wang [23]. Using the ranking and the decoding algorithm for these sequences as suggested by Kociumaka, Radoszewski, and Rytter [19] for decode the lexicographic least de Bruijn sequence, we can decode the generated (n, s)-sequence which will be constructed in this section. For ordering the necklaces, it is worthwhile to know their number. The total number of necklaces of order n, Z(n), is a well-known result [8, 13]. It involves the well-known arithmetic function, the M\u00a8 obius function [17]. Theorem 5.1. The number of necklaces of order n is Z(n) = 1 n X d|n \u00b5(d)2n/d, where \u00b5(\u00b7) is the M\u00a8 obius function. The Lyndon words have some simple properties. Lemma 5.2. The Lyndon word of a necklace starts with the longest run of zeros and ends with a one. Lemma 5.3. If the Lyndon word X1 of a necklace has a larger run of consecutive zeros than in the Lyndon word X2 of another necklace, than x1 has a smaller value than X2. Lemma 5.2 is not a precise characterization of a Lyndon word since for example there might be a few runs with the longest run of zeros. The least lexicographic de Bruijn sequence is generated based on the following celebrated lemma [12]. 16 \fLemma 5.4. Let X1, X2, X3, \u00b7 \u00b7 \u00b7 , XZ(n) be the ordering of the Lyndon words of order n from the smallest one lexicographically to the largest one. The concatenation of these words in this order is a de Bruijn sequence of order n, i.e., an Eulerian circuit in Gn\u22121 which is also a Hamiltonian cycle in Gn. The next step is to show that lemma 5.4 is also true if we restrict ourselves to the Lyndon words of the (n, s)-necklaces. Lemma 5.5. Let X1, X2, X3, \u00b7 \u00b7 \u00b7 , XZ(n) be the ordering of the Lyndon words of order n from the smallest one lexicographically to the largest one. There exists an index k such that X1, X2, \u00b7 \u00b7 \u00b7 , Xk are Lyndon words which are not contained in (n, s)-necklaces, while Xk+1, Xk+2, \u00b7 \u00b7 \u00b7 , XZ(n) are Lyndon words which are contained in (n, s)-necklaces. Proof. Since the Lyndon words are ordered by their lexicographic value from the smallest to the largest, it follows that they are \ufb01rst ordered by the number of leading zeros (from the largest number to the smallest number of leading zeros) in each word. By Lemma 5.2, this number of leading zeros is also the longest run of zeros in the necklace. By Lemma 5.3, all Lyndon words that start with more than s zeros are contained in necklaces that are not (n, s)-necklaces, while all the Lyndon words that start with at most s zeros are contained in (n, s)-necklaces which imply the claim of the lemma. \u25a0 Corollary 5.6. The last Lyndon word, which is not an (n, s)-word, in the order from the smallest one to the largest one, is 0s+11n\u2212s\u22121. Proof. By Lemma 5.3, the Lyndon words with a smaller number of leading zeros have larger values than the ones with a larger number of leading zeros. The word 0s+11n\u2212s\u22121 is a Lyndon word with the largest value among the words starting with s + 1 zeros. The next Lyndon word in the order has s leading zeros and the claim of the lemma follows. \u25a0 Corollary 5.7. In the lexicographic ordering of the Lyndon words of order n, all Lyndon words after the Lyndon word 0s+11n\u2212s\u22121 are contained in (n, s)-necklaces. Proof. All Lyndon words after 0s+11n\u2212s\u22121 start with at most s zeros and end with a one and hence their necklaces cannot contain any run of more than s zeros. This implies that all these words are contained in (n, s)-necklaces. \u25a0 The algorithm which generates all the Lyndon words that follow the last Lyndon word 0s+11n\u2212s\u22121 which is not contained in an (n, s)-necklace is given as follows. Algorithm for the lexicographic generation of Lyndon words in (n, s)-necklaces: The \ufb01rst word is de\ufb01ned as X0 = 0s+11n\u2212s\u22121 (this word is not considered for the (n, s)necklaces); set Y := X0 and i := 0. (E1) If Y = 1n, goto (E5); otherwise set i := i + 1. (E2) Find the largest index j in Y = y1y2 \u00b7 \u00b7 \u00b7 yn such that yj = 0. (E3) Let Z = y1y2 \u00b7 \u00b7 \u00b7 yj\u221211 and set Y to be the \ufb01rst n bits of the in\ufb01nite sequence ZZZ \u00b7 \u00b7 \u00b7 . (E4) If j divides n, then set Xi := Z and goto (E1); otherwise goto (E2). 17 \f(E5) X1, X2, . . . , Xi is the sequence of Lyndon words of the (n, s)-necklaces from the smallest to the largest. The concatenation of X1X2 \u00b7 \u00b7 \u00b7 Xi is the required maximum length (n, s)-sequence that contains exactly all the words of the (n, s)-necklaces. The correctness of the algorithm is shown in the same way as it is proved in the FKM algorithm. To \ufb01nd the position of a given word v of length n in the concatenation of the Lyndon words we have to apply the algorithm proposed by Kociumaka, Radoszewski, and Rytter [19] which \ufb01nds the position of v in the concatenation of all the Lyndon word of order n. After the position of v was found we have to subtract the position of the last entry in the word 0s+11n\u2212s\u22121 since our sequence starts after this last entry. It is worthwhile and more e\ufb03cient in the long run to compute this position in advance and save it as it will be used in every application of the algorithm. The complexity of the decoding algorithm is the same as in [19] since the ordering of the Lyndon words is the same with the exception that we do not start from the \ufb01rst one but with the one after 0s+11n\u2212s\u22121. 6"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2308.15884v3",
+                    "title": "Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry",
+                    "abstract": "Determining the optimal fidelity for the transmission of quantum information\nover noisy quantum channels is one of the central problems in quantum\ninformation theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical\nProgramming, 2021] introduced an asymptotically converging semidefinite\nprogramming hierarchy of outer bounds for this quantity. However, the size of\nthe semidefinite programs (SDPs) grows exponentially with respect to the level\nof the hierarchy, thus making their computation unscalable. In this work, by\nexploiting the symmetries in the SDP, we show that, for a fixed output\ndimension of the quantum channel, we can compute the SDP in time polynomial\nwith respect to the level of the hierarchy and input dimension. As a direct\nconsequence of our result, the optimal fidelity can be approximated with an\naccuracy of $\\epsilon$ in $\\mathrm{poly}(1/\\epsilon, \\text{input dimension})$\ntime.",
+                    "authors": "Yeow Meng Chee, Hoang Ta, Van Khu Vu",
+                    "published": "2023-08-30",
+                    "updated": "2024-03-21",
+                    "primary_cat": "quant-ph",
+                    "cats": [
+                        "quant-ph",
+                        "cs.DS",
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "Introduction One of the central problems in information theory is determining the minimum error probability that arises during data transmission through a noisy channel or when storing it on an unreliable storage medium. This task can be formulated as an optimization problem that maximizes the probability of determining the sent messages exactly over all valid encoders and decoders. This problem was initially investigated by Shannon [Sha48]. There, the author showed that when considering large numbers of independent copies of the same channel, this quantity can be characterized by a simple expression known as the channel capacity of the channel. In terms of the algorithmic point of view regarding this problem, the study conducted in [BF17] focused on investigating the problem of determining the optimal encoder and decoder that leads to the maximum success probability over a noisy channel in the non-asymptotic regime. In this setting, given NX\u2192Y that is a noisy channel from X to Y and M \u2208N, let p(N, M) to be the maximum success probability for transmitting a uniform M-dimensional message through the channel NX\u2192Y . The study in [BF17] showed that the computation of p(N, M) can be formulated as an optimization problem involving a submodular function. Building upon this insight, a simple and efficient greedy algorithm was introduced in [BF17] to find a code that achieves a (1\u2212e\u22121)-approximation of p(N, M). Furthermore, the study in [BF17] provided an efficiently computable linear programming relaxation denoted by LP(N, M) (also referred to as meta converse [Hay09, PPV10]). This relaxation provides upper bounds on p(N, M), and it was proven that p(N, M) \u2264LP(N, M) \u2264(1 \u2212e\u22121)\u22121p(N, M). However, it was highlighted in [BF17] that the problem of obtaining an approximate solution for p(N, M) with a better constant factor than 1 \u2212e\u22121 is classified as NP-Hard. Further research on this problem on variant models has been conducted in [BFS16, FSS19, BFGG22, FF23a]. The problem in the analog quantum setting involves determining quantum channel fidelity (or short channel fidelity) F(N, M) for transmitting one part of a maximally entangled state with dimension 1 arXiv:2308.15884v3  [quant-ph]  21 Mar 2024 \fM through a noisy quantum channel NA\u2192B from A to B, where A and B are finite dimensional complex Hilbert spaces. Similar to the classical case, this problem can be formulated as a bilinear optimization problem with matrix-valued variables. In order to approximate F(N, M), an efficiently computable semidefinite programming relaxation was proposed in [LM15]. However, unlike the classical case, the gap between this relaxation and F(N, M) is not well understood in the quantum setting. In other directions, numerical methods based on iterative seesaw techniques, such as those proposed in [RW05, FSW07, TKL10, JRO+17], have been developed to provide lower bounds for F(N, M). These methods are computationally tractable semidefinite programs that often converge in practice. In different settings, several works in [TBR16, WD16, WFD18, KDWW19] have focused on the problem of determining the size of a maximally entangled state that can be transmitted through a noisy quantum channel while maintaining a fixed fidelity of 1 \u2212\u03f5. In a recent work [BBFS21], the authors introduced an asymptotically converging semidefinite programming hierarchy {SDPn(N, M)}n\u2208N of outer bounds for F(N, M) (defined in Section 2.1). The hierarchy provides a measure of convergence speed and approaches the channel fidelity F(N, M) as the parameter n tends to infinity. In particular, the authors showed that the inequality 0 \u2264 OPT(SDPn(N, M)) \u2212F(N, M) \u2264poly(d) \u221an holds, where OPT(SDPn(N, M)) is the optimal value of the program SDPn(N, M), and d depends only on the input dimension, output dimension, M. This result implies that for any \u03f5 > 0, estimating F(N, M) with an additive error of \u03f5 can be achieved by solving the semidefinite program SDPn(N, M), where n = poly(d) \u03f52 . However, it is important to note that the size of the matrix variables in SDPn(N, M) grows exponentially with respect to n. Therefore, direct computation of the optimal value of the program SDPn(N, M) is inefficient. Main result and techniques In this paper, we aim to propose an efficient way to compute the optimal value of the program SDPn(N, M), and therefore, have an efficient approximate algorithm for computing channel fidelity F(N, M), where N is a quantum channel, and M, n \u2208N. In particular, our main result is stated as follows. Theorem. For M, n \u2208N\u22651, let N \u00af A\u2192B be a quantum channel, and let d \u00af A denote the input dimension of N \u00af A\u2192B. The optimal value of the program SDPn(N, M) can be computed in poly(n, d \u00af A) time, for a fixed output dimension of the quantum channel N \u00af A\u2192B. As a direct consequence, the channel fidelity F(N, M) can be estimated with an additive error of \u03f5 in poly(1/\u03f5, d \u00af A) time. To achieve this goal, we exploit the symmetries of the semidefinite program SDPn(N, M) to obtain an equivalent representation and solve it efficiently. Specifically, in Lemma 3.1, we demonstrate that the search space of the program can be restricted to an invariant subspace EndSn(Hn), which is obtained by taking a natural action of the symmetric group Sn on the Hn the original search space. Utilizing tools from representation theory, particularly the representation theory of the symmetric group, we can construct a bijective linear map \u03c8 from EndSn(Hn) to Lt i=1 Cmi\u00d7mi for some integers t and mi, i = 1, . . . , t. This map ensures that for any X \u2208EndSn(Hn), the matrix \u03c8(X) takes on a block diagonal form with poly(n) number of blocks, i.e., t = poly(n), and each of which has a size bounded by a polynomial in d \u00af A and n, i.e., mi = poly(d \u00af A, n) for all i \u2208{1, . . . , t}. In addition, the bijective map \u03c8 preserves positive semidefiniteness, i.e., for any X \u2208EndSn(Hn), X \u2ab00 if and only if \u03c8(X) \u2ab00. Therefore, for any X \u2208EndSn(Hn), checking whether X is a positive semidefinite matrix can be reduced to checking if the smaller matrices [\u03c8(X)]1, . . . , [\u03c8(X)]t are all positive semidefinite, where [\u03c8(X)]i is the i-th block of \u03c8(X) for i = 1, . . . , t. From the construction of \u03c8, we can build a transformation \u03a6 that converts the program SDPn(N, M) into an equivalent semidefinite program, denoted as \u03a6(SDPn(N, M)), which has poly(d \u00af A, n) number of constraints, and each matrix variable has a polynomial size in d \u00af A and n (see Theorem 3.4). Additionally, by utilizing the representation of the symmetric group and exploiting specific properties of the constraints involved in the program SDPn(N, M), we show in Theorem 3.7 that the transformation can be implemented in poly(d \u00af A, n) time, while a direct implementation of the transformation would require exponential computation in n. Therefore, we can determine the optimal value of the program 2 \fSDPn(N, M) by solving the program \u03a6(SDPn(N, M)), and this can be done in poly(d \u00af A, n) time. The Table 1 shows the reduction in the size of the matrix variables in the program \u03a6(SDPn(N, 2)) compared to the program SDPn(N, 2) (for certain values of n) for arbitrary qubit quantum channels N. n size.matrix in SDPn(N, 2) max.size.matrix in \u03a6(SDPn(N, 2)) number of blocks 2 64 \u00d7 64 40 \u00d7 40 2 3 256 \u00d7 256 80 \u00d7 80 3 4 1024 \u00d7 1024 180 \u00d7 180 5 5 4096 \u00d7 4096 336 \u00d7 336 6 6 16384 \u00d7 16384 560 \u00d7 560 9 7 65536 \u00d7 65536 896 \u00d7 896 11 8 262144 \u00d7 262144 1440 \u00d7 1440 15 9 1048576 \u00d7 1048576 2160 \u00d7 2160 18 10 4194304 \u00d7 4194304 3080 \u00d7 3080 23 Table 1: Comparing the sizes of two semidefinite programs: SDPn(N, 2) vs. \u03a6(SDPn(N, 2)) for arbitrary qubit quantum channels N In the case where the input dimension d \u00af A of the channel N \u00af A\u2192B is fixed, we will consider another semidefinite programming hierarchy, which is also an asymptotically converging semidefinite programming hierarchy of outer bounds for determining F(N, M) (see Section 2.1 for more details). By exploiting symmetries with the same strategy as in this work for the program, we can show that the channel fidelity F(N, M) can be estimated with an additive error of \u03f5 in poly(1/\u03f5, dB), where dB is the output dimension of the quantum channel N \u00af A\u2192B. Related work Algorithmic aspects of optimal channel coding have been extensively studied in various settings. In classical channels, the problem was explored in [BF17] for point-to-point channels, and a generalization was investigated in [BFGG22]. For entanglement-assisted classical channels, the research can be found in [BFS16], and for the multiple-access channel, the study is investigated in [FF23b], and for broadcast channels, it is discussed in [FF23a]. In terms of quantum channels, the problem has been examined in [LM15, BBFS21, KDWW21, HSW23] for general quantum channels and in [FSS19] for classical-quantum channels. In addition, in recent work [HSW23], authors exploited the unitary covariance symmetry of the identity channel to reduce the complexity of the SDPn(N, M) when n = 2 compared with the original form in [BBFS21]. Exploiting symmetries to simplify convex programs, especially semidefinite programs, has been done for various problems and applications [LM11, Chapter 9]. This approach has been seen in other surveys 3 \fas well [Val09, DK10]. Specifically, it has been applied to the mutually unbiased bases problem [GP21] and the classical (quantum) capacity of quantum channels in quantum information theory [FST22], quantum separability problem [DPS04]. In coding theory, this method has been used to provide upper bounds for nonbinary codes [Sch05, GST06, Lau07, LPS17, Pol19a]. Furthermore, this approach has also proven beneficial in various other fields and problems, such as queueing theory [BLS21, Pol22], crossing numbers for graphs [DKPS07, BP22], truss topology optimization [BdKPS09], and polynomial optimization [GP04, RTAL13, RSST18]. Our approach in this paper is to exploit symmetries under the symmetric group action, inspired by some prior works in [LPS17, GP21, FST22]. Organization The rest of this paper is organized as follows. Section 2 provides an introduction to some fundamental notation and a brief overview of the problem of approximate quantum error correction, and presents tools for exploiting symmetries in semidefinite programs. Finally, our proof for the main result is presented in Section 3. 2 Preliminaries Basic notation Let H be a finite dimensional complex Hilbert space; we denote by L (H) the set of linear operators on H, P(H) denotes the set of positive semidefinite operators on H, and S(H) := {\u03c1 \u2208P(H) : tr (\u03c1) = 1} is the set of density operators on H. For any two Hermitian operators \u03c1, \u03c3 \u2208L (H), we write \u03c3 \u2ab0\u03c1 if \u03c3 \u2212\u03c1 \u2208P(H). Let A, B be finite dimensional complex Hilbert spaces, we denote dA, dB as the dimension of A and B, respectively. For X \u2208P(A \u2297B), we often explicitly indicate the quantum systems as a subscript by writing XAB. The marginal on the subsystem A is denoted XA = tr B(XAB) := P i(IA \u2297\u27e8i|B)XAB(IA \u2297|i\u27e9B), where {|i\u27e9B}i is an orthogonal basis of B and IA denotes the identity map on L (A). Let {|i\u27e9}i and {|j\u27e9}j be the standard bases for A and B, respectively. We will use a correspondence between the linear operator in L (B, A) and vectors in A \u2297B, given by the linear map vec : L (B, A) \u2192A \u2297B, defined as vec (|i\u27e9 \u27e8j|) = |i\u27e9|j\u27e9. In the case of n copies of the same Hilbert space H, we denote by H\u2297n = H \u2297H \u2297\u00b7 \u00b7 \u00b7 \u2297H, we also use the notation Hn 1 or Hn to indicate H1 \u2297\u00b7 \u00b7 \u00b7 \u2297Hn, where Hi \u223c = H for all i \u2208[n]. For i < j, we denote by Hj i := Hi \u2297\u00b7 \u00b7 \u00b7 \u2297Hj. We denote by CP(A : B) the set of completely positive (CP) maps from L (A) to L (B). A quantum channel NA\u2192B is a CP and trace-preserving linear map from L (A) to L (B). Let A\u2032 be isomorphic to A and |\u03a6\u27e9AA\u2032 = 1 \u221adA P i |i\u27e9A|i\u27e9A\u2032 be the maximally entangled state. For a linear map NA\u2032\u2192B, we denote by JN AB \u2208P(A \u2297B) the corresponding Choi matrix defined as JN AB = (IA \u2297N)(|\u03a6\u27e9 \u27e8\u03a6|AA\u2032). For n \u2208N, we use the notation Sn to denote the symmetric group on n symbols. This group consists of the permutations that can be performed on the n symbols, and its group operation involves the composition of permutations. For every \u03c0 \u2208Sn, we define the action of \u03c0 on n copies of a finite dimensional Hilbert space H\u2297n as \u03c0 \u00b7 (h1 \u2297\u00b7 \u00b7 \u00b7 \u2297hn) = h\u03c0\u22121(1) \u2297\u00b7 \u00b7 \u00b7 \u2297h\u03c0\u22121(n) , hi \u2208H , \u2200\u03c0 \u2208Sn . For every \u03c0 \u2208Sn, we denote by UHn(\u03c0) is a permutation matrix which corresponds to the action of \u03c0 on H\u2297n. For any \u03c1Hn \u2208H\u2297n, we write UHn(\u03c0) (\u03c1Hn) := UHn(\u03c0)\u03c1HnUHn(\u03c0)\u2217, where UHn(\u03c0)\u2217is conjugate transpose of UHn(\u03c0). Moreover, because the permutation matrices are real, thus UHn(\u03c0)\u2217= UHn(\u03c0)T for all \u03c0 \u2208Sn. A multipartite operator \u03c1Hn \u2208L (H\u2297n) is called symmetric if \u03c1Hn = UHn(\u03c0) (\u03c1Hn) for all \u03c0 \u2208Sn . 4 \f2.1 Approximate quantum error correction In this section, we briefly present the mathematical setting of approximate quantum error correction. For further information, we refer to [BBFS21, Bor22]. Let A, B, \u00af A, \u00af B be finite dimensional complex Hilbert spaces. Let N \u00af A\u2192B (N in short) be a quantum channel. The quantum channel fidelity (or short channel fidelity) for message dimension M \u2208N is defined as F(N, M) := max DB\u2192\u00af B ,EA\u2192\u00af A F (\u03a6 \u00af BR, ((DB\u2192\u00af B \u25e6N \u00af A\u2192B \u25e6EA\u2192\u00af A) \u2297IR)(\u03a6AR)) s.t. DB\u2192\u00af B , EA\u2192\u00af A quantum channels , where F(\u03c1, \u03c3) := \u2225\u221a\u03c1\u221a\u03c3\u22252 1 denotes the fidelity, \u03a6AR denotes the maximally entangled state on AR, and M = dA = d \u00af B = dR. Using the Choi-Jamiolkowski isomorphism, the authors in [BBFS21] demonstrated that the channel fidelity can be reformulated as a bilinear optimization problem as follows. F(N, M) = max EA \u00af A,DB \u00af B d \u00af AdB \u00b7 tr \u0002 (JN \u00af AB \u2297\u03a6A \u00af B)(EA \u00af A \u2297DB \u00af B) \u0003 s.t. EA \u00af A \u2ab00, , DB \u00af B \u2ab00 EA = IA dA , DB = IB dB . (1) By the linearity of the objective function, we can write the above program in an equivalent form that optimizes over a convex hull of feasible solutions [BBFS21] as follows. F(N, M) = max d \u00af AdB \u00b7 tr \" \u0000JN \u00af AB \u2297\u03a6A \u00af B \u0001  X i\u2208I Ei A \u00af A \u2297Di B \u00af B !# s.t. pi \u22650 , \u2200i \u2208I, X i\u2208I pi = 1 , Ei A \u00af A \u2ab00, Di B \u00af B \u2ab00 \u2200i \u2208I , Ei A = IA dA , Di B = IB dB \u2200i \u2208I . (2) Program (2) can be viewed as the maximization of a linear function over a subset of quantum bipartite states known as separable quantum states. Approximating the set of separable states within the set of bipartite states is a hard problem in quantum information, as showed in [Gha10]. However, it is possible to approximate the set of separable states using a semidefinite programming hierarchy [DPS02, DPS04], and the convergence of this approximation can be analyzed with the finite quantum de Finetti theorem [CKMR07]. Similarly, in [BBFS21], authors considered the problem of approximating feasible solutions of Program (2) and introduced a semidefinite programming hierarchy SDPn(N, M) for this purpose as follows. OPT(SDPn(N, M)) := max \u03c1A \u00af A(B \u00af B)n 1 d \u00af AdB \u00b7 tr h (JN \u00af AB1 \u2297\u03a6A \u00af B1)\u03c1A \u00af AB1 \u00af B1 i s.t. \u03c1A \u00af A(B \u00af B)n 1 \u2ab00 , tr h \u03c1A \u00af A(B \u00af B)n 1 i = 1 , \u03c1A \u00af A(B \u00af B)n 1 = \u0010 IA \u00af A \u2297U(B \u00af B)n 1 (\u03c0) \u0011 \u0010 \u03c1A \u00af A(B \u00af B)n 1 \u0011 \u2200\u03c0 \u2208Sn , \u03c1A(B \u00af B)n 1 = IA dA \u2297\u03c1(B \u00af B)n 1 , \u03c1A \u00af A(B \u00af B)n\u22121 1 Bn = \u03c1A \u00af A(B \u00af B)n\u22121 1 \u2297IBn dB . (SDPn(N, M)) 5 \fNote that we identified B1 := B in Program SDPn(N, M) and recall that B1 \u223c = B2 \u223c = . . . \u223c = Bn and \u00af B1 \u223c = \u00af B2 \u223c = . . . \u223c = \u00af Bn. In addition, by introducing novel finite quantum de Finetti theorems that allow imposing linear constraints on the approximating states, in [BBFS21] provided a quantification of the convergence of Program SDPn(N, M) to channel fidelity F(N, M). Theorem 2.1 ([BBFS21]). Let N \u00af A\u2192B be a quantum channel and n, M \u2208N. Then, we have 0 \u2264OPT(SDPn(N, M)) \u2212F(N, M) \u2264poly(d) \u221an , where d = max{dA = d \u00af B = M, d \u00af A, dB}. Noting that instead of extending the B-systems we could alternatively extend the A-systems, which leads to the following asymptotically converging semidefinite program hierarchy SDPn(N, M) for approximating F(N, M) [BBFS21]. OPT(SDPn(N, M)) := max \u03c1(A \u00af A)n 1 B \u00af B d \u00af AdB \u00b7 tr h (JN \u00af AB1 \u2297\u03a6A \u00af B1)\u03c1A1 \u00af A1B \u00af B i s.t. \u03c1(A \u00af A)n 1 B \u00af B \u2ab00 , tr h \u03c1(A \u00af A)n 1 B \u00af B i = 1 , \u03c1(A \u00af A)n 1 B \u00af B = \u0010 U(A \u00af A)n 1 (\u03c0) \u2297IB \u00af B \u0011 \u0010 \u03c1(A \u00af A)n 1 B \u00af B \u0011 \u2200\u03c0 \u2208Sn , \u03c1(A \u00af A)n 1 B = \u03c1(A \u00af A)n 1 \u2297IB dB , \u03c1(A \u00af A)n\u22121 1 AnB \u00af B = IAn dA \u2297\u03c1(A \u00af A)n\u22121 1 B \u00af B . (SDPn(N, M)) The Program SDPn(N, M) and Program SDPn(N, M) are non-equivalent. However, the convergence guarantees in Theorem 2.1 still hold, i.e., 0 \u2264OPT(SDPn(N, M)) \u2212F(N, M) \u2264poly(d) \u221an , where d = max{dA = d \u00af B = M, d \u00af A, dB}. In this work, given M = dA = d \u00af B \u2208N, we consider the problem of computing SDPn(N, M) where the output dimension dB of N \u00af A\u2192B is fixed. In this case, follow the Theorem 2.1, one as 0 \u2264OPT(SDPn(N, M)) \u2212F(N, M) \u2264poly(d \u00af A) \u221an , where d \u00af A is the input dimension of the channel N. In particular, poly(d \u00af A) is at most O(d \u00af A p log d \u00af A) follows [Bor22, Theorem 4.2.1]. Therefore, by choosing n = poly(d \u00af A) \u03f52 , we can estimate F(N, M) with an accuracy of \u03f5 by solving SDPn(N, M). However, the size of the matrix variables in SDPn(N, M) grows exponentially with n. Thus, solving SDPn(N, M) directly would require exponential time in terms of n. In this paper, we provide an efficient method to compute the optimal value of the program SDPn(N, M), which can be done in poly(d \u00af A, n) time. Remark 2.2. For a fixed M and the output dimension dB of N \u00af A\u2192B, the channel fidelity F(N, M) can be determined in exponential time in terms of d \u00af A by solving Program (1) using a brute-force algorithm, as it forms a constrained polynomial optimization problem. Some methods for solving constrained polynomial optimization problems can be found at [Las15]. Remark 2.3. Noting that our method in this work can be adapted to the case where M and the input dimension d \u00af A of N \u00af A\u2192B are fixed. In this scenario, the same strategy can be employed to exploit the symmetries in Program SDPn(N, M), yielding poly(n, dB) time complexity for solving SDPn(N, M). 2.2 Elementary Representation Theory In this subsection, we present the essential mathematical foundation for exploiting symmetries in a semidefinite program to represent the program effectively. For further information, readers interested in this topic can refer to references such as [LM11, Chapter 9]. 6 \fRepresentation theory We recall some basic facts and notions in the representation theory of finite groups. For further information, we refer the reader to Refs. [Ser77] and [FH13]. Let G be a finite group and H be a finite dimensional complex Hilbert space. A group homomorphism \u03f1 : G \u2192GL(H) is called a linear representation of G on H, where GL(H) is the general linear group on H. The space H is called a G-module. For v \u2208H and g \u2208G, we write g \u00b7 v as shorthand for \u03f1(g)v. For X \u2208L (H), the action of g \u2208G on X is given by \u03f1(g)X\u03f1(g)\u2217. A representation \u03f1 : G \u2192GL(H) of G is called irreducible if for any subspace H\u2032 of H then {gH\u2032 : g \u2208G} \u228aH\u2032, in other words, it contains no proper submodule H\u2032 of H such that gH\u2032 \u2286H\u2032. Let H and H\u2032 be G-modules, a G-equivariant map from H to H\u2032 is a linear map \u03d5 : H \u2192H\u2032 such that g \u00b7 \u03d5(v) = \u03d5(g \u00b7 v) for all g \u2208G, v \u2208H. Two G-modules H and H\u2032 are called G-isomorphic, write H \u223c = H\u2032, if there is a bijective equivariant map from H to H\u2032. We denote by EndG(H), the set of all G-equivariant maps from H to H, i.e., EndG(H) = {T \u2208L (H) : T(g \u00b7 v) = g \u00b7 T(v), \u2200v \u2208H, g \u2208G}. Note that EndG(H) is known to form a matrix \u2217-algebra [LM11], that is, a set of complex matrices that is closed under addition, scalar multiplication, matrix multiplication, and taking the conjugate transpose. Let G be a finite group acting on a finite dimensional complex vector space H. Then the space H can be decomposed into a direct sum of subspaces as H = H1 \u2295\u00b7 \u00b7 \u00b7 \u2295Ht, where H1, . . . , Ht are Gmodules and called the G-isotypical component. In more detail, every Hi is a direct sum of irreducible G-modules denoted as Hi,1 \u2295\u00b7 \u00b7 \u00b7 \u2295Hi,mi and note that two irreducible G-modules Hi,j and Hi\u2032,j\u2032 are isomorphic (i.e., Hi,j \u223c = Hi\u2032,j\u2032) if and only if i = i\u2032. The tuple (m1, . . . , mt) are called multiplicities of the corresponding irreducible representations. For each i \u2208[t] and j \u2208[mi], let ui,j \u2208Hi,j be a nonzero vector such that for each i and all j, j\u2032 \u2208[mi], there is a bijective G-equivariant map from Hi,j to Hi,j\u2032 that maps ui,j to ui,j\u2032. For i \u2208[t], we define a matrix Ui as [ui,1, . . . , ui,mi], with ui,j forming the j-th column of Ui. The matrix set {U1, . . . , Ut} obtained in this way is called a representative matrix set for the action of G on H. The columns of the matrices Ui can be viewed as elements of the dual space H\u2217(by taking the standard inner product). Then each Ui is an ordered set of linear functions on H. Since Hi,j is the linear space spanned by G \u00b7 ui,j (for each i, j), we have H = t M i=1 mi M j=1 CG \u00b7 ui,j , where CG = nP g\u2208G \u03b1gg : \u03b1g \u2208C o denotes the complex group algebra of G. Furthermore, we have dim EndG(H) = dim EndG \uf8eb \uf8ed t M i=1 mi M j=1 Hi,j \uf8f6 \uf8f8= t X i=1 m2 i . (3) Note that with the action of the finite group G on the space H, any inner product \u27e8, \u27e9on H gives rise to a G-invariant inner product \u27e8, \u27e9G on H via the rule \u27e8x, y\u27e9G := 1 |G| P g\u2208G\u27e8g \u00b7 x, g \u00b7 y\u27e9. Let \u27e8, \u27e9be a G-invariant inner product on H and {U1, . . . , Ut} be a representative matrix set for the action of G on H. Consider the linear map \u03c8 : EndG(H) \u2192Lt i=1 Cmi\u00d7mi defined as \u03c8(X) := t M i=1 (\u27e8Xui,j\u2032, ui,j\u27e9)mi j,j\u2032=1 , \u2200X \u2208EndG(H) . (4) For i \u2208[t] and X \u2208EndG(H), we denote the matrix (\u27e8Xui,j\u2032, ui,j\u27e9)mi j,j\u2032=1 corresponding to the i-th block of \u03c8(X) by J\u03c8(X)Ki. 7 \fLemma 2.4 (Proposition 2.4.4, [Pol19b]). The linear map \u03c8 of Eq. (4) is bijective and for every X \u2208EndG(H), we have X \u2ab00 if and only if \u03c8(X) \u2ab00. Moreover, there is a unitary matrix U such that U \u2217XU = t M i=1 mi M j=1 J\u03c8(X)Ki , \u2200X \u2208EndG(H) , where mi = dim(Hi,1), for every i \u2208[t]. Lemma 2.4 plays a crucial role in our symmetry reductions. Noting that dim(EndG(H)) = Pt i=1 m2 i can be significantly smaller than the dimension of H. Furthermore, thanks to this lemma, we can simplify the task of verifying whether a matrix X \u2208EndG(H) is positive semidefinite. This simplification involves checking if the smaller mi \u00d7 mi matrices J\u03c8(X)Ki are positive semidefinite for each i \u2208[t]. The following result, which was mentioned in [Pol19b], helps construct a representative set of direct product groups. Lemma 2.5. Let G1 and G2 be two finite groups. Let {U (1) 1 , . . . , U (1) k1 } and {U (2) 1 , . . . , U (2) k2 } be the representative matrix sets that correspond to the action of G1 and G2 on H1 and H2, respectively. Then {U (1) i \u2297U (2) j : i = 1, . . . , k1, j = 1, . . . , k2} (5) is representative matrix set for the action of G1 \u00d7 G2 on H1 \u2297H2. Proof. Let {U (1) 1 , . . . , U (1) k1 } and {U (2) 1 , . . . , U (2) k2 } be the representative matrix sets that correspond to the action of G1 and G2 on H1 and H2, respectively. That is, let H1 = \u2295k1 i=1 \u2295mi j=1 H(1) i,j and H2 = \u2295k2 i=1\u2295si j=1H(2) i,j be decompositions of H1 and H2 into irreducible modules, such that H(1) i,j \u223c = H(1) i\u2032,j\u2032 if and only if i = i\u2032 (for i \u2208[k1] and j \u2208[mi]), and H(2) i,j \u223c = H(2) i\u2032,j\u2032 if and only if i = i\u2032 (for i \u2208[k2] and j \u2208[si]). For each i \u2208[k1] and j \u2208[mi], let u(1) i,j \u2208H(1) i,j be a nonzero vector such that for each i \u2208[k1] and j, j\u2032 \u2208[mi] there is a bijective G1-equivariant linear map \u03d5(1) i,j,j\u2032 from H(1) i,j to H(1) i,j\u2032 mapping u(1) i,j to u(1) i,j\u2032. Assume that the matrix U (1) i := (u(1) i,j |j \u2208[mi]) is the i-th matrix in the representative matrix set for the action of G1 on H1, for i \u2208[k1]. Similarly, For each i \u2208[k2] and j \u2208[si], let u(2) i,j \u2208H(2) i,j be a nonzero vector such that for each i \u2208[k2] and j, j\u2032 \u2208[si] there is a bijective G2-equivariant linear map \u03d5(2) i,j,j\u2032\u2032 from H(2) i,j to H(2) i,j\u2032 mapping u(2) i,j to u(2) i,j\u2032. Assume that the matrix U (2) i := (u(2) i,j |j \u2208[si]) is the i-th matrix in the representative matrix set for the action of G2 on H2, for i \u2208[k2]. According to [Ser77, Theorem 10], the decomposition of H1 \u2297H2 into irreducible representations for G1 \u00d7 G2 is \u2295k1 i=1 \u2295k2 i\u2032=1 \u2295mi j=1 \u2295si j\u2032=1 H(1) i,j \u2297H(2) i\u2032,j\u2032, and H(1) i,j \u2297H(2) i\u2032,j\u2032 \u223c = H(1) i\u2032\u2032,j\u2032\u2032 \u2297H(2) i\u2032\u2032\u2032,j\u2032\u2032\u2032 if and only if i = i\u2032\u2032 and i\u2032 = i\u2032\u2032\u2032. Now, if u(1) i,j \u2297u(2) i\u2032,j\u2032 \u2208H(1) i,j \u2297H(2) i\u2032,j\u2032 and u(1) i,j\u2032\u2032 \u2297u(2) i\u2032,j\u2032\u2032\u2032 \u2208H(1) i,j\u2032\u2032 \u2297H(2) i\u2032,j\u2032\u2032\u2032, the map \u03d5(1) i,j,j\u2032 \u2297\u03d5(2) i\u2032,j\u2032\u2032,j\u2032\u2032\u2032 is a G1 \u00d7G2-equivariant map from H(1) i,j \u2297H(2) i\u2032,j\u2032 to H(1) i,j\u2032\u2032 \u2297H(2) i\u2032,j\u2032\u2032\u2032 mapping u(1) i,j \u2297u(2) i\u2032,j\u2032 to u(1) i,j\u2032\u2032 \u2297u(2) i\u2032,j\u2032\u2032\u2032. It follows that the representative vectors for the action of G1 \u00d7G2 on H1 \u2297H2 are the vectors u(1) i,j \u2297u(2) i\u2032,j\u2032, and that the representative matrix set is {U (1) i \u2297U (2) j : i = 1, . . . , k1, j = 1, . . . , k2}, as desired. Representation theory of the symmetric group Fix n \u2208N and a finite-dimensional vector space H with dim(H) = d. We consider the natural action of the symmetric group Sn on H\u2297n by permuting the indices, i.e., \u03c0 \u00b7 (h1 \u2297\u00b7 \u00b7 \u00b7 \u2297hn) = h\u03c0\u22121(1) \u2297\u00b7 \u00b7 \u00b7 \u2297h\u03c0\u22121(n) , hi \u2208H , \u2200\u03c0 \u2208Sn . (6) 8 \fBased on representation theory of the symmetric group, we describe a representative set for the action of Sn on H\u2297n. Many of the concepts and notation are based on the method for symmetry reduction from [LPS17]. The concepts we describe in this section will be used throughout this paper A partition \u03bb of n is a sequence (\u03bb1, . . . , \u03bbd) of natural numbers with \u03bb1 \u2265. . . \u03bbd > 0 and \u03bb1 + \u00b7 \u00b7 \u00b7 + \u03bbd = n. The number d is called the height of \u03bb. We write \u03bb \u22a2d n if \u03bb is a partition of n with height d. Let Par(d, n) := {\u03bb : \u03bb \u22a2d n}. The Young shape Y (\u03bb) of \u03bb is the set Y (\u03bb) := {(i, j) \u2208N2 : 1 \u2264j \u2264d, 1 \u2264i \u2264\u03bbj} . Following the French notation [Pro07], for an index j0 \u2208[d], the j0-th row of Y (\u03bb) is set of elements (i, j0) in Y (\u03bb). Similarly, fixing an element i0 \u2208[\u03bb1], the i0-th column of Y (\u03bb) is set of elements (i0, j) in Y (\u03bb). We label the elements in Y (\u03bb) from 1 to k according the lexicographic order on their positions. Then the row stabilizer R\u03bb of \u03bb is the group of permutations \u03c0 of Y (\u03bb) with \u03c0(L) = L for each row L of Y (\u03bb). Similarly, the column stabilizer C\u03bb of \u03bb is the group of permutations \u03c0 of Y (\u03bb) with \u03c0(L) = L for each column L of Y (\u03bb). For \u03bb \u22a2d n, a \u03bb-tableau is a function \u03c4 : Y (\u03bb) \u2192N. A \u03bb-tableau is semistandard if the entries are non-decreasing in each row and strictly increasing in each column. Let T\u03bb,d be the collection of semistandard \u03bb-tableaux with entries in [d]. We write \u03c4 \u223c\u03c4 \u2032 for \u03bb-tableaux \u03c4, \u03c4 \u2032 if \u03c4 \u2032 = \u03c4r for some r \u2208R\u03bb. Let e1, . . . , ed be the standard basis of H. For any \u03c4 \u2208T\u03bb,d, define u\u03c4 \u2208H\u2297n as u\u03c4 := X \u03c4 \u2032\u223c\u03c4 X c\u2208C\u03bb sgn(c) O y\u2208Y (\u03bb) e\u03c4 \u2032(c(y)) . (7) Here the Young shape Y (\u03bb) is ordered by concatenating its rows. Then the matrix set {U\u03bb : \u03bb \u22a2d n} with U\u03bb = [u\u03c4 : \u03c4 \u2208T\u03bb,d] (8) is a representative matrix set for the natural action of Sn on H\u2297n [LPS17, Section 2.1]. Moreover, we have |Par(d, n)| \u2264(n + 1)d and |T\u03bb,d| \u2264(n + 1)d(d\u22121)/2 , \u2200\u03bb \u2208Par(d, n) . (9) 3 Efficient approximation of channel fidelity Given Hilbert space H, we follow the notation in Section 2 and consider the action of symmetric group Sn on H\u2297n defined in Eq. (6). Throughout this paper, we work with spaces of the form A\u2297H\u2297n\u2297B for some Hilbert spaces A, B. A multipartite operator \u03c1AHnB \u2208L (A \u2297H\u2297n \u2297B) is said to be symmetric with respect to A and B, if it is invariant under permutation of H-systems while keeping A and B fixed. In particular, \u03c1AHnB = (IA \u2297UHn(\u03c0) \u2297IB) (\u03c1AHnB) := (IA \u2297UHn(\u03c0) \u2297IB) \u03c1AHnB (IA \u2297UHn(\u03c0)\u2217\u2297IB) \u2200\u03c0 \u2208Sn . The invariant subspace under the action of Sn is given by EndSn \u0000A \u2297H\u2297n \u2297B \u0001 := \b \u03c1 \u2208L \u0000A \u2297H\u2297n \u2297B \u0001 : \u03c1 = (IA \u2297UHn(\u03c0) \u2297IB) (\u03c1) for all \u03c0 \u2208Sn \t . We first show in the following lemma that the search space of Program SDPn(N, M) can be restricted to the invariant subspaces that arising under the action of Sn. Lemma 3.1. For n \u2208N\u22651, if \u03c1A \u00af A(B \u00af B)n 1 \u2208EndSn \u0000A \u00af A \u2297(B \u00af B)\u2297n\u0001 , then \u03c1A(B \u00af B)n 1 , IA dA \u2297\u03c1(B \u00af B)n 1 \u2208EndSn \u0000A \u2297(B \u00af B)\u2297n\u0001 , (10) \u03c1A \u00af A(B \u00af B)n\u22121 1 Bn , \u03c1A \u00af A(B \u00af B)n\u22121 1 \u2297IBn dB \u2208EndSn\u22121 \u0010 A \u00af A \u2297(B \u00af B)\u2297(n\u22121) \u2297Bn \u0011 . (11) 9 \fProof. The proof can be found in Appendix A. From Lemma 3.1, we will use the tools from Section 2.2 to simplify Program SDPn(N, M). Let Par(dH, n) := {\u03bb : \u03bb \u22a2dH n} and m\u03bb(H) := |T\u03bb,dH| for each \u03bb \u2208Par(dH, n). From Lemma 2.5, the representative matrix set for the action of Sn on the space A \u2297H\u2297n \u2297B is described as follows, {IA \u2297U\u03bb \u2297IB}\u03bb\u2208Par(dH,n) , (12) where {U\u03bb}\u03bb\u2208Par(dH,n) are real matrices defined in Eq. (8). For each \u03bb \u2208Par(dH, n), let m\u03bb(A, Hn, B) be the number of rows of the matrix (IA \u2297U\u03bb \u2297IB). From Lemma 2.4 and the representative matrix set constructed in Eq. (12), we have the following proposition. Proposition 3.2. The following map \u03a8A,Hn,B : EndSn(A \u2297H\u2297n \u2297B) \u2192 M \u03bb\u22a2dHn Cm\u03bb(A,Hn,B)\u00d7m\u03bb(A,Hn,B) X 7\u2192 M \u03bb\u22a2dHn \u0000IA \u2297U T \u03bb \u2297IB \u0001 X (IA \u2297U\u03bb \u2297IB) , (13) is a bijective linear map that preserve the positive semidefinitess property, i.e., X \u2ab00 if and only if \u03a8A,Hn,B(X) \u2ab00. Moreover, as a consequences of Eqs. (9) and (3), we have the following proposition. Proposition 3.3. We have: |Par(dH, n)| \u2264(n + 1)dH , (14) m\u03bb(A, Hn, B) \u2264dAdB(n + 1)dH(dH\u22121)/2 : \u2200\u03bb \u2208Par(dH, n) , (15) m(A, Hn, B) := dim h EndSn \u0000A \u2297H\u2297n \u2297B \u0001i \u2264d2 Ad2 B(n + 1)d2 H . (16) A basis for the invariant subspace. We can construct the canonical basis that consists of zeroone incidence matrices of the EndSn (A \u2297H\u2297n \u2297B) from the orbits of the group action of Sn on pairs [dA \u00d7 (dH)n \u00d7 dB]2(see [KPS07] or [LM11, Chapter 9] for more information). In particular, let i \u2208[dA \u00d7 (dH)n \u00d7 dB] be the index of the standard basis of A \u2297H\u2297n \u2297B. Then the orbit of the pair (i, j) \u2208[dA \u00d7 (dH)n \u00d7 dB]2 under the action of the group Sn is given by O(i, j) = {(\u03c0(i), \u03c0(j)) : \u03c0 \u2208Sn}, where \u03c0(i) is the index of the basis vector (IA \u2297UHn(\u03c0) \u2297IB) |i\u27e9. The set [dA \u00d7 (dH)n \u00d7 dB]2 decomposes into orbits O1, . . . , Om(A,Hn,B) under the action of Sn, recall that m(A, Hn, B) is the dimension of the invariant space EndSn (A \u2297H\u2297n \u2297B). Moreover, from Eq. (16), one has m(A, Hn, B) \u2264 d2 Ad2 B(n + 1)d2 H. For each r \u2208[m(A, Hn, B)], we construct a zero-one matrix Cr of size (dA \u00d7 (dH)n \u00d7 dB) \u00d7 (dA \u00d7 (dH)n \u00d7 dB) given by (Cr)ij = ( 1 if (i, j) \u2208Or , 0 otherwise. (17) We have the set {C1, . . . , Cm(A,Hn,B)} forms a canonical basis of EndSn (A \u2297H\u2297n \u2297B) [LM11, Chapter 9]. 10 \fNext, we will show how to utilize the symmetries in Program SDPn(N, M) to simplify it. Let H := B \u00af B. For each D \u2208{A \u00af A \u2297H\u2297n, A \u2297H\u2297n, A \u00af A \u2297H\u2297n\u22121 \u2297Bn}, we set t(D) = n \u22121 if D = A \u00af A \u2297H\u2297n\u22121 \u2297Bn, otherwise t(D) = n. Let \u03c8D : EndSt(D) (D) \u2192L \u03bb\u2208Par(dH,t(D)) Cm\u03bb(D)\u00d7m\u03bb(D) be the bijective linear map defined in Eq. (13), where dH = dB \u00af B = dBd \u00af B. For any X \u2208EndSt(D) (D) and \u03bb \u2208Par(dH, t(D)), we write J\u03c8D(X)K\u03bb for the block of \u03c8D(X) indexed by \u03bb. Let m(D) denote the dimension of the invariant subspace EndSt(D) (D), i.e., m(D) dim := EndSt(D) (D), and {C1(D), . . . , Cm(D)(D)} denote the canonical basis of EndSt(D) (D) defined in Eq. (17). For D1 := A \u00af A \u2297H\u2297n, D2 := A \u2297H\u2297n and D3 := A \u00af A \u2297H\u2297n\u22121 \u2297Bn. Since \u03c1A \u00af A(B \u00af B)n 1 \u2208EndSn (D1), thus we can write \u03c1A \u00af A(B \u00af B)n 1 = Pm(D1) i=1 xiCi(D1) for some xi \u2208C, i = 1, . . . , m(D1). From Lemma 3.1 and Proposition 3.2, we can construct a transformation \u03a6 that maps Program SDPn(N, M) to an equivalent semidefinite program \u03a6(SDPn(N, M)) as follows. maximize : d \u00af AdB \u00b7 m(D1) X i=1 xi \u00b7 tr h\u0010 J N \u00af AB1 \u2297\u03a6A \u00af B1 \u0011 Ci(D1)A \u00af AB1 \u00af B1 i s.t. m(D1) X i=1 xi \u00b7 J\u03c8D1(Ci(D1))K\u03bb \u2ab00 , \u2200\u03bb \u2208Par(dBd \u00af B, n) , m(D1) X i=1 xi \u00b7 tr [Ci(D1)] = 1 , dA \u00b7 m(D1) X i=1 xi \u00b7 J\u03c8D2 (tr \u00af A(Ci(D1)))K\u03bb = m(D1) X i=1 xi \u00b7 J\u03c8D2 (IA \u2297tr A \u00af A(Ci(D1)))K\u03bb \u2200\u03bb \u2208Par(dBd \u00af B, n) , dB \u00b7 m(D1) X i=1 xi \u00b7 q \u03c8D3 \u0000tr \u00af Bn(Ci(D1)) \u0001y \u03bb = m(D1) X i=1 xi \u00b7 q \u03c8D3 \u0000tr Bn \u00af Bn(Ci(D1)) \u2297IB \u0001y \u03bb \u2200\u03bb \u2208Par(dBd \u00af B, n \u22121) , x1, . . . , xm(D1) \u2208C . (\u03a6(SDPn(N, M))) Theorem 3.4. For M, n \u2208N\u22651. Let N \u00af A\u2192B be a quantum channel. The semidefinite program \u03a6(SDPn(N, M)) has O \u0010 d2 Ad2 \u00af A(n + 1)d2 Bd2 \u00af B \u0011 variables and O \u0000(n + 1)dBd \u00af B\u0001 positive semidefinite constraints involving matrices of size at most O \u0000dAd \u00af AdB(n + 1)dBd \u00af B(dBd \u00af B\u22121)/2\u0001 . As a consequence, the size of the Program \u03a6(SDPn(N, M)) is poly(d \u00af A, n), for fixed dA = d \u00af B = M and dB. Proof. The proof of the theorem follows directly by using the results from Proposition 3.3. Firstly, the number of variables in Program \u03a6(SDPn(N, M)) is m(D1) = dim EndSn \u0000A \u00af A \u2297(B \u00af B)\u2297n\u0001 . Thus, from Eq. (16), it is bounded by O \u0010 d2 Ad2 \u00af A(n + 1)d2 Bd2 \u00af B \u0011 . Since |Par(dBd \u00af B, n)| \u2264(n + 1)d2 Bd2 \u00af B according to Eq. (14), so the number of positive semidefinite (PSD) constraints is bounded by O \u0000(n + 1)dBd \u00af B\u0001 . Finally, for every q \u2208[3] and \u03bb \u2208Par(dBd \u00af B, n), by Eq. (15), we have m\u03bb(Dq) \u2264dAd \u00af AdB(n + 1)dBd \u00af B(dBd \u00af B\u22121)/2. This implies that the size of matrices involved in PSD constraints is at most O \u0000dAd \u00af AdB(n + 1)dBd \u00af B(dBd \u00af B\u22121)/2\u0001 . Therefore, when dA = d \u00af B = M and dB are fixed, the size of Program \u03a6(SDPn(N, M)) is bounded by poly(d \u00af A, n), which completes the proof. 11 \fFrom Theorem 3.4, we have the size of the Program \u03a6(SDPn(N, M)) is bounded by a polynomial in n and d \u00af A, therefore it can be solved in poly(d \u00af A, n) time. However, directly applying the transformation \u03a6 that maps Program SDPn(N, M) into the equivalent Program \u03a6(SDPn(N, M)) requires exponential computations in n. Next, we will provide an efficient method to accomplish this task. In particular, let {C1(D), . . . , Cm(D)(D)} be the canonical basis of EndSt(D) (D) defined in Eq. (17). Given z1, . . . , zm(D) \u2208C, we show that for any \u03bb \u2208Par(dH, t(D)), the block r \u03c8D \u0010Pm(D) i=1 ziCi(D) \u0011z \u03bb can be computed in poly(d \u00af A, n) time, where dH is fixed. We then use this result to apply it to our specific spaces to reach the desired conclusion. Let CH 1 , . . . , CH m(H) denote the canonical basis of EndSn (H\u2297n) as defined in Eq. (17). Furthermore, we can explicitly express this canonical basis as follows, {|iA\u27e9 \u27e8jA| \u2297CH t \u2297|iB\u27e9 \u27e8jB|}iA,jA\u2208[dA] iB,jB\u2208[dB] t\u2208[m(H)] . (18) Let \u03c8A,H,B : EndSn (A \u2297H\u2297n \u2297B) \u2192L \u03bb\u2208Par(dH,n) Cm\u03bb(A,Hn,B)\u00d7m\u03bb(A,Hn,B) be a bijective linear map defined in Eq. (13). For any \u03bb \u2208Par(dH, n), we have q \u03c8A,H,B \u0000|iA\u27e9 \u27e8jA| \u2297CH t \u2297|iB\u27e9 \u27e8jB| \u0001y \u03bb := \u0000IA \u2297U T \u03bb \u2297IB \u0001 \u0000|iA\u27e9 \u27e8jA| \u2297CH t \u2297|iB\u27e9 \u27e8jB| \u0001 (IA \u2297U\u03bb \u2297IB) = |iA\u27e9 \u27e8jA| \u2297U T \u03bb CH t U\u03bb \u2297|iB\u27e9 \u27e8jB|. (19) We are going to show U T \u03bb CH t U\u03bb can be computed in poly(n) time, for any \u03bb \u2208Par(dH, n) and t \u2208[m(H)]. More generally, given z1, . . . , zm(H) \u2208C, for any \u03bb \u2208Par(dH, n), we will show that Pm(H) i=1 ziU T \u03bb CH i U\u03bb can be computed in poly(n) time. In particular, from Eqs. (8) and (9), we just need to show that we can compute Pm(H) i=1 ziuT \u03c4 CH i u\u03b3 in poly(n) time for any \u03c4, \u03b3 \u2208TdH,\u03bb. We note that u\u03c4, u\u03b3 and CH r all have exponential size in n. As a direct consequence from [LPS17, Lemma 2] (see also in [FST22, Lemma 4.5]), we have the following lemma. Lemma 3.5. Let \u03bb \u2208Par(dH, n), and \u03c4, \u03b3 \u2208T\u03bb,dH. Given z1, . . . , zm(H) \u2208C. Then Pm(H) r=1 zruT \u03c4 CH r u\u03b3 can be computed in polynomial time in n, for fixed dH. As a direct consequence, for any iA, jA \u2208 [dA]; iB, jB \u2208[dB]; t \u2208[m(H)], we can determine the matrix q \u03c8A,H,B \u0000|iA\u27e9 \u27e8jA| \u2297CH t \u2297|iB\u27e9 \u27e8jB| \u0001y \u03bb in poly(dA, dB, n) time for all \u03bb \u2208Par(dH, n). The proof of Lemma 3.5 is based on [LPS17] and [Pol19b, pp. 30-31] (see also [FST22]), but we provide it here for the reader\u2019s convenience. We first introduce some notation and basic results. For a finite dimensional complex vector space H, the dual vector space H\u2217of H is the vector space of all linear transformations \u03c6 : H \u2192C. The coordinate ring of H, denoted as O(H), is the algebra consisting of all C-linear combinations of products of elements from H\u2217. An element of O(H) is called a polynomial on H. A polynomial p \u2208O(H) is called homogeneous if it is a C-linear combination of a product of n non-constant elements of H\u2217(for a fixed non-negative integer n). We denote by On(H) the set of all homogeneous polynomials of degree n. Set WH := H \u2297H, for each p := (x, y) \u2208[dH] \u00d7 [dH], define ap := x \u2297y \u2208WH , then the set W := {ap : p \u2208[dH] \u00d7 [dH]} is a basis of WH. Let W\u2217:= {a\u2217 p : p \u2208[dH] \u00d7 [dH]} be the corresponding dual basis for W \u2217 H. Using the natural identification of ([dH] \u00d7 [dH])n and [(dH)n]2, for any r \u2208[m(H)], we have CH r := X (p1,...,pn)\u2208OH r ap1 \u2297\u00b7 \u00b7 \u00b7 \u2297apn \u2208W \u2297n H . 12 \fNote that CH r can be obtained from vec \u0000CH r \u0001 by applying the permutation operator which maps (H\u2297n)\u22972 to \u0000H\u22972\u0001\u2297n. For every (p1, . . . , pn) \u2208[(dH)2]n, let \u00b5(p1, . . . , pn) := a\u2217 p1 \u00b7 \u00b7 \u00b7 a\u2217 pn \u2208On(WH) (20) be a degree n monomial expressed in the basis W\u2217. Note that, for a fixed r \u2208[m(H)], \u00b5(p1, . . . , pn) is the same monomial, for every (p1, . . . , pn) \u2208OH r . We denote this monomial by \u00b5 \u0000OH r \u0001 . Moreover, {OH r }r\u2208[m(H)] partitions \u0002 (dH)2\u0003n into disjoint subsets. Therefore, there exists a bijection between {OH r }r\u2208[m(H)] and the set of degree n monomials expressed in the basis W\u2217. Let \u03b6 : (W \u2217 H)\u2297n \u2192On(WH) be a linear function defined by \u03b6(w\u2217 1 \u2297\u00b7 \u00b7 \u00b7 \u2297w\u2217 n) := w\u2217 1 \u00b7 \u00b7 \u00b7 w\u2217 n for all w\u2217 1, . . . , w\u2217 n \u2208W \u2217 H . We denote w = \u03b6(w) for all w \u2208(W \u2217 H)\u2297n. For any \u03bb \u22a2dH n and \u03c4, \u03b3 \u2208T\u03bb,dH, define the polynomial G\u03c4,\u03b3 \u2208C[xi,j : i, j = 1, . . . , dH] by G\u03c4,\u03b3(X) := X \u03c4 \u2032\u223c\u03c4 \u03b3\u2032\u223c\u03b3 X c,c\u2032\u2208C\u03bb sgn(cc\u2032) Y y\u2208Y (\u03bb) x\u03c4 \u2032c(y),\u03b3\u2032c\u2032(y) , (21) for X = (xi,j)dH i,j=1 \u2208CdH\u00d7dH. Ref [LPS17, Proposition 3] and [Gij09, Theorem 7] show that the polynomial in Eq (21) can be computed in polynomial time, i.e., expressed as a linear combination of monomials in variables xi,j in poly(n) time. Lemma 3.6 ([Gij09, LPS17]). Let \u03bb \u2208Par(dH, n) and every \u03c4, \u03b3 \u2208T\u03bb,dH. Expressing the polynomial G\u03c4,\u03b3(X) as a linear combination of monomials can be done in poly(n) time, for fixed dH. We now prove Lemma 3.5. Proof Lemma 3.5. For r \u2208[m(H)], utilizing the fact that CH r \u2208H\u2297n \u2297H\u2297n and u\u03c4, u\u03b3 \u2208(H\u2297n)\u2217(via the standard inner product), we can write uT \u03c4 CH r u\u03b3 = (u\u03c4 \u2297u\u03b3)(CH r ). Set g := u\u03c4 \u2297u\u03b3 = X \u03c4 \u2032\u223c\u03c4 \u03b3\u2032\u223c\u03b3 X c,c\u2032\u2208C\u03bb sgn(cc\u2032) O y\u2208Y (\u03bb) (F)\u03c4 \u2032c(y),\u03b3\u2032c\u2032(y) , where F \u2208(W \u2217)dH\u00d7dH with (F)x,y = a\u2217 (x,y). Then m(H) X r=1 (u\u03c4 \u2297u\u03b3)(CH r )\u00b5(OH r ) = m(H) X r=1 g(CH r )\u00b5(OH r ) = X (p1,...,pn)\u2208([dH]\u00d7[dH])n g(ap1 \u2297\u00b7 \u00b7 \u00b7 \u2297apn)a\u2217 p1 \u00b7 \u00b7 \u00b7 a\u2217 pn = X (p1,...,pn)\u2208([dH]\u00d7[dH])n g(ap1 \u2297\u00b7 \u00b7 \u00b7 \u2297apn)a\u2217 p1 \u2297\u00b7 \u00b7 \u00b7 \u2297a\u2217 pn = g = X \u03c4 \u2032\u223c\u03c4 \u03b3\u2032\u223c\u03b3 X c,c\u2032\u2208C\u03bb sgn(cc\u2032) Y y\u2208Y (\u03bb) (F)\u03c4 \u2032c(y),\u03b3\u2032c\u2032(y) = G\u03c4,\u03b3(F) . By the Lemma 3.6, one computes the entry Pm(H) r=1 zruT \u03c4 CH r u\u03b3 by replacing each monomial \u00b5(OH r ) in G\u03c4,\u03b3(F) with the variable zr. Thus, given z1, . . . , zm(H) we can compute the value Pm(H) r=1 zruT \u03c4 CH r u\u03b3 in poly(n) time. Finally, from Eq. (19) we can determine the matrix q \u03c8A,H,B \u0000|iA\u27e9 \u27e8jA| \u2297CH t \u2297|iB\u27e9 \u27e8jB| \u0001y \u03bb in poly(dA, dB, n) time for all \u03bb \u2208Par(dH, n) and iA, jA \u2208[dA]; iB, jB \u2208[dB]; t \u2208[m(H)], which completes the proof. 13 \fConsider the Program \u03a6(SDPn(N, M)). For each D \u2208A \u00af A \u2297H\u2297n, A \u2297H\u2297n, A \u00af A \u2297H\u2297n\u22121 \u2297Bn and any X \u2208EndSt(D)(D), if all coordinates in the expansion of X in the canonical basis EndSt(D)(D) are given, we can compute the block J\u03c8D(X)K\u03bb in poly(d \u00af A, n) time for all \u03bb \u2208Par(dH, t(D)), as pointed out in Lemma 3.5. With this result, we demonstrate in the following theorem that the transformation \u03a6 can be implemented in polynomial time with respect to d \u00af A and n. As a conclusion, Program SDPn(N, M) can be solved in poly(d \u00af A, n) time. Theorem 3.7. For M, n \u2208N\u22651. Let N \u00af A\u2192B be a quantum channel. The transformation \u03a6 that maps Program SDPn(N, M) to Program \u03a6(SDPn(N, M)) can be done in poly(d \u00af A, n) time for fixed dH = dBd \u00af B. Before proving Theorem 3.7, we present some properties of the orbits that were generated from the natural action symmetric group. Enumerating all orbits. Let {CH 1 , . . . , CH m(H)} be the canonical basis of EndSn (H\u2297n) defined in Eq. (17). For each r = {1, . . . , m(H)}, let OH r be an orbit that corresponds to the matrix CH r , we need to compute a representative element of OH r . In order to do so, we define a matrix E(i,j) \u2208ZdH\u00d7dH \u22650 (E(i,j))a,b := |{v \u2208[n] : iv = a, jv = b}| , \u2200a, b \u2208[dH] . (22) Based on the construction given in Eq. (22), for any two pairs (i, j), (i\u2032, j\u2032) belonging to the sets [dH]n \u00d7 [dH]n, it holds that (i\u2032, j\u2032) = (\u03c0(i), \u03c0(j)) for some permutation \u03c0 \u2208Sn if and only if E(i,j) = E(i\u2032,j\u2032). Therefore, there exists a direct one-to-one mapping between the orbits \b OH r \t r\u2208[mH] and matrices E \u2208ZdH\u00d7dH \u22650 where the sum of all elements in E is equal to n, i.e., P a,b Ea,b = n. Thus, we can efficiently find a representative element for each OH r in polynomial time by listing all non-negative integer solutions of the equation P a,b\u2208[dH] Ea,b = n. For convenience, we will recall some notations. For each D \u2208{A \u00af A\u2297H\u2297n, A\u2297H\u2297n, A \u00af A\u2297H\u2297n\u22121 \u2297 Bn}, we set t(D) = n\u22121 iff D = A \u00af A\u2297H\u2297n\u22121 \u2297Bn. Let {C1(D), . . . , Cm(D)(D)} denote the canonical basis of EndSt(D) (D) defined in Eq. (17). Following the notation in the proof of Theorem 3.4, let \u03c8D : EndSt(D) (D) \u2192L \u03bb\u2208Par(dH,t(D)) Cm\u03bb(D)\u00d7m\u03bb(D) be the bijective linear map defined in Eq. (13), where m\u03bb(D) is the size of the block indexed by \u03bb. Recall that, for any \u03bb \u2208Par(dH, t(D)), one has m\u03bb(D) \u2264poly(d \u00af A, n) by Theorem 3.4. Proof of the Theorem 3.7. For D1 := A \u00af A\u2297H\u2297n and note that dH = dBd \u00af B is fixed. Let {CH 1 , . . . , CH m(H)} denote the canonical basis of EndSn (H\u2297n) defined in Eq. (17) and let {OH 1 , . . . , OH m(H)} denote the set of orbits of pairs that corresponding to {CH 1 , . . . , CH m(H)}. The following set of matrices B1 = {|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t } i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] is the canonical basis of the EndSn (D1). Since \u03c1A \u00af A(B \u00af B)n 1 \u2208EndSn (D1), we can express it in terms of the canonical basis B1 as \u03c1A \u00af A(B \u00af B)n 1 = X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t , where vi,j,x,y,t \u2208C for all i, j \u2208[dA]; x, y \u2208[d \u00af A]; t \u2208[m(H)], note that m(H) \u2264poly(n). We first write the Program \u03a6(SDPn(N, M) in terms of variables vi,j,x,y,t for i, j \u2208[dA]; x, y \u2208[d \u00af A]; t \u2208[m(H). 14 \fFor simplicity of notation, for any i, j \u2208[dA], x, y \u2208[d \u00af A], and t \u2208[m(H)], we denote Z(i, j, x, y, t) = |i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t . For D2 := A \u2297H\u2297n and D3 := A \u00af A \u2297H\u2297n\u22121 \u2297Bn. Program \u03a6(SDPn(N, M)) can be rewritten as maximize d \u00af AdB \u00b7 X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t \u00b7 tr h\u0010 J N \u00af AB1 \u2297\u03a6A \u00af B1 \u0011 tr (B \u00af B)n 2 (Z(i, j, x, y, t)) i s.t. X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t J\u03c8D1 (Z(i, j, x, y, t))K\u03bb \u2ab00 , \u2200\u03bb \u2208Par(dBd \u00af B, n) , X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t \u00b7 tr (Z(i, j, x, y, t)) = 1 , dA \u00b7 X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t J\u03c8D2 (tr \u00af A (Z(i, j, x, y, t)))K\u03bb = X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t J\u03c8D2 (IA \u2297tr A \u00af A (Z(i, j, x, y, t)))K\u03bb \u2200\u03bb \u2208Par(dBd \u00af B, n) , dB \u00b7 X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t q \u03c8D3 \u0000tr \u00af Bn (Z(i, j, x, y, t)) \u0001y \u03bb = X i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[m(H)] vi,j,x,y,t q \u03c8D3 \u0000tr Bn \u00af Bn (Z(i, j, x, y, t)) \u2297IB \u0001y \u03bb \u2200\u03bb \u2208Par(dBd \u00af B, n \u22121) , vi,j,x,y,t \u2208C for all i, j \u2208[dA]; x, y \u2208[d \u00af A], and t \u2208[m(H)] . We will demonstrate that the objective function and constraints in Program \u03a6(SDPn(N, M)) can be determined explicitly in poly(d \u00af A, n) time. Firstly, for any t \u2208[m(H)], let (a, b) be an representative element of OH t . We can compute the trace of each matrix in the canonical basis B1 of EndSn(D1) efficiently. Indeed, tr \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 = ( |OH t |, if i = j, x = y and a = b , 0, otherwise. Therefore, we can compute tr (Z(i, j, x, y, t)) efficiently for all i, j \u2208[dA]; x, y \u2208[d \u00af A]; t \u2208[m(H). In addition, from Lemma 3.5, we can determine J\u03c8D1 (Z(i, j, x, y, t))K\u03bb in poly(d \u00af A, n) time for all \u03bb \u2208Par(dBd \u00af B, n). Next, for each q \u2208[3], given z1, . . . , zm(Dq) \u2208C and consider a matrix X = Pm(Dq) i=1 ziCi(Dq). From Lemma 3.5, we can determine all blocks in the diagonal matrix \u03c8Dq(X) in poly(d \u00af A, n) time. Using this observation, to show J\u03c8D2 (tr \u00af A (Z(i, j, x, y, t)))K\u03bb , J\u03c8D2 (IA \u2297tr A \u00af A (Z(i, j, x, y, t)))K\u03bb for \u03bb \u2208Par(dBd \u00af B, n) and q \u03c8D3 \u0000tr \u00af Bn (Z(i, j, x, y, t)) \u0001y \u03bb , q \u03c8D3 \u0000tr Bn \u00af Bn (Z(i, j, x, y, t)) \u2297IB \u0001y \u03bb for \u03bb \u2208 Par(dBd \u00af B, n\u22121) can be computed in poly(d \u00af A, n) time. It suffices to show that, for all i, j \u2208[dA] ; x, y \u2208 [d \u00af A] ; t \u2208[m(H)], expanding tr A \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 , IA \u2297tr A \u00af A \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 in the canonical basis of EndSn(D2) and tr \u00af Bn \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 ; tr Bn \u00af Bn \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 in the canonical basis EndSn\u22121(D3) can be done in poly(d \u00af A, n) time. 15 \fWe have tr \u00af A \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 = ( |i\u27e9 \u27e8j| \u2297CH t , if x = y 0, otherwise. And, tr A \u00af A \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 = ( CH t , if i = j and x = y 0, otherwise. On the other hand, the canonical basis of the space EndSn (D2) is described as follows, B2 = {|i\u27e9 \u27e8j| \u2297CH t } i,j\u2208[dA] t\u2208[m(H)] . Therefore, for any i, j \u2208[dA]; x, y \u2208[d \u00af A], and t \u2208[m(H)], we can efficiently compute coordinates of the matrices tr \u00af A \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 and IA \u2297tr A \u00af A \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 over the canonical basis B2. Let \u2113(H) := dim EndSn\u22121 \u0000H\u2297n\u22121\u0001 , note that from Eq. (16), one has \u2113(H) \u2264poly(n). Consider the invariant subspace EndSn\u22121 (D3), let {KH 1 , . . . , KH \u2113(H)} be the canonical basis of the space EndSn\u22121 \u0000H\u2297n\u22121\u0001 defined in Eq. (17). The following set of matrices B3 = {|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297KH t \u2297|u\u27e9 \u27e8v|} i,j\u2208[dA] x,y\u2208[d \u00af A] t\u2208[\u2113(H)] u,v\u2208[dBn] is the canonical basis of the space EndSn\u22121 (D3). We will show that all coordinates of tr \u00af Bn \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 over B3 can be computed in poly(d \u00af A, n) time for any i, j \u2208[dA]; x, y \u2208[d \u00af A], and t \u2208[m(H)]. Firstly, for any t \u2208[m(H)], we can write CH t = X (a,b)\u2208OH t |a1\u27e9 \u27e8b1| \u2297|a2\u27e9 \u27e8b2| \u2297\u00b7 \u00b7 \u00b7 \u2297|an\u27e9 \u27e8bn| . Let (a, b) \u2208[dB \u00af B]n \u00d7 [dB \u00af B]n be a representative element of OH t following the construction in Eq. (22). Let S be the set of all distinct pairs in n pairs (a1, b1), (a2, b2), . . . , (an, bn), i.e., S = {(a1, b2)\u222a(a2, b2)\u222a \u00b7 \u00b7 \u00b7 \u222a(an, bn)}. Under the action of Sn\u22121, the set OH t can be partitioned into |S| sets {Ot (c,d)}(c,d)\u2208S, where Ot (c,d) := {(u, v) \u2208OH t : un = c, vn = d}. Note that the size of Ot (c,d) can be computed in poly(n) time by determining the frequency for all pairs in S in (a, b). For any c \u2208[dB \u00af B], using the natural identification between [dB \u00af B] and [dB]\u00d7[d \u00af B] ([dB \u00af B] \u223c = [dB]\u00d7[d \u00af B]), and write c = (cB, c \u00af B) \u2208[dB]\u00d7[d \u00af B]. We have CH t = X (c,d)\u2208S X (u,v)\u2208Ot (c,d) |u1\u27e9 \u27e8v1| \u2297\u00b7 \u00b7 \u00b7 \u2297|un = c\u27e9 \u27e8vn = d| = X (c,d)\u2208S X (u,v)\u2208Ot (c,d) |u1\u27e9 \u27e8v1| \u2297\u00b7 \u00b7 \u00b7 \u2297|cBn\u27e9 \u27e8dBn| \u2297|c \u00af Bn\u27e9 \u27e8d \u00af Bn| . Therefore, tr \u00af Bn(|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t ) = tr \u00af Bn \uf8eb \uf8ec \uf8ed|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297 X (c,d)\u2208S X (u,v)\u2208Ot (c,d) |u1\u27e9 \u27e8v1| \u2297\u00b7 \u00b7 \u00b7 \u2297|cBn\u27e9 \u27e8dBn| \u2297|c \u00af Bn\u27e9 \u27e8d \u00af Bn| \uf8f6 \uf8f7 \uf8f8 = X (c,d)\u2208S c \u00af Bn=d \u00af Bn X (u,v)\u2208Ot (c,d) |i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297|u1\u27e9 \u27e8v1| \u2297\u00b7 \u00b7 \u00b7 \u2297|un\u22121\u27e9 \u27e8vn\u22121| \u2297|cBn\u27e9 \u27e8dBn| 16 \fIn addition, we can determine an index t\u2032 \u2208[\u2113(H)] such that Kt\u2032 = X (u,v)\u2208Ot (c,d) c \u00af Bn=d \u00af Bn |u1\u27e9 \u27e8v1| \u2297\u00b7 \u00b7 \u00b7 \u2297|un\u22121\u27e9 \u27e8vn\u22121| in poly(n) time by considering all representative elements that corresponding to KH 1 , . . . , KH \u2113(H). Thus, the coordinates of tr \u00af Bn \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 over the canonical basis B3 can be computed in poly(n) time. The same method can be used for tr Bn \u00af Bn \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 \u2297IB. Finally, we will show that the objective function can be explicitly described by using variables vi,j,x,y,t for i, j \u2208[dA] ; x, y \u2208[d \u00af A], and t \u2208[m(H)], in poly(d \u00af A, n) time. In particular, the expression \u03c1A \u00af AB1 \u00af B1 can be described using variables vi,j,x,y,t, for vi,j,x,y,t for i, j \u2208[dA] ; x, y \u2208[d \u00af A], and t \u2208 [m(H)], in poly(d \u00af A, n) time. Similarly to the previous argument, we need to show for any i, j \u2208 [dA] ; x, y \u2208[d \u00af A], and t \u2208[m(H)], the value tr (B \u00af B)n 2 \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 can be computed efficiently. Firstly, recall that for x, y \u2208[m]n, their Hamming distance H(x, y) is defined as H(x, y) := |{i \u2208 [n] : xi \u0338= yi}|. Let (a, b) be a representative element of OH t . Let C(t) be the size of set {(u, v) \u2208OH t : u1 = a1, v1 = b1}. We have tr (B \u00af B)n 2 \u0000|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297CH t \u0001 = tr (B \u00af B)n 2 \uf8eb \uf8ed X (c,d)\u2208OH t |i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297|c1\u27e9 \u27e8d1| \u2297|c2\u27e9 \u27e8d2| \u2297\u00b7 \u00b7 \u00b7 \u2297|cn\u27e9 \u27e8dn| \uf8f6 \uf8f8 = ( C(t) (|i\u27e9 \u27e8j| \u2297|x\u27e9 \u27e8y| \u2297|a1\u27e9 \u27e8b1|) if H(a, b) = 1 and a1 \u0338= b1 , 0 otherwise. Here we used the fact that the Hamming distance is invariant under the action of the symmetric group, which is defined in Eq. (6). In addition, C(t) can be determined in poly(n) time, which completes the proof. Acknowledgements. We would like to thank Sven Polak for his useful comments and help in the proof of Lemma 2.5. HT would like to thank Omar Fawzi and Mario Berta for the useful discussions. 17"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2308.07187v3",
+                    "title": "On the Asymptotic Nonnegative Rank of Matrices and its Applications in Information Theory",
+                    "abstract": "In this paper, we study the asymptotic nonnegative rank of matrices, which\ncharacterizes the asymptotic growth of the nonnegative rank of fixed\nnonnegative matrices under the Kronecker product. This quantity is important\nsince it governs several notions in information theory such as the so-called\nexact R\\'enyi common information and the amortized communication complexity. By\nusing the theory of asymptotic spectra of V. Strassen (J. Reine Angew. Math.\n1988), we define formally the asymptotic spectrum of nonnegative matrices and\ngive a dual characterization of the asymptotic nonnegative rank. As a\ncomplementary of the nonnegative rank, we introduce the notion of the subrank\nof a nonnegative matrix and show that it is exactly equal to the size of the\nmaximum induced matching of the bipartite graph defined on the support of the\nmatrix (therefore, independent of the value of entries). Finally, we show that\ntwo matrix parameters, namely rank and fractional cover number, belong to the\nasymptotic spectrum of nonnegative matrices.",
+                    "authors": "Yeow Meng Chee, Quoc Tung Le, Hoang Ta",
+                    "published": "2023-08-14",
+                    "updated": "2024-01-29",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "cs.CC",
+                        "math.AC",
+                        "math.IT"
+                    ],
+                    "main_content": "Introduction The nonnegative rank of a nonnegative matrix A of size m \u00d7 n, denoted by R+(A), is the smallest integer number r such there exist two nonnegative matrices B and C of size m \u00d7 r and r \u00d7 n respectively such that A = BC (BC is the usual matrix product between B and C). The notion of nonnegative rank can be found in various \ufb01elds such as combinatorial optimization [Yan91] and communication complexity [Kus97, Lov90, LS+09, RY20]. For instances, in two parties communication complexity, two players compute a boolean function f : X \u00d7Y \u2192{0, 1} on inputs (x, y) \u2208X \u00d7Y where each party holds only one argument (either x or y). In this setting, the communication complexity D(f) of the function f is the minimum number of exchanging bits to evaluate f. Results in [Lov90, LS+09] show that the logarithm of the nonnegative rank of the associated matrix Mf of size |X| \u00d7 |Y| de\ufb01ned by Mf(x, y) = f(x, y) is a lower bound for the communication complexity, i.e., log R+(Mf) \u2264D(f). Another application of nonnegative rank is in combinatorial optimization. In [Yan91], it is shown that the minimum number of linear inequalities representing a given polytope can be determined by the nonnegative rank of a particular matrix. This quantity plays an important role in solving linear programming because the complexity of interior-point methods depends on the number of linear inequalities. Finally, it is demonstrated in [Zha12, JSWZ13] that the minimum amount of information exchanged by two parties in order to sample from the bivariate 1 \fprobability distribution p(x, y) = Ax,y represented by a matrix A (assuming A is normalized so that P x,y Ax,y = 1) can be determined by the nonnegative rank of A. The asymptotic nonnegative rank, which is de\ufb01ned as the asymptotic growth of the nonnegative rank of a matrix under Kronecker product (see Equation (1) for its mathematical de\ufb01nition), is equally important since it is also related to many quantities in information theory and communication complexity such as the exact R\u00b4 enyi common information [YT20] and the amortized communication complexity [FKNN95]. However, in comparison to the nonnegative rank, the asymptotic counterpart is not equally well-studied. For this reason, in this paper, we make several contributions to the study of asymptotic nonnegative rank: \ufb01rst, we give a dual characterization of this notion as well as additional results that relate to it via the theory of asymptotic spectra. Historically, this theory was developed by V. Strassen in [Str88] (see also the exposition in [WZ22, Part I]) to understand the arithmetic complexity of matrix multiplication (see, e.g., [BCS97]). More recently, the theory of asymptotic spectra has been applied to a range of other problems in combinatorics [Zui19], quantum information [LZ20, FFHT23], circuit complexity [RZ22] and other domains [JV19, Fri21, Vra22]. Two related notions to the nonnegative rank and the asymptotic nonnegative rank, the nonnegative subrank and its asymptotic version, are also introduced and studied. 1.1 Notations and De\ufb01nitions We de\ufb01ne some basic notions that will be used in the rest of this paper. Let A \u2208RmA\u00d7nA + , B \u2208 RmB\u00d7nB + be two nonnegative matrices, the direct sum A \u2295B and Kronecker product A \u2297B are de\ufb01ned respectively by: A \u2295B := \u0012A 0 0 B \u0013 \u2208R(mA+nA)\u00d7(mB+nB) + , A \u2297B := \uf8eb \uf8ec \uf8ed A1,1B . . . A1,nAB . . . ... . . . AmA,1B . . . AmA,nAB \uf8f6 \uf8f7 \uf8f8\u2208RmAmB\u00d7nAnB + , where Ai,j is the value of A at the index (i, j). For n \u2208N, denote In the identity matrix of size n and GL+(n) := {A \u2208Rn\u00d7n + | A\u22121 \u2208Rn\u00d7n + } a subgroup of GL(n) whose elements and their inverses are nonnegative matrices. Given a set S equipped with a binary operator \u00b7 : S \u00d7 S, we say (S, \u00b7) is a commutative monoid if: (1) The operator \u00b7 is associative: (a \u00b7 b) \u00b7 c = a \u00b7 (b \u00b7 c), \u2200a, b, c \u2208S. (2) The operator \u00b7 is commutative: a \u00b7 b = b \u00b7 a, \u2200a, b \u2208S. (3) There exists an identity element 0: a \u00b7 0 = 0 \u00b7 a = a, \u2200a \u2208S. For a nonnegative matrix A, supp(A) := {(i, j) : Ai,j > 0} the support of A is the set of indices whose coe\ufb03cients in A is strictly positive. In this work, supp(A) has two interpretations: either a subset of the index set or a binary matrix of the same size as A whose coe\ufb03cients at (i, j) are equal to 1 if (i, j) \u2208supp(A) and 0 otherwise. Asymptotic nonnegative rank The nonnegative rank is sub-multiplicative under Kronecker product. Indeed, given two nonnegative matrices A, B, suppose that A = UV T and B = XY T , where U, V, X, Y are nonnegative matrices, then A \u2297B = (U \u2297X)(V \u2297Y )T . Therefore R+(A \u2297 2 \fB) \u2264R+(A) R+(B). In general, R+ is not multiplicative since there exists A \u2208R4\u00d74 + such that R+(A) = 4 and R+(A \u2297A) = 15 \u0338= R+(A)2 [BKLT13]. For more similar examples, we refer to [BKLT13, VGGT16]). Therefore, the asymptotic nonnegative rank is interesting to consider. Mathematically, the asymptotic nonnegative rank of nonnegative matrices is de\ufb01ned as follows. e R+(A) := lim n\u2192\u221eR+(A\u2297n)1/n . (1) Since the sequence R+(A\u2297)1/n is sub-multiplicative, by Fekete\u2019s lemma [Fek23], we can prove that e R+(A) is well-de\ufb01ned and the limit can be replaced by the in\ufb01mum over n. Namely, e R+(A) = inf n R+(A\u2297n)1/n . We also note that the approximated version of the asymptotic nonnegative rank was studied in [BJLP17]. In that work, the authors showed that this notion captures the common information [Wyn75]. 1.2 Applications of asymptotic nonnegative rank The asymptotic nonnegative rank can be interpreted as the exact R\u00b4 enyi common information of order zero [YT20]. This information is de\ufb01ned based on correlated sources (joint distribution), which is represented as a nonnegative matrix with normalized rows and columns. Speci\ufb01cally, given correlated sources \u03c0XY over X \u00d7Y . The exact R\u00b4 enyi common information of order zero, denoted by T 0 Ex(\u03c0XY ), is the minimum common randomness rate for exact generation of the target distribution, with the constraint that common randomness can only be compressed using \ufb01xed-length codes. In [YT20], the authors showed that this information can be characterized by asymptotic nonnegative rank of \u03c0XY . More precisely, they proved T 0 Ex(\u03c0XY ) = log e R+(\u03c0XY ). For more information about the exact R\u00b4 enyi common information, we refer to [YT22, Section 5 and Section 7]. In addition, the asymptotic nonnegative rank can be used as a lower bound for the amortized communication complexity. Unlike the original communication complexity (that we brie\ufb02y introduce in the precedent paragraph), this model is interested in the minimum number of exchanging bits required to calculate a boolean function f multiple times (on di\ufb00erent inputs (xi, yi) | i = 1, . . . , m). A trivial upper bound for this quantity is m times the complexity of calculating f once. In other words, we obtain this complexity by simply calculating f(xi, yi) separately, one after another. However, it turns out that in many computational models, the amortized communication complexity is smaller than this natural upper bound. It implies that the gain of complexity by performing calculation in batch can be signi\ufb01cant. A notable example can be found in [FKNN95], where the author constructs a speci\ufb01c function f whose communication complexity D(f) and its amortization \u00af D(f) are totally di\ufb00erent, i.e., \u00af D(f)/m < D(f). Using the result log R+(Mf) \u2264D(f) of [Lov90], we can prove a similar amortized counterpart, which states: log e R+(Mf) \u2264\u00af D(f)/m (remind that Mf is of size |X| \u00d7 |Y|, Mf(x, y) = f(x, y)). Therefore, the asymptotic nonnegative rank e R+ is important for the study of amortized communication complexity. 1.3 Contribution In this paper, we consider the asymptotic spectra theory in the context of asymptotic nonnegative rank. Our main contributions are: 1. By introducing a nonnegative matrix semiring M+ with two operators: \u2295(the direct sum) and \u2297 (the Kronecker product) and a preorder \u2a7d+ (cf. De\ufb01nition 2.1), we de\ufb01ne X(M+, \u2a7d+), the asymptotic spectrum of M+. The set X(M+, \u2a7d+) consists of monotone semiring homomorphisms \u03c6 from 3 \fthe set of nonnegative matrices M+ to the set of nonnegative numbers R\u22650, i.e., if A \u2a7d+ B, then \u03c6(A) \u2264\u03c6(B) (A, B are two nonnegative matrices). In fact, X(M+, \u2a7d+) is the nonnegative matrix version of other existing asymptotic spectra such as the asymptotic spectrum of tensors [Str88], graphs [Zui19] and LOCC transformations [JV19], which are studied in the literature. 2. The de\ufb01nition of X(M+, \u2a7d+) allows us to establish another characterization of the asymptotic nonnegative rank of a matrix A in Theorem 3.7, which states: e R+(A) = max \u03c6\u2208X(S,\u2a7d+) \u03c6(A) 3. Similar to previous works on asymptotic spectra, we de\ufb01ne and study two quantities: the subrank (Q+) and asymptotic subrank (e Q+) of a nonnegative matrix. While Q+ is a natural lower bound of R+, e Q+ is the dual of e R+. In this work, we prove that Q(A) is exactly equal to the induced matching number of the bipartite graph de\ufb01ned on the support of A (see Theorem 3.9) and thus, independent of the coe\ufb03cients of A. Moreover, similar to the asymptotic nonnegative rank, the asymptotic subrank can also be characterized by using X(M+, \u2a7d+) (cf. Theorem 3.7) as: e Q+(A) = min \u03c6\u2208X(S,\u2a7d+) \u03c6(A) 4. Since X(M+, \u2a7d+) allows us to characterize precisely e R+ and e Q+, we study in detail the set X(M+, \u2a7d+). Our investigation of X(M+, \u2a7d+) consists of showing its two elements: the rank function and the fractional cover number (cf. Section 4). We note that the construction of explicit elements in the asymptotic spectrum of a speci\ufb01c instance is nontrivial. For example, the construction of nontrivial (universal) elements in the asymptotic spectrum of tensors has been an open problem for more than thirty years until it was solved by recent work in [CVZ18]. In addition, we also showed that for all \u03c6 \u2208X(M+, \u2a7d+), for all non-degenerate lower triangular nonnegative matrix A \u2208Rm\u00d7n, we have: \u03c6(A) = min(m, n). Organization The paper is organized as follows: In Section 2, we brie\ufb02y present the theory of asymptotic spectra of Strassen. We then introduce the asymptotic spectrum on nonnegative matrices and give a dual characterization of the asymptotic nonnegative rank of nonnegative matrices in Section 3. Finally, in Section 4, we show two elements in the asymptotic spectrum of nonnegative matrices. 2 Asymptotic spectra and Strassen\u2019s spectral theorem In this section, we present brie\ufb02y the abstract theory of asymptotic spectra which was introduced by V. Strassen [Str88]. For a more complete presentation, we refer the readers to [Str88, Zui18, WZ22]. Let (S, +, \u00b7, 0S, 1S) be a commutative semiring, i.e., a set S equipped with a binary addition operation +, a binary multiplication \u00b7, and having two elements 0S, 1S \u2208S such that for all a, b, c \u2208S: (1) (S, +) is a commutative monoid with the identity element 0S. (2) (S, \u00b7) is a commutative monoid with the identity element 1S. (3) \u00b7 distributes over +: a \u00b7 (b + c) = (a \u00b7 b) + (a \u00b7 c) (4) 0S \u00b7 a = 0S. 4 \fFor n \u2208N we denote the sum of n elements 1S, i.e., 1S + \u00b7 \u00b7 \u00b7 + 1S \u2208S by nS. A preorder \u2a7don S is a relation on S such that for all a, b, c \u2208S 1) \u2a7dis re\ufb02exive: a \u2a7da 2) \u2a7dis transitive: a \u2a7db and b \u2a7dc implies a \u2a7dc. De\ufb01nition 2.1. A preorder \u2a7don S is a Strassen preorder if (i) For all n, m \u2208N n \u2264m i\ufb00nS \u2a7dmS in S. (ii) For all a, b, c, d \u2208S if a \u2a7db and c \u2a7dd, then a + c \u2a7db + d and a \u00b7 c \u2a7db \u00b7 d. (iii) For all a, b \u2208S, b \u0338= 0 there is r \u2208N such that a \u2a7drS \u00b7 b. Let S be a commutative semiring and let \u2a7dbe a Strassen preorder on S. Throughout the paper, we will use \u2264to denote the usual order on R. Let R\u22650 be the semiring of nonnegative real numbers. A semiring homomorphism from S to R\u22650 (denoted by Hom (S, R\u22650)) is a map \u03c6 : S \u2192R\u22650 such that: (1) \u03c6(a \u00b7 b) = \u03c6(a)\u03c6(b) (2) \u03c6(a + b) = \u03c6(a) + \u03c6(b) (3) \u03c6(1S) = 1. Let X(S, \u2a7d) be the set of \u2a7d-monotone semiring homomorphisms from S to R\u22650, i.e: X(S, \u2a7d) := {\u03c6 \u2208Hom (S, R\u22650) : \u2200a, b \u2208S, a \u2a7db \u21d2\u03c6(a) \u2264\u03c6(b)} , we note that for every \u03c6 \u2208X(S, \u2a7d) holds \u03c6(1S) = 1 and thus \u03c6(nS) = n for all n \u2208N. We call X(S, \u2a7d) the asymptotic spectrum of (S, \u2a7d) and elements of X(S, \u2a7d) spectral points. Let a \u2208S, the rank of a is de\ufb01ned as R(a) := min{n \u2208N : a \u2a7dnS}. The subrank of a is de\ufb01ned as Q(a) := max{n \u2208N : nS \u2a7da}. The asymptotic subrank and asymptotic rank of a are de\ufb01ned as e R(a) := lim N\u2192\u221eR(aN)1/N and e Q(a) := lim N\u2192\u221eQ(aN)1/N (2) From Fekete\u2019s lemma, the limits in (2) indeed exist and can be replaced by an in\ufb01mum and a supremum, that is, e R(a) = inf N R(aN)1/N, and e Q(a) = sup N Q(aN)1/N . The notions of asymptotic rank and asymptotic subrank have been extensively studied in various \ufb01elds. For instance, in [Str88], the author showed that the asymptotic rank of matrix multiplication tensors corresponds to the complexity of matrix multiplication, more examples can be found in [WZ22]. Beside that, the notion of asymptotic subrank for tensors can be utilized to establish upper bounds for several combinatorial problems, such as capset and sun\ufb02ower problems [CVZ18], as well as the Shannon capacity of hypergraphs [CFTZ22]. Furthermore, research has also focused on exploring structures of the set of nonnegative real numbers obtained from the asymptotic rank or asymptotic subrank of complex tensors, as seen in [BDR22, CGZ22, GZ23]. In [Str88], the author showed that the asymptotic rank and asymptotic subrank have the following dual characterization in terms of the asymptotic spectrum. 5 \fTheorem 2.2 ([Str88, Th. 3.8] and [Zui18, Cor. 2.14]). Let S be a commutative semiring and let \u2a7dbe a Strassen preorder on S. For any a \u2208S such that 1S \u2a7da and 2S \u2a7dak for some k \u2208N, we have that e R(a) = max \u03c6\u2208X(S,\u2a7d) \u03c6(a), and e Q(a) = min \u03c6\u2208X(S,\u2a7d) \u03c6(a) . The theory of asymptotic spectra, as well as Theorem 2.2, was \ufb01rst applied to study the complexity of the matrix multiplication problem [Str88, Str91]. Subsequently, in [Zui19, LZ20], the authors used this theory to introduce the (quantum) asymptotic spectrum of graphs, allowing them to establish a dual characterization for the (quantum) Shannon capacity on graphs. In the context of quantum information theory, [JV19] used this theory to provide a characterization of the converse error exponents for asymptotic entanglement transformations. In the following section, we will build a commutative semiring for the set of nonnegative matrices in which the theory of asymptotic spectra can be applied. This will allow us to establish a characterization for the asymptotic nonnegative rank. 3 Asymptotic spectrum of nonnegative matrices and duality 3.1 Asymptotic spectrum of nonnegative matrices In this section, our goal is to apply the theory of the previous section to give the dual characterization of the asymptotic nonnegative rank of matrices. We remind the readers that \u2297denotes the Kronecker product, \u2295denotes the direct sum of matrices, and In denotes the identity matrix of size n \u00d7 n. A natural semiring can be de\ufb01ned by considering the set of nonnegative matrices equipped with the binary operators \u2295and \u2297. However, the Kronecker product \u2297is not commutative. Therefore, in the following construction, rather than working with the set of nonnegative matrices, we will work with its quotient space where \u2295and \u2297are commutative. In fact, we will introduce an equivalence relation in which equivalent elements share the same nonnegative rank, and the induced operators \u2295/\u2297on the corresponding quotient space are commutative. This allows us to bypass the technical requirement and apply the theory of asymptotic spectra. Denote by M(m, n) the set of all nonnegative matrices of size m\u00d7n, for two nonnegative matrices A, B \u2208M(m, n), we write A \u223c = B if there is a pair of matrices (X, Y ) \u2208GL+(m)\u00d7GL+(n) such that A = XBY \u22a4(or equivalently B = X\u22121A(Y \u22121)\u22a4). It is clear that \u223c = is an equivalence relation since if A = X1BY T 1 , B = X2CY \u22a4 2 , then A = (X1X2)C(Y1Y2)\u22a4with X1X2 \u2208GL+(m), Y1Y2 \u2208GL+(n). In addition, the relation \u223c = enjoys other nice properties, as shown in Lemma 3.1. Lemma 3.1. Given four matrices A, B \u2208M(m1, n1), C, D \u2208M(m2, n2) such that A \u223c = B, C \u223c = D, we have: 1. R+(A) = R+(B) (and R+(C) = R+(D)). 2. A \u2295C \u223c = B \u2295D. 3. A \u2297C \u223c = B \u2297D. Proof. Since A \u223c = B, so there are X \u2208GL+(m1) and Y \u2208GL+(n1) such that XAY T = B. This implies that R+(A) = R+(B) since X, Y are invertible matrices. Since C \u223c = D, so there are U \u2208GL+(m2) and V \u2208GL+(n2) such that UCV T = D. We have (X \u2295U)(A \u2295C)(Y \u2295V )T = B \u2295D, therefore A \u2295C \u223c = B \u2295D since X \u2295U \u2208GL+(m1 + m2) 6 \fand Y \u2295V \u2208GL+(n1 + n2). Similarly, one has (X \u2297U)(A \u2297C)(Y \u2297V )T = B \u2297D, this implies A \u2297C \u223c = B \u2297D. Let A \u2208M(mA, nA) and B \u2208M(mB, nB), we say A and B are equivalent, and write A \u223cB, if there are zero matrices A0 = 0 \u2208Ra1\u00d7a2 + and B0 = 0 \u2208Rb1\u00d7b2 + such that A \u2295A0 \u223c = B \u2295B0. In fact, \u223cis also an equivalence relation and elements in the same equivalence class shares a common nonnegative rank, as shown below. Lemma 3.2. The binary relation \u223cis an equivalence one. Moreover, the nonnegative rank is invariant under \u223c. Proof. We \ufb01rst prove that \u223cis an equivalence class. Since the re\ufb02exivity and symmetry of \u223cis trivial, we focus on proving the transitivity of \u223c, i.e, for three matrices A, B, C, if A \u223cB, B \u223cC, then A \u223cC. Indeed, by de\ufb01nition of \u223c, there exists zero matrices A0, B0, B1, C0 such that: A \u2295A0 \u223c = B \u2295B0, B \u2295B1 \u223c = C \u2295C0 Therefore, (A \u2295A0) \u2295B1 \u223c = (B \u2295B0) \u2295B1, (B \u2295B1) \u2295B0 \u223c = (C \u2295C0) \u2295B0, using Lemma 3.1. Moreover, B0 \u2295B1 = B1 \u2295B0 since both are zero matrix of the same sizes. This implies B \u2295B1 \u2295B0 = B \u2295B0 \u2295B1. Using the fact that \u223c = is an equivalence relation, we have: A \u2295(A0 \u2295B1) \u223c = C \u2295(C0 \u2295B0) Notice that both A0 \u2295B1 and C0 \u2295B0 are zero matrices. This proves the \ufb01rst claim. Moreover, the direct sum of a matrix with a zero matrix does not a\ufb00ect the nonnegative rank. Therefore, if A \u223cB, then R+(A) = R+(A \u2295A0) = R+(B \u2295B0) = R+(B). This concludes the proof. Let M+ be the set of \u223c-equivalence classes of nonnegative matrices (which is now well-de\ufb01ned thanks to Lemma 3.2) , we de\ufb01ne the addition and multiplication on M+ as follow [A] \u2295[B] = [A \u2295B] [A] \u2297[B] = [A \u2297B] . where [A] denotes the equivalence class of the matrix A. These two operations are also well-de\ufb01ned, as shown in the following lemma. Lemma 3.3. Consider two pairs of matrices (A, B) and (A\u2032, B\u2032) such that A \u223cA\u2032 and B \u223cB\u2032, then: [A \u2295B] = [A\u2032 \u2295B\u2032] [A \u2297B] = [A\u2032 \u2297B\u2032] Proof. Due to the de\ufb01nition of the equivalence relation \u223c, there exists zero matrices A0, B0, A\u2032 0, B\u2032 0 such that: A \u2295A0 \u223c = A\u2032 \u2295A\u2032 0, B \u2295B0 \u223c = B\u2032 \u2295B\u2032 0 . By Lemma 3.1, we have: (A \u2295A0) \u2295(B \u2295B0) \u223c = (A\u2032 \u2295A\u2032 0) \u2295(B\u2032 \u2295B\u2032 0) (3) (A \u2295A0) \u2297(B \u2295B0) \u223c = (A\u2032 \u2295A\u2032 0) \u2297(B\u2032 \u2295B\u2032 0) (4) 7 \f1. For addition (\u2295): The key observation for this part is (A\u2295A0)\u2295(B \u2295B0) \u223c = (A\u2295B)\u2295(A0 \u2295B0). Indeed, we have: (A \u2295A0) \u2295(B0 \u2295B0) = \uf8eb \uf8ec \uf8ec \uf8ed A 0 0 0 0 A0 0 0 0 0 B 0 0 0 0 B0 \uf8f6 \uf8f7 \uf8f7 \uf8f8, (A \u2295A0) \u2295(B0 \u2295B0) = \uf8eb \uf8ec \uf8ec \uf8ed A 0 0 0 0 B 0 0 0 0 A0 0 0 0 0 B0 \uf8f6 \uf8f7 \uf8f7 \uf8f8 Therefore, the matrix (A \u2295B) \u2295(A0 \u2295B0) can be obtained by permuting rows and columns of (A \u2295A0) \u2295(B0 \u2295B0). This implies there exists two permutation matrices X, Y such that X((A\u2295A0)\u2295(B0\u2295B0))Y \u22a4= (A\u2295B)\u2295(A0\u2295B0). Similarly, we also have: (A\u2032\u2295A\u2032 0)\u2295(B\u2032\u2295B\u2032 0) \u223c = (A\u2032 \u2295B\u2032)\u2295(A\u2032 0 \u2295B\u2032 0). Thus, we have (A\u2295B)\u2295(A0 \u2295B0) \u223c = (A\u2032 \u2295B\u2032)\u2295(A\u2032 0 \u2295B\u2032 0) (by combining with (3)). The proof is concluded by noticing that A0 \u2295B0 and A\u2032 0 \u2295B\u2032 0 are zero matrices. 2. For multiplication (\u2297): we have: (A \u2295A0) \u2297(B \u2295B0) = [A \u2297(B \u2295B0)] \u2295[A0 \u2297(B \u2295B0)] \u223c = [(B \u2295B0) \u2297A] \u2295[(B \u2295B0) \u2297A0] \u223c = (A \u2297B) \u2295(A \u2297B0 \u2295A0 \u2297B \u2295A0 \u2297B0) Here we use the fact (A \u2295B) \u2297C = (A \u2297C) \u2295(B \u2297C) and A \u2297B \u223c = B \u2297A (note that A \u2297(B \u2295C) \u0338= (A \u2297B) \u2295(A \u2297C) in general). Similarly, we have (A\u2032 \u2295A\u2032 0) \u2297(B\u2032 \u2295B\u2032 0) \u223c = (A\u2032 \u2297B\u2032) \u2295(A\u2032 \u2297B\u2032 0 \u2295A\u2032 0 \u2297B\u2032 \u2295A\u2032 0 \u2297B\u2032 0) . Note that the second terms of both above equations are zero matrices. Combining with (4), we deduce that [A \u2297B] = [A\u2032 \u2297B\u2032]. From above arguments, we conclude that the elements [A] \u2295[B] and [A] \u2297[B] does not depend on the choice of representative A, B. Hence, these two operations are well-de\ufb01ned in the quotient space M+. Moreover, one can easy to verify that [A \u2295B] = [B \u2295A], [A \u2297B] = [B \u2297A]. Thus, (M+, \u2295, \u2297, [0], [I1]) is a commutative semiring where [0] is the equivalence class of all zero matrices. Lemma 3.4. The set M+ with addition \u2295, multiplication \u2297, additive unit [0] and multiplicative unit [I1] is commutative semiring. We write A \u2a7d+ B if there are two nonnegative matrices X \u2208RmA\u00d7mB + and Y \u2208RnA\u00d7nB + such that A = XBY T . This preorder ensures that if A \u2a7d+ B, then R+(A) \u2264R+(B). Note that \u2a7d+ is less restrictive than \u223c = (which requires X \u2208GL+(mA), Y \u2208GL+(nA)). This also implies that if A \u223c = B, then A \u2a7d+ B and B \u2a7d+ A. The natural induced preorder of \u2a7d+ on M+ is also well-de\ufb01ned M+, which is shown as follow: Lemma 3.5. If A \u2a7d+ B, then A\u2032 \u2a7d+ B\u2032, for all A\u2032 \u223cA, B\u2032 \u223cB. Proof. It is su\ufb03cient to show that if A \u223cA\u2032, then A \u2a7d+ A\u2032, A\u2032 \u2a7d+ A. Indeed, combining with the transitivity of \u2a7d+, we can conclude: A\u2032 \u2a7d+ A \u2a7d+ B \u2a7d+ B\u2032 By de\ufb01nition of \u223c, there exists two zero matrices A0, A\u2032 0 such that A \u2295A0 \u223c = A\u2032 \u2295A\u2032 0. Assume that A \u2208Rm\u00d7n + and A0 = 0 \u2208Rp\u00d7q + . De\ufb01ne matrices X = [ Im 0 ] \u2208Rm\u00d7(m+p) + and Y = [ In 0 ] \u2208Rn\u00d7(n+q) + . We have, A = X(A \u2295A0)Y T and A \u2295A0 = XT AY . Therefore, A \u2a7d+ A \u2295A0 A \u2295A0 \u2a7d+ A . 8 \fSimilarly, we also have: A\u2032 \u2a7d+ A\u2032 \u2295A\u2032 0, A\u2032 \u2295A\u2032 0 \u2a7d+ A\u2032. As such, A \u2a7d+ A \u2295A0 \u223c = A\u2032 \u2295A\u2032 0 \u2a7d+ A\u2032. The other direction can be proved similarly. This yields the desired result. Hereinafter, to ease the notation, we abuse the notations by writing A to denote both the matrix A and also the equivalence class [A]. The meaning should be clear from the context. The following lemma proves that \u2a7d+ is also a Strassen preorder. Lemma 3.6. The preorder \u2a7d+ is a Strassen preorder on M+. That is, for nonnegative matrices A, B, C, D \u2208M+ we have the following. (i) For all n, m \u2208N, n \u2264m if and only if In \u2a7d+ Im (ii) If A \u2a7d+ B and C \u2a7d+ D then A \u2295C \u2a7d+ B \u2295D and A \u2297C \u2a7d+ B \u2297D (iii) If B \u0338= 0, there is an r \u2208N such that A \u2a7d+ Ir \u2297B. Proof. The statement ((i)) is trivial. We prove ((ii)). Let U, V, X, Y such that A = UBV T and C = XDY T . Then A \u2295C = (U \u2295X)(A \u2295C)(V \u2295Y )T and A \u2297C = (U \u2297X)(B \u2297D)(V \u2297Y )T . This proves ((ii)). We prove ((iii)). Let r = R+(A). Then A \u2a7d+ Ir. By assumption, B \u0338= 0, so I1 \u2a7d+ B. Therefore A \u2a7d+ Ir \u223c = Ir \u2297I1 \u2a7d+ Ir \u2297B (where r = R+(A)). This proves ((iii)). We thus have a semiring M+ with a Strassen preorder \u2a7d+. Thus, we can apply the theory of asymptotic spectra in this context. Let us translate the abstract terminology to this setting. Let S \u2286M+ be a semiring. For example, one may take S = M+, or one may choose any set X \u2286M+ and let S = N[X] be the subsemiring of M+ generated by X under \u2295and \u2297. The asymptotic spectrum of S is the set X(S, \u2a7d+) of \u2a7d+-monotone semiring homomorphism S \u2192R\u22650, i.e., all maps \u03c6 : S \u2192R\u22650 such that. for all A, B \u2208S (1) \u03c6(A \u2297B) = \u03c6(A)\u03c6(B) (2) \u03c6(A \u2295B) = \u03c6(A) + \u03c6(B) (3) \u03c6(In) = n for all n \u2208N (4) If A \u2a7d+ B then \u03c6(A) \u2264\u03c6(B). We call X(M+, \u2a7d+) the asymptotic spectrum of nonnegative matrices. In the next subsection, we use the asymptotic spectrum of nonnegative matrices to give a dual characterization for asymptotic nonnegative rank. 3.2 Asymptotic nonnegative rank and asymptotic nonnegative subrank From Subsection 3.1, we have shown that the preorder \u2a7d+ is a Strassen preorder. Thus, similar to Section 2, using the preorder \u2a7d+, we can rewrite the nonnegative rank and de\ufb01ne the nonnegative subrank of a nonnegative matrix A as R+(A) := min{r \u2208N : A \u2a7d+ Ir} Q+(A) := max{q \u2208N : Iq \u2a7d+ A} . Similarly to the asymptotic nonnegative rank, the asymptotic nonnegative subrank is naturally de\ufb01ned as follows. e Q+(A) := lim n\u2192\u221eQ+(A\u2297n)1/n = sup n Q+(A\u2297n)1/n . 9 \fUsing the abstract theory of Strassen, the asymptotic spectrum of nonnegative matrices can be used to characterize the asymptotic nonnegative rank and the asymptotic nonnegative subrank. Theorem 3.7. Let A be a nonnegative matrix. Then e R+(A) = max \u03c6\u2208X(M+,\u2a7d+) \u03c6(A), and e Q+(A) = min \u03c6\u2208X(M+,\u2a7d+) \u03c6(A) . Before proceeding to the proof of Theorem 3.7, we need to show \ufb01rst a combinatorial interpretation of subrank Q+. It is easy to see that the nonnegative rank of matrix A depends on its entries. In the other direction, we show in the following that the nonnegative subrank of a nonnegative matrix depends only on its support. We \ufb01rst introduce some notions and preliminary results. Let A be a nonnegative matrix, we remind that supp(A) = {(i, j) : Ai,j > 0} the support of A. Two pairs (a, b), (c, d) \u2208supp(A) is called totally pairwise independent if Aa,d = Ac,b = 0. A subset \u03a6 of supp(A) is called induced matching of A if each pair (a, b) \u0338= (c, d) \u2208\u03a6 is totally pairwise independent. The name induced matching here come from the fact that the totally pairwise independent set of matrix A is an induced matching of the bipartite graph that construct from the support of A. We denote \u03b3(A) to be the largest induced matching of A. Let us illustrate this notion by the following example. Example 3.8. A = \uf8eb \uf8ec \uf8ec \uf8ed 1 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8. It is to easy to verify that in this example, \u03c8 = {(1, 1), (2, 3), (3, 2), (4, 4)} is not an induced matching of A (since A2,1 = 1 \u0338= 0), and \u03a6 = {(4, 1), (3, 2)} is an induced matching of A. We show that for every nonnegative matrix, the largest size of an induced matching is equal to the nonnegative subrank, i.e., Theorem 3.9. For all nonnegative matrices A, Q+(A) = \u03b3(A). To prove Theorem 3.9, we \ufb01rst prove two results: Proposition 3.10 and Proposition 3.11 Proposition 3.10. For any nonnegative matrix A, we have \u03b3(A) \u2264Q+(A). Proof. We prove: \u03b3(A) \u2264Q+(A), \u2200A \u2208Rm\u00d7n + . Let {(x1, y1), . . . , (xt, yt)} \u2286supp(A) be the maximum induced matching set of A. Due to the invariance of \u03b3 and Q+ under the permutation of rows and columns, we can assume WLOG that xi = yi = i, \u22001 \u2264i \u2264t (otherwise, we permute the rows and columns correspondingly). This allows us to write the matrix A in the form of block matrices: A = \u0012 D A1 A2 A3 \u0013 , where D \u2208Rt\u00d7t + is a positive diagonal matrix and Ai, i = 1, 2, 3 are some nonnegative matrices. This claim holds since {(i, i) | i = 1, . . . , t} \u2286supp(A) is an induced matching. Consider: \u0000D\u22121/2 0 \u0001 A \u0012 D\u22121/2 0 \u0013 = \u0000D\u22121/2 0 \u0001 \u0012 D A1 A2 A3 \u0013 \u0012 D\u22121/2 0 \u0013 = It, (5) where D\u22121/2 \u2208Rt\u00d7t + is a positive diagonal matrix obtained by taking the root square of diagonal elements of D. Since [ D\u22121/2 0 ] , \u0002 D\u22121/2 0 \u0003 are nonnegative, Equation (5) shows that t = \u03b3(A) \u2264 Q+(A). 10 \fProposition 3.11. Let A, B, C be three nonnegative matrices such that A = BC, we have: \u03b3(A) \u2264min(\u03b3(B), \u03b3(C)) . Proof. It is su\ufb03cient to prove that \u03b3(A) \u2264\u03b3(B) and \u03b3(A) \u2264\u03b3(C). We will show the \ufb01rst claim. The second one can be dealt with similarly. Let {(x1, y1), . . . , (xt, yt)} be one of the maximum induced matchings of A. Using the same argument as in the proof of Proposition 3.10, we can assume WLOG that xi = yi = i, \u2200i = 1, . . . , t. Therefore, the top left t \u00d7 t submatrix of A is a diagonal matrix. Since A = BC, then the ith column of A is equal to P k Ck,iB:k, where B:k is the column k-th of B and Ck,i is the coe\ufb03cients of C at the index (k, i). For each 1 \u2264i \u2264t, we de\ufb01ne Si = {k | Ck,i > 0, Bi,k > 0}. It is clear that Si is non-empty (otherwise, the value Ai,i is zero, a contradiction). Moreover, \u2200k \u2208Si, \u2200j \u2264t, j \u0338= i, Bj,k = 0 to keep the submatrix at the top left corner of the matrix A diagonal. As a consequence, Si \u2229Sj = \u2205. Hence, we have {(1, y1), . . . , (t, yt)} is an induced matching of B where yi is an arbitrary element of Si. Therefore \u03b3(A) \u2264\u03b3(B). We can now proceed to the proof of Theorem 3.9. Proof of Theorem 3.9. From Proposition 3.10 we have \u03b3(A) \u2264Q+(A) for any nonnegative matrix A. Suppose that Q+(A) = m, we have Im = XAY T , from Proposition 3.11, one has m = \u03b3(Im) = \u03b3(XAY T ) \u2264\u03b3(XA) \u2264\u03b3(A). That implies \u03b3(A) = Q+(A) = m. Theorem 3.9 characterizes a very nice property of nonnegative subrank: it is exactly equal to the maximum induced matching of the bipartite graph whose adjacency matrix is the support of its matrix. Finding the subrank of a matrix is thus reducible to the problem of \ufb01nding the maximum induced matching of a bipartite graph. Unfortunately, the maximum induced matching problem is NP-hard, even in the setting of bipartite graph [Cam89]. Even so, by using the properties of nonnegative matrices, we provide a combinatorial interpretation of the subrank of a nonnegative matrix. The subrank or symmetric subrank studied in other works [Str91, CFTZ21, CGZ22], however, do not have this property. These quantities depend also on their coe\ufb03cients. With this characterization of Q+, we can now prove Theorem 3.7. Proof of Theorem 3.7. There are two cases to consider: 1. If A = 0, then \u03c6(A) = 0 for any \u03c6 \u2208X(M+, \u2a7d+). The result follows trivially. 2. If A \u0338= 0, it is di\ufb03cult to directly apply Theorem 2.2 since it is non-trivial to verify the condition: There exists an integer k such that I2 \u2a7d+ Ak. In this proof, we provide a simple workaround: Instead of considering A, we consider B := (A \u2295A). It is trivial that I2 \u2a7d+ Bk with k = 1 because we assume that A is non-zero. Therefore, using Theorem 2.2, we obtain: e R(B) = max \u03c6\u2208X(S,\u2a7d) \u03c6(B), and e Q(B) = min \u03c6\u2208X(S,\u2a7d) \u03c6(B) . Due to the de\ufb01nition of X(S, \u2a7d), we have: \u03c6(B) = 2\u03c6(A), \u2200\u03c6 \u2208X(S, \u2a7d). Thus, we can rewrite the above equation as: e R(B) = 2 max \u03c6\u2208X(S,\u2a7d) \u03c6(A), and e Q(B) = 2 min \u03c6\u2208X(S,\u2a7d) \u03c6(A) . (6) 11 \fIt is thus, su\ufb03cient to show the two following equalities: R+(Bk) = 2k R+(Ak) Q+(Bk) = 2k Q+(Ak). Using de\ufb01nitions of e R+ and e Q+, we deduce that e R+(B) = 2 e R+(A) and e Q+(B) = 2 e Q+(A). Combining with Equation (6), we prove the theorem. To this end, we \ufb01rst observe that: Bk = (A \u2295A)k = Ak \u2295. . . \u2295Ak | {z } 2k times . (7) For the case of R+, we will use the following claim: for every pair of nonnegative matrices (X, Y ), we have: R+(X \u2295Y ) = R+(X) + R+(Y ). The proof of this claim can be found in A. As a consequence of Equation (7), we have: R+(Bk) = 2k R+(Ak). For the case of Q+, thanks to Theorem 3.9, it is also clear that Q+ is additive under \u2295, i.e., Q+(X \u2295Y ) = Q+(X) + Q+(Y ) for all nonnegative matrices X, Y . Indeed, the bipartite graph (corresponding to the support) of (X \u2295Y ) is the union of two disconnected bipartite graphs (corresponding to the support) of X and Y respectively. Thus, the maximum induced matching of (X \u2295Y ) is equal to the sum of these of X and Y . Using the similar argument for R+, we also have Q+(Bk) = 2k Q+(Ak). This concludes the proof. 4 Some elements in the asymptotic spectrum of nonnegative matrices From Theorem 3.7, an explicit description of the asymptotic spectrum of nonnegative matrices is desired to understand the asymptotic nonnegative rank. However, the abstract theory of asymptotic spectra in Section 2 does not give an explicit construction the elements of X(M+, \u2a7d+). Previous works for graphs and tensors, thus, are devoted to show some elements in their asymptotic spectrum. Namely, for the (quantum) asymptotic spectrum of graphs, several spectral points were introduced in [Zui19, LZ20, GGL22, GL20] such as Lov\u00b4 asz theta number [Lov79]. In the case of tensors, [Str91] provided a universal spectral point for the asymptotic spectrum of oblique tensors (oblique is a strict subset of all tensors). Recently, in [CVZ18] constructed an in\ufb01nite family of the asymptotic spectrums of tensors over complex \ufb01eld. We recall that known all elements of the asymptotic spectrum of tensors can be used to determine the complexity of the matrix multiplication problem. In the same spirit, in this section, we show two elements that belong asymptotic spectrum of nonnegative matrices X(M+, \u2a7d+) introduced in Section 3, namely rank and fractional cover number. In addition, in the last subsection, we show that the asymptotic spectrum of lower triangular nonnegative matrices with non-zero values on their diagonal contains exactly a single point. 4.1 Rank We \ufb01rst remind readers of the de\ufb01nition of rank of a matrix. De\ufb01nition 4.1. Given a matrix A \u2208Rm\u00d7n, the rank of the matrix A (denoted as rank(A)) is the largest integer r such that there exists an invertible square submatrix of A of size r \u00d7 r. The following lemma shows that rank is indeed a spectral point of X(M+, \u2a7d+). 12 \fLemma 4.2. The matrix rank is an element of the asymptotic spectrum X(M+, \u2a7d+). Proof. Matrix rank is well-known to be multiplicative under \u2297and additive under \u2295. Also clearly rank(In) = n for all n \u2208N. Finally, it holds for any two matrices X, Y that if A = XBY , using the fact that rank(AB) \u2264min{rank(A), rank(B)}, then rank(A) \u2264rank(B). In particular, this is true if X, Y are nonnegative matrices, which proves rank is monotone. 4.2 Fractional cover number The fractional cover number is a parameter of nonnegative matrices introduced by Lov\u00b4 asz [Lov75] to study the chromatic number of uniform hypergraphs. Furthermore, in [KKN95], the authors showed that this parameter gives a characterization for amortized nondeterministic communication complexity. In this subsection, we show that the fractional cover number is a spectral point of X(M+, \u2a7d+). We \ufb01rst remind readers of the de\ufb01nition of fractional cover number. Let A \u2208Rm\u00d7n + be a nonnegative matrix, a subset X \u00d7 Y \u2286[m] \u00d7 [n] is called monochromatic rectangle of the matrix A if X \u00d7 Y \u2286supp(A). Let R1, . . . , Rk be all monochromatic rectangle of A. We de\ufb01ne the fractional cover number of A, denoted F(A), as follows minimize: k X i=1 \u03c4(Ri) s.t X i:(a,b)\u2208Ri \u03c4(Ri) \u22651 for all (a, b) \u2208supp(A) 0 \u2264\u03c4(Ri) \u22641 for all i \u2208[k] . The dual of the above linear program is given by: maximize: X (a,b)\u2208supp(A) \u00b5(a, b) s.t X (a,b)\u2208Ri \u00b5(a, b) \u22641 for all i \u2208[k] 0 \u2264\u00b5(a, b) \u22641 for all (a, b) \u2208supp(A) . In [Lov75], the author showed that the fractional cover number has the following properties. (i) Multiplicative: F(A \u2297B) = F(A)F(B). (ii) Additive: F(A \u2295B) = F(A) + F(B). It is easy to verify that F(In) = n for all n \u2208N. To prove that F is a spectral point, it is su\ufb03cient to prove its monotonicity with respect to the preorder \u2a7d+, that is, if A \u2a7d+ B then F(A) \u2264F(B). To do this, we \ufb01rst recall another characterization of fractional cover. We begin by reminding the notions of t-fold coverings and t-fold covering number of nonnegative A [SU11]. Let t be a positive integer, a t-fold covering of A is a multiset {R1, . . . , Rk} (unlike set, a multi-set allows an element to appear multiple times) where each Ri is a monochromatic rectangle of A and each (i, j) \u2208supp(A) is covered by at least t of the Ri\u2019s. The smallest cardinality of such a multiset is called the t-fold covering number of A and is denoted Ft(A). One can describe Ft(A) 13 \fas an integer linear programming as: minimize: k X i=1 \u03c4(Ri) s.t X i:(a,b)\u2208Ri \u03c4(Ri) \u2265t for all (a, b) \u2208supp(A) \u03c4(Ri) \u2208N for all i \u2208[k] . In particular, F(A) is simply a linear relaxation of the integer programming de\ufb01nition of F1(A). It is clear that Ft is subadditive in its subscript, that is, Fs+t(A) \u2264Fs(A) + Ft(A), because the sum of s and t-fold coverings yields a (s + t)-fold coverings. By Fekete\u2019s lemma, we can de\ufb01ne the asymptotic fractional covering number of A as follow F \u2217(A) := lim t\u2192\u221e Ft(A) t = inf t Ft(A) t . Proposition 4.3 ([SU11]). For any nonnegative matrix A, the fractional cover number is equal to the asymptotic fractional cover number, that is, F \u2217(A) = F(A). Proof. The proof can be found in [SU11], but we do it here for self-containedness. Let R := {R1, . . . , Rk} be the set of all monochromatic rectangles of A. The fractional cover F(A) can be described in the following compact form: minimize: k X i=1 xi s.t. Mx \u22651 x \u22650 . (8) where M is a binary matrix dependent on the set R. First we prove F(A) \u2264F \u2217(A). Given an arbitrary positive integer a, let {R1, . . . , Rv} be the smallest a-fold covering of A and let x = (x1, . . . , xk) the vector encoding the number of times Ri appears in this covering. Then Mx \u2265a, x \u22650. This implies that M(x/a) \u22651. Thus x/a is a feasible solution of the linear program (8), hence F(A) \u2264Pk i=1 xi/a = Fa(A)/a. Since this holds for all positive integers a, we have F(A) \u2264F \u2217(A). Next we prove that F(A) \u2265F \u2217(A). Since all coe\ufb03cients of the linear programming (8) are integers, there exists at least one rational vector x that yields the minimum value of the linear program (8). Let b be an integer such that bx is an integer vector, we have M(bx) \u2265b, bx \u22650. This shows that we can form an b-fold cover of A by choosing a monochromatic rectangle Ri with multiplicity bxi. Therefore bF(A) = Pk i=1 bxi \u2265Fb(A). By de\ufb01nition of F \u2217(A), we have F(A) \u2265Fb(A)/b \u2265F \u2217(A). Using Proposition 4.3, we obtain the following result. Proposition 4.4. Let A, B, C be nonnegative matrices, if A = BC then F(A) \u2264min(F(B), F(C)). 14 \fProof. It is su\ufb03cient to prove that F(A) \u2264F(B), F(A) \u2264F(C). We will show the former, i.e. F(A) \u2264F(B). The latter can be demonstrated similarly. In this proof, we represent a monochromatic rectangle by a rank-one binary matrix. In fact, given a subset X \u00d7 Y \u2282[m] \u00d7 [n], we represent it by the binary matrix 1X \u00d7 1\u22a4 Y \u2208Rm\u00d7n + where 1X \u2208Rm + (resp. 1Y \u2208Rn +) is the indication vector of the set X (resp. Y ). If A = BC, we will show that Ft(A) \u2264Ft(B) for any positive integer t. That concludes the proof by using the characterization of Proposition 4.3. Let {R1, . . . , Rk} be a multiset of monochromatic rectangles that yields the minimum value for t-fold covering of B. As discussed, there exist two binary vectors xi, yi such that Ri = xiy\u22a4 i . We de\ufb01ne R\u2032 i = supp(RiC) (RiC is the product between two nonnegative matrices Ri and C). We prove that the set {R\u2032 1, . . . , R\u2032 k} is a t-fold coverings of A. It is immediate that Ft(A) \u2264k = Ft(B). To do so, we \ufb01rst demonstrate that R\u2032 i is a monochromatic rectangle of A. Indeed, we use the following property of nonnegative matrices: for M, N two nonnegative matrices, then: supp(MN) = supp(supp(M)supp(N)), (9) where supp(M) is the binary matrix representation of the support of the matrix M. In other words, the support of the product of MN is determined by the product of supports of M and N (in their binary matrix forms). This claim only holds for nonnegative matrices and it is incorrect for general matrices. First, it is clear that R\u2032 i \u2286supp(A). Indeed, R\u2032 i = supp(RiC) = supp(supp(Ri)supp(C)) \u2286supp(supp(B)supp(C)) (since Ri \u2286supp(B)) (9) = supp(BC) = supp(A). Moreover, R\u2032 i = supp(RiC) = supp(xiy\u22a4 i C) = supp(supp(xi)supp(y\u22a4 i C)) = supp(xi)supp(y\u22a4 i C) since supp(xi)supp(y\u22a4 i C) is a binary matrix (product of two binary vectors). Thus, R\u2032 i is indeed a monochromatic rectangle of A. For any entry (i, j) \u2208supp(A), since 0 < Aij = P z=1 Bi,zCz,j, there is an index z such that Bi,z > 0 and Cz,j > 0. Suppose that (i, z) \u2208supp(B) is covered by a multiset {Ri1, . . . , Rit} then the entry (i, j) \u2208supp(A) is covered by the multiset {R\u2032 i1, . . . , R\u2032 it}. Therefore, {R\u2032 1, . . . , R\u2032 k} are indeed a t-fold coverings of A. That concludes the proof. Proposition 4.5. If A \u2a7d+ B then F(A) \u2264F(B). Proof. From A \u2a7d+ B, we have A = XBY T . By the Proposition 4.4, one has F(A) = F(XBY T ) \u2264F(XB) \u2264F(B) . We have the following lemma. Lemma 4.6. The fractional cover number F is an element of the asymptotic spectrum X(M+, \u2a7d+). 15 \fRemark 4.7. Rank and fractional cover number are incomparable. We have two elements in asymptotic spectrum X(M+, \u2a7d+) are rank and fractional cover number F. The following examples show that two quantities are incomparable, i.e., there are nonnegative matrices A, B with F(A) > rank(A) and F(B) < rank(B). A = \uf8eb \uf8ec \uf8ec \uf8ed 1 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 We have F(A) = 4, but rank(A) = 3. B = \uf8eb \uf8ec \uf8ec \uf8ed 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 On can compute that F(B) = 3.5, and rank(B) = 4. 4.3 Asymptotic spectrum of triangular nonnegative matrices As de\ufb01ned in Section 3, asymptotic spectrum is a \u2a7d+ monotone semiring homomorphism from S to R\u22650 that satis\ufb01es the constraint \u03c6(In) = n, \u2200n \u2208N, \u2200\u03c6 \u2208X(M+, \u2a7d+). In other words, the set of {\u03c6(In) | \u03c6 \u2208X(M+, \u2a7d+)} collapses to a single value n. In this section, we prove that this phenomenon also happens to the set of lower (square) triangular matrices with nonzero diagonal. Let A \u2208Rm\u00d7n + be a nonnegative matrix, we call the matrix A is a nonnegative rectangle lower triangular matrix if Ai,j = 0 for all i > j. Lemma 4.8. Let A \u2208Rm\u00d7n + be a rectangle lower triagular nonnegative matrix. De\ufb01ne V = {i \u2264 min{m, n} : Ai,i \u0338= 0}, then e Q+(A) \u2265|V |. Proof. Let d = |V | and let N be any multiple of d. Our proof is constructive, i.e, for any N multiple of d, we can \ufb01nd a diagonal submatrix of A\u2297N whose coe\ufb03cients on the diagonal are nonzero of size dN such that limN\u2192\u221ed1/N N = d. Our argument is valid since Proposition 3.9 assure that Q+(A\u2297N) \u2265dN. We argue that one can assume that the matrix A is square A, i.e. m = n. Indeed, if A is not square, one can take the principle submatrix B of size p \u00d7 p of the matrix A where p = min(m, n) and prove that e Q+(B) \u2265|V |. Since e Q+(A) \u2265e Q+(B) (by Proposition 3.9), we have: e Q+(A) \u2265|V |. Hence, in the remaining proof, we assume m = n and the size of the matrix A\u2297N is thus nN \u00d7 nN. In this proof, we will use I := {0, . . . , nN \u22121} as the set of row (or column) index of A\u2297N. It is convenient to represent each element of I in n-base (or radix), i.e, given i \u2208IR, we write i = (i1, . . . , iN), 0 \u2264i \u2264n \u22121 to implies: i = PN k=1 iknk\u22121. This representation is useful since the coe\ufb03cient at the index (i, j) of A\u2297N has the form: A\u2297N i,j = N Y k=1 Aik,jk. (10) De\ufb01ne S(N) the subset of supp(A\u2297N) on the diagonal such that for all (i, i) \u2208S(N), i = (i1, . . . , iN) \u2208S(N), we have: S(N) := {(i, i) \u2208supp(A\u2297N) | |{k \u2208[N] : ik = p}| = N/d for all p \u2208V } 16 \fIn words, if (i, i) \u2208S(N), then the n-basis representation of i contains exactly N/d digits p for every p \u2208V . The size of S(N) is thus, |S(N)| = \u0012 N N/d, . . . , N/d \u0013 \u2265 dN (N + 1)d . The inequality follows from the fact that the largest multinomial coe\ufb03cient is the central one, that is, \u0000N a1,...,ad \u0001 \u2264 \u0000N N d ,..., N d \u0001 and the number of possible partitions of N into d parts is at most (N + 1)d. This proves e Q+(A) \u2265limN\u2192\u221e|S(N)|1/N = d. Thus, it is su\ufb03cient to show that the principle submatrix of A restricted to indices S(N) is a diagonal matrix with nonzero coe\ufb03cients in its diagonal. Consider two indices i, j \u2208S(N) with i = (i1, . . . , iN), j = (j1, . . . , jN) and i \u0338= j. Since PN k=1 ik = PN k=1 yk = (N/d) P p\u2208V p, there must exist an index \u2113\u2264N such that i\u2113> j\u2113. By Equation 10, we deduce that Ai,j = 0 (since Ai\u2113,j\u2113= 0). In addition, Ai,i = QN k=1 Aik,ik \u0338= 0, \u2200i \u2208 S(N) because ik \u2208V . That shows elements on the diagonal are nonzero. That concludes the proof. Denote T \u2286M+ is a subsemiring which generated by all lower triangular matrices with nonzero entries on diagonal under operation \u2295and \u2297. From Lemma 4.8, the asymptotic spectrum X(T, \u2a7d+) contains only a single point. It is a corollary of Lemma 4.8. Corollary 4.9. Let A \u2208Rm\u00d7n + be a lower rectangular triangular nonnegative matrix with nonzero on its diagonal. Then \u03c6(A) = min(m, n) for any \u03c6 \u2208X(M+, \u2a7d+). Proof. Let d = min(m, n), since A \u2208Rm\u00d7n + with nonzero on its diagonal, it implies rank(A) = R+(A) = d, therefore e R+(A) \u2264d. By Lemma 4.8 one has e Q+(A) \u2265d. Using Theorem 2.2 (which states e R+(A) = max\u03c6\u2208X(M+,\u2a7d+) \u03c6(A) \u2265min\u03c6\u2208X(M+,\u2a7d+) \u03c6(A) = e Q+(A)), we have e R+(A) = e Q+(A) = d and \u03c6(A) = d, \u2200\u03c6 \u2208X(M+, \u2a7d+). This proves the corollary. Acknowledgments HT would like to thank Jeroen Zuiddam for the useful discussions about the theory of asymptotic spectra and asymptotic nonnegative rank. We also thank Yaroslav Shitov and Vincent Y. F. Tan for the useful discussions about the non-multiplicativity of nonnegative rank and the exact R\u00b4 enyi common information. 17"
+                },
+                {
+                    "url": "http://arxiv.org/abs/2109.09932v1",
+                    "title": "Endurance-Limited Memories: Capacity and Codes",
+                    "abstract": "\\emph{Resistive memories}, such as \\emph{phase change memories} and\n\\emph{resistive random access memories} have attracted significant attention in\nrecent years due to their better scalability, speed, rewritability, and yet\nnon-volatility. However, their \\emph{limited endurance} is still a major\ndrawback that has to be improved before they can be widely adapted in\nlarge-scale systems.\n  In this work, in order to reduce the wear out of the cells, we propose a new\ncoding scheme, called \\emph{endurance-limited memories} (\\emph{ELM}) codes,\nthat increases the endurance of these memories by limiting the number of cell\nprogramming operations. Namely, an \\emph{$\\ell$-change $t$-write ELM code} is a\ncoding scheme that allows to write $t$ messages into some $n$ binary cells\nwhile guaranteeing that each cell is programmed at most $\\ell$ times. In case\n$\\ell=1$, these codes coincide with the well-studied \\emph{write-once memory}\n(\\emph{WOM}) codes. We study some models of these codes which depend upon\nwhether the encoder knows on each write the number of times each cell was\nprogrammed, knows only the memory state, or even does not know anything. For\nthe decoder, we consider these similar three cases. We fully characterize the\ncapacity regions and the maximum sum-rates of three models where the encoder\nknows on each write the number of times each cell was programmed. In\nparticular, it is shown that in these models the maximum sum-rate is $\\log\n\\sum_{i=0}^{\\ell} {t \\choose i}$. We also study and expose the capacity regions\nof the models where the decoder is informed with the number of times each cell\nwas programmed. Finally we present the most practical model where the encoder\nread the memory before encoding new data and the decoder has no information\nabout the previous states of the memory.",
+                    "authors": "Yeow Meng Chee, Michal Horovitz, Alexander Vardy, Van Khu Vu, Eitan Yaakobi",
+                    "published": "2021-09-21",
+                    "updated": "2021-09-21",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION Emerging resistive memory technologies, such as resistive random access memories (ReRAM) and phase-change memories (PCM), have the potential to be the future\u2019s universal memories. They combine several important attributes starting from the speed of SRAM, the density of DRAM, and the non-volatility of \ufb02ash memories. However, they fall short in their write endurance, which signi\ufb01cantly increases their bit error rate (BER). Hence, solving the limited endurance of these memories is crucial before they can be widely adapted in large-scale systems [9], [18], [28]. Resistive memories are non-volatile memories which are composed of cells. The information is stored in the cells by changing their resistance. They combine the following two properties of DRAM and \ufb02ash memories. Similarly to \ufb02ash memories and unlike DRAM they are non-volatile memories and thus they don\u2019t require refresh operations. Furthermore, like DRAM and unlike \ufb02ash memories they are rewritable without an erase operation. The main challenge that has remained to be solved in order to make these memories a legitimate candidate as a universal memory is their limited write endurance, which is the goal of this paper. Endurance is de\ufb01ned as the number of set/reset cycles to switch the state of cells in ReRAM while it is still reliable. Owing to its importance, there are many researches which test and characterize the endurance of ReRAM in order to show a strong dependence on material of cells, cell size, [16], [26], and program operation [22]. To improve the endurance of ReRAM, recent research have focused on the structure and material of devices [16], [25] and programming schemes [6], [22]. In this work, we present a scheme to use rewriting code to improve the endurance lifetime of ReRAM. Previous works have offered different solutions to combat the write endurance of resistive memories. In [12], the authors proposed to use locally repairable codes (LRC) in order to construct codes with small rewriting locality in order to mitigate both the problems of endurance and power consumption. In [27], the authors proposed mellow writes, a technique which is targeted to reduce the wear-out of the writes rather than reducing the number of writes. Lastly, several other works proposed coding schemes which correct stuck-at cells; see e.g. [14], [19], [24]. In order to combat the limited write endurance in resistive memories, this paper proposes to study the following new family of codes, called endurance-limited memory (ELM) codes. Assume there are n binary cells and t messages that are required to be stored in these cells sequentially. Assume also that each cell can be programmed at most \u2113\u2a7e1 times. Then, we seek to \ufb01nd the set of achievable rates, i.e., the capacity region, and design code constructions for this model. Note that for \u2113= 1, we get the classical problem of write-once memory (WOM) codes [2], [10], [11], [17], [21], [23]. Besides that, if t \u2a7d\u2113, the coding scheme is trivial. Hence, in this work, we only focus on the cases where t > \u2113> 1. Note that a trivial lower bound on the maximum sum-rate is min{t, \u2113}, which achieved by writing with rate 1 in the \ufb01rst min{t, \u2113} writes and with rate 0 in the remaining writes. Parts of the results in this paper were presented in the International Symposium on Information Theory and Its Applications, Oct 2018 [4], and in IEEE Information Theory Workshop, August 2019 [5]. Yeow Meng Chee is with the Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore (email: ymchee@nus.edu.sg). Michal Horovitz is with the Department of Computer Science, Tel-Hai College, and The Galilee Research Institute Migal, Kiryat Shmona 11016, Upper Galilee Israel (e-mail: horovitzmic@telhai.ac.il). Alexander Vardy is with the Department of Electrical and Computer Engineering and the Department of Computer Science and Engineering, University of California San Diego, La Jolla, CA 92093 USA (e-mail: avardy@ucsd.edu). Van Khu Vu is with the Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore (email: isevvk@nus.edu.sg). Eitan Yaakobi is with the Computer Science Department, Technion \u2014 Israel Institute of Technology, Haifa 3200003, Israel (e-mail: yaakobi@cs.technion.ac.il). arXiv:2109.09932v1  [cs.IT]  21 Sep 2021 \fLet us consider \ufb01rst the case where \u2113= 2 and t = 3. A naive solution is to use a two-write WOM code for the \ufb01rst two writes and then write n more bits on the third write. The maximum sum-rate using this solution will be log(3) + 1 = log(6), while, as will be shown in the paper, the maximum sum-rate in this case is log(7). The intuition behind this is as follows. Let p1 be the probability to program a cell on the \ufb01rst write, so we assume that p1n cells are programmed. Then, on the second and third writes we have a two-write WOM code problem for the p1n programmed cells, and for the (1 \u2212p1)n non-programmed cells, we can write twice on them so no coding is needed. The maximum sum-rate is achieved for p1 = 3/7. However, it is still a challenging task to design speci\ufb01c code constructions that can approach sum-rate of log(7). There are several models of ELM codes which can be studied. These models are distinguishable by the information that is available to the encoder and the decoder. In particular, for the encoder, we consider three cases which depend upon whether the encoder knows the number of times each cell was programmed, encoder informed all (EIA), only the current state of the cell, encoder informed partially (EIP), or no information about the cells state, encoder uninformed (EU). The decoder will also have three cases, corresponding to the same information that is available to the encoder. Thus, by considering all combinations of the above three cases for the encoder and the decoder, it is possible to de\ufb01ne and study nine models, EX : DY , where X, Y \u2208{IA, IP, U}. The rest of this paper is organized as follows. In Section II, we formally de\ufb01ne the models studied in this paper and discuss some basic observations. In Section III, we study the capacity regions and the maximum sum-rates of the EIA models, and also present capacity achieving codes. We prove that the capacity region of all EIA models, are the same, for both \u03f5-error and zero-error cases. In the next two sections, we discuss the EIP:DIA model. In Section IV, we study the capacity region of this model for the \u03f5-error case, and in Section V, we compare between this model and the EIA models. Then, we discuss the EU:DIA and the EIP:DU models in Sections VI and VII, respectively. Finally, we conclude our results and discuss a future work in Section VIII. II. DEFINITIONS AND PRELIMINARIES In this section, we formally de\ufb01ne the nine models of ELM codes, and we state some simple propositions. Assume that each cell can be programmed at most \u2113times, so if the encoder attempts to program a cell more than \u2113times then it will not change its value. For the EIA models, we assume that the encoder will not try to program a cell that has already been programmed \u2113 times before the current write. We see this as an extension of the WOM model for \u2113= 1. These models will be de\ufb01ned both for the zero-error and the \u03f5-error cases. For a positive integer a, the set {0, . . . , a\u22121} is de\ufb01ned by [a]. Throughout this paper, we assume that the number of cells is n. We use the vector notation c \u2208[2]n to represent the cell-state vector of the n memory cells, and the vector v \u2208[\u2113+ 1]n, which will be called the cell-program-count vector, to represent the number of times each cell was programmed. Note that the state of a cell is the parity of the number of times it was programmed. Thus, if the encoder (or the decoder) knows the cell-program-count vector v, in particular it knows the cell-state vector c as well. For a vector v \u2208[\u2113+ 1]n, we denote by \u27e8v\u27e92 the length-n binary vector which satis\ufb01es \u27e8v\u27e92,k = vk(mod 2) for all k \u2208[n], and we say that \u27e8v\u27e92 equals to v modulo 2. The complement of a binary vector c is denoted by c. The all ones, zeros vector will be denoted by 1, 0, respectively. For two length-n vectors a and b, a + b is the vector obtained by pointwise addition. If a and b are binary vectors a \u2295b is the vector obtained by pointwise addition modulo 2. For a cell-program-count vector v \u2208[\u2113+ 1]n and a new cell-state vector c \u2208[2]n to be programmed to the cells, we de\ufb01ne by N(v, c) \u2208[\u2113+ 1]n and f(v, c) \u2208[2]n the result of programming the new cell-state vector c. That is, N(v, c) is the new cell-program-count vector after programming c, f(v, c) is the new cell-state vector, and they are formally de\ufb01ned as follows. N(v, c)k = vk if ck = vk(mod 2), and otherwise N(v, c)k = min{\u2113, vk + 1}, where the index k for a vector means its k-th element. Similarly, f(v, c)k = ck if vk < \u2113, and otherwise f(v, c)k = vk(mod 2). Note that \u27e8N(v, c)\u27e92 = f(v, c), i.e., f(v, c) equals to N(v, c) modulo 2. We are ready now to de\ufb01ne all models studied in this paper. De\ufb01nition 1. An [n, t, \u2113; M1, . . . , Mt]EX:DY,pe \u2113-change t-write Endurance-Limited Memory (ELM) code with error-probability vector pe = (pe1, . . . , pet), where X, Y \u2208{IA, IP, U}, is a coding scheme comprising of n binary cells and is de\ufb01ned by t encoding and decoding maps (Ej, Dj) for 1 \u2a7dj \u2a7dt. For the map Ej, Im(Ej) is its image, where by de\ufb01nition Im(E0) = {(0, . . . , 0)}. Furthermore, for j \u2208[t + 1], let Nj and Im\u2217(Ej) be the sets of all state-program-count vectors, cell-state vectors which can be obtained after the \ufb01rst j writes, respectively. Formally, for 1 \u2a7dj, Nj = {N(v, c) : c \u2208Im(Ej), v \u2208Nj\u22121}, where N0 = {(0, . . . , 0)}, and Im\u2217(Ej) = {\u27e8v\u27e92 : v \u2208Nj}. Note that for the EIA models Im\u2217(Ej) = Im(Ej). The domain and the range of the encoding maps are de\ufb01ned as follows: (1) for the EIA models, Ej : [Mj] \u00d7 Nj\u22121 \u2192[2]n, such that for all (m, v) \u2208[Mj] \u00d7 Nj\u22121 it holds that v + (\u27e8v\u27e92 \u2295Ej(m, v)) \u2208 [\u2113+ 1]n. (2) for the EIP models, Ej : [Mj] \u00d7 Im\u2217(Ej\u22121) \u2192[2]n. (3) for the EU models, Ej : [Mj] \u2192[2]n. For a message m we denote by Im(x) the indicator function, where Im(x) = 0 if m = x, and otherwise Im(x) = 1. Additionally, for a message m \u2208[Mj], Pr(m) is the probability of programming a message m on the j-th write, and for v \u2208Nj\u22121, Pr(v) is the probability to have cell-program-count vector v before the j-th write. The nine models are de\ufb01ned as follows. For all 1 \u2a7dj \u2a7dt, (1) if (X, Y ) = (IA, IA) then Dj : {(Ej(m, v), v) : m \u2208[Mj], v \u2208Nj\u22121}\u2192[Mj], \fX (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(Ej(m, v), v)) \u2a7dpej, (2) if (X, Y ) = (IA, IP) then Dj : {(Ej(m, v), \u27e8v\u27e92) : m \u2208[Mj], v \u2208Nj\u22121}\u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(Ej(m, v), \u27e8v\u27e92)) \u2a7dpej, (3) if (X, Y ) = (IA, U) then Dj : Im(Ej) \u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(Ej(m, v))) \u2a7dpej, (4) if (X, Y ) = (IP, IA) then Dj : {(f(v, Ej(m, \u27e8v\u27e92)), v) : m \u2208[Mj], v \u2208Nj\u22121}\u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(f(v, Ej(m, \u27e8v\u27e92)), v)) \u2a7dpej, (5) if (X, Y ) = (IP, IP) then Dj : {(f(v, Ej(m, \u27e8v\u27e92)), \u27e8v\u27e92) : m \u2208[Mj], v \u2208Nj\u22121}\u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(f(v, Ej(m, \u27e8v\u27e92)), \u27e8v\u27e92)) \u2a7dpej, (6) if (X, Y ) = (IP, U) then Dj : Im\u2217(Ej) \u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(f(v, Ej(m, \u27e8v\u27e92)))) \u2a7dpej, (7) if (X, Y ) = (U, IA) then Dj : {(f(v, Ej(m)), v) : m \u2208[Mj], v \u2208Nj\u22121}\u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(f(v, Ej(m))v)) \u2a7dpej, (8) if (X, Y ) = (U, IP) then Dj : {(f(v, Ej(m)), \u27e8v\u27e92) : m \u2208[Mj], v \u2208Nj\u22121}\u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(f(v, Ej(m))\u27e8v\u27e92)) \u2a7dpej, (9) if (X, Y ) = (U, U) then Dj : Im\u2217(Ej) \u2192[Mj], X (m,v)\u2208[Mj]\u00d7Nj\u22121 Pr(m)Pr(v) \u00b7 Im (Dj(f(v, Ej(m)))) \u2a7dpej. If pej = 0 for all 1 \u2a7dj \u2a7dt, then the code is called a zero-error ELM code and is denoted by [n, t, \u2113; M1, . . . , Mt]EX:DY,z. The rate on the j-th write of an [n, t, \u2113; M1, . . . , Mt]EX:DY,pe ELM code, X, Y \u2208{IA, IP, U}, is de\ufb01ned as Rj = log Mj n , and the sum-rate is the sum of the individual rates on all writes, Rsum = Pt j=1 Rj. A rate tuple R = (R1, . . . , Rt) is called \u03f5-error achievable in model EX : DY , if for all \u03f5 > 0 there exists an [n, t, \u2113; M1, . . . , Mt]EX:DY ,pe ELM code with error-probability vector pe = (pe1, . . . , pet) \u2a7d(\u03f5, . . . , \u03f5), such that log Mj n \u2a7eRj \u2212\u03f5. The rate tuple R will be called zero-error achievable if for all 1 \u2a7dj \u2a7dt, pej = 0. The \u03f5-error capacity region of the EX:DY model is the set of all \u03f5-error achievable rate tuples, that is, CEX:DY,\u03f5 t,\u2113 = {(R1, . . . , Rt)|(R1, . . . , Rt) is \u03f5-error achievable}, and the \u03f5-error maximum sum-rate will be denoted by REX:DY,\u03f5 t,\u2113 . The zero-error capacity region CEX:DY,z t,\u2113 and the zero-error maximum sum-rate REX:DY,z t,\u2113 are de\ufb01ned similarly. We say that R \u2a7dR\u2032 for R = (R1, . . . , Rt) and R\u2032 = (R\u2032 1, . . . , R\u2032 t) if Rj \u2a7dR\u2032 j for all 1 \u2a7dj \u2a7dt, and R < R\u2032 if R \u2a7dR\u2032 and R \u0338= R\u2032. \fAccording to these de\ufb01nitions it is easy to verify the following relations. For g \u2208{z, \u03f5} and X, Y \u2208{IA, IP, U} it holds that CEU:DY,g t,\u2113 \u2286CEIP :DY,g t,\u2113 \u2286CEIA:DY,g t,\u2113 , CEX:DU,g t,\u2113 \u2286CEX:DIP,g t,\u2113 \u2286CEX:DIA,g t,\u2113 , and CEX:DY,z t,\u2113 \u2286CEX:DY,\u03f5 t,\u2113 . Similar connections hold for the maximum sum-rates. Note that if \u2113\u2a7et then all problems are trivial since it is possible to program all cells on each write, so the capacity region in all models is [0, 1]t and the maximum sum-rate is t. For \u2113= 1, we get the classical and well-studied WOM codes [2], [10], [11], [17], [21], [23]. In this case, we also notice that the IA and IP models are the same for both the encoder and the decoder. The capacity region and the maximum sum-rate in most of these cases are known; see e.g. [10], [11], [17], [23]. In the rest of this paper, and unless stated otherwise, we assume that 1 \u2a7d\u2113< t. A. Related Work The EIA models of ELM codes studied in this paper are strongly related to non-binary WOM codes and their modi\ufb01ed versions studied in [3], [8], [11], [13]. In these EIA models, we can treat every cell as an (\u2113+ 1)-ary cell, where it is only possible to increase its level by one on each write, while its maximum level is \u2113. In non-binary WOM codes, each cell has q levels and its level can not be decreased [8], [11]. In a write \u2113-step-up memory [3], a special version of the non-binary WOM code, each cell has q levels and each time we write a cell, its level can be increased by at most some value \u2113. Recently, Kobayashi et al. [13] also studied a modi\ufb01ed version of non-binary WOM codes, called write constrained memories, where there is a cost on each state transition. Yet, these codes are not identical and we can not apply previous results to solve the models of ELM codes. In fact, our results on EIA models are useful to obtain an explicit formula of the capacity region of write \u2113-step-up memory codes. We also note that some models of ELM codes, such as the EIP:DU model, are much different from previous models and are dif\ufb01cult to solve. Moreover, our proposed ELM coding scheme is also related to the cooling code which is used to control the peak temperature of an interconnect [1], [2]. In [1], cooling codes are proposed to avoid all hottest wires and in [2], cooling codes are shown to be equivalent to two-write binary WOM codes. Later in this work, we will use cooling codes as two-write binary WOM codes in order to construct our ELM codes. Now, we discuss the ability of our ELM codes to control the peak temperature of an interconnection. In fact, using our coding scheme in this work, we can control the maximal number of switches in each wire. We note that the temperature of each wire is closely related to the number of switches in each wire. And thus, we can control the peak temperature of each wire. Recently, the EIA:DIA model of ELM is shown to be useful for two-dimensional (2D) weight-constrained code scheme [15]. In a 2D weight-constrained code, each codeword is an array of size m \u00d7 n where the number of 1 symbols in each row and each column is limited by pn and qm, respectively. The encoder and decoder know the values of all the mn bits. We can view this 2D weight-constrained code as we write m messages in ReRAM where each cell can be switched at most qm times and both encoder and decoder know all previous messages. B. Our Contribution In this work, we propose a novel scheme of rewriting code to improve the endurance lifetime of ReRAM, called endurance limited memories (ELM) code. In \u2113-change t-write ELM codes, each cell can be programmed at most \u2113times during the writing of t messages. In the case that \u2113is much smaller than t, we can signi\ufb01cantly improve the lifetime of the memories. Depending upon whether the encoder (E) and decoder (D) know the number of times each cell is programmed (IA), the current state of each cell (IP), or have no information on the state of each cell (U), we present and investigate all nine models EX : DY where X, Y \u2208{IA, IP, U}. We note that the most practical model to increase the endurance of ReRAM is the EIP:DU model. However, for the theoretical interest, we study all nine models in this paper. Furthermore, although the EIA models are not suitable for improving the endurance of ReRAM, they have several applications, including write \u2113-step-up memories and two-dimensional weight-constrained codes. We expect that we can see other applications of all nine models of ELM codes in near future. In Section III, all three EIA models are investigated in both cases, \u03f5-error and zero-error. We \ufb01rst provide the capacity region and the maximum sum-rate of these models. The techniques are used to achieve the results on the capacity regions in Theorems 2 and 3 are similar to those used in [11] for WOM codes. To achieve the explicit formula of the maximum sum-rate in Theorem 4, we need a new simple technique. We then present several constructions of ELM codes for the EIA models. To construct these codes, we need some new ideas even though we also use several special families of WOM codes as components of our ELM codes. In Section IV, we use a known technique in information theory to obtain the capacity region of the EIP:DIA model. The capacity region of all EIA models is compared to the EIP:DIA model in Section V. Using the same technique carefully, we can also \ufb01nd the capacity region of the EU:DIA model in Section VI. Finally, in Section VII, we study the most practical model for ReRAM, the EIP:DU model. Although several good bounds on the maximum sum-rate are presented, these bounds are not tight and \ufb01nding an exact formula of the maximum sum-rate of the EIP:DU model is still an open problem. To achieve some good constructive lower bounds, we provide several constructions of EIP:DU ELM codes for the zero-error case. These results are novel and we need different original methods to achieve them. \fIII. THE EIA MODELS In this section, we explore the capacity region and the maximum sum-rate of the EIA models for both the \u03f5-error and the zero-error cases. We also propose capacity achieving codes for these cases. For each j \u2208[t + 1], we let cj denote the binary length-n vector which represents the cell-state vector after the j-th write, where c0 = 0. Recall that in the EIA models, on the j-th write the encoder knows the number of times each cell was programmed before the current write. That is, the encoder receives as an input a length-n cell-program-count vector N(cj\u22121) \u2208[\u2113+ 1]n that represents the number of times each cell was programmed so far. Next, for all t and \u2113, we de\ufb01ne the region Ct,\u2113and in Theorem 2, we prove that this is the capacity region of all the EIA models. For 1 \u2a7dj \u2a7dt and i \u2208[\u2113+ 1], i \u2a7dj, let pj,i \u2208[0, 0.5], be the probability to program a cell on the j-th write, given that this cell has been already programmed i times. We de\ufb01ne pj,j = pj,\u2113= 0 for 1 \u2a7dj \u2a7dt, and let Qj,i be the probability that a cell has been programmed exactly i times on the \ufb01rst j writes. Formally, Qj,i is de\ufb01ned recursively by using pj,i and pj,i\u22121 as follows: Qj,i = ( Qj\u22121,i(1 \u2212pj,i) + Qj\u22121,i\u22121pj,i\u22121, if i > 0, Qj\u22121,i(1 \u2212pj,i), if i = 0, (1) where for j = 0 we set Q0,0 = 1 otherwise Q0,i = 0. The rates region Ct,\u2113is de\ufb01ned as follows: Ct,\u2113= n (R1, . . . , Rt)|\u22001\u2a7dj\u2a7dt : Rj \u2a7dPmin{\u2113,j}\u22121 i=0 Qj\u22121,ih(pj,i), \u2200i \u2208[\u2113] : pj,i \u2208[0, 0.5], and Qj,i is de\ufb01ned in (1) o , (2) where in this paper h(x), H(X) is the binary entropy function where 0 \u2a7dx \u2a7d1, X is a random variable, respectively. Note that for \u2113= 1, it is possible to verify that we get the capacity region of WOM [10], [17], [23]. It is also readily veri\ufb01ed that the maximum sum-rate is achieved with pj,i = 0.5 for all t \u2212j + 1 \u2a7e\u2113\u2212i, since t \u2212j + 1 is the number of the remaining writes, and \u2113\u2212i is the number of times the cell can be programmed. Thus, if t \u2212j + 1 \u2a7e\u2113\u2212i then we can program the cell with probability 0.5 to obtain the maximum rate. The next theorem proves that for 2 \u2a7d\u2113\u2a7dt \u22121, Ct,\u2113is the capacity region of the \u2113-change t-write ELM in all EIA models, and thus we denote this capacity by CEIA t,\u2113 , and the maximum sum-rate by REIA t,\u2113 . Theorem 2. The rates region Ct,\u2113is the capacity region of the \u2113-change t-write ELM in all EIA models for both \u03f5-error and zero-error cases. That is, for all g \u2208{z, \u03f5} and Y \u2208{IA, IP, U}, CEIA:DY,g t,\u2113 = Ct,\u2113. Proof: Recall that by the de\ufb01nitions of the models CEIA:DU,z t,\u2113 \u2286CEIA:DIP,z t,\u2113 \u2286CEIA:DIA,z t,\u2113 \u2286CEIA:DIA,\u03f5 t,\u2113 , and CEIA:DU,z t,\u2113 \u2286CEIA:DU,\u03f5 t,\u2113 \u2286CEIA:DIP,\u03f5 t,\u2113 \u2286CEIA:DIA,\u03f5 t,\u2113 . The rest of the proof consists of two parts. The \ufb01rst part, called the direct part, proves that Ct,\u2113\u2286CEIA:DU,z t,\u2113 , and in the second, called the converse part, we prove that CEIA:DIA,\u03f5 t,\u2113 \u2286Ct,\u2113. The direct part is proved in Subsection III-A for the zero-error case of the EIA:DU model, while the converse part is proved in Subsection III-B for the \u03f5-error case of the EIA:DIA model. Next, we seek to present the capacity region of the EIA models in a recursive form. While we see this representation of the capacity region more intuitive, it will also help us in \ufb01nding the maximum sum-rate of this model. For all t \u2a7e1 and \u2113\u2a7e1, let b Ct,\u2113be the following region which is de\ufb01ned recursively as follows. For t > \u2113\u2a7e1 b Ct,\u2113= n (R1, . . . , Rt)|R1 \u2a7dh(p), p \u2208[0, 0.5], for 2 \u2a7dj \u2a7dt, Rj \u2a7dpR\u2032 j + (1 \u2212p)R\u2032\u2032 j , (R\u2032 2, . . . , R\u2032 t) \u2208b Ct\u22121,\u2113\u22121 and (R\u2032\u2032 2, . . . , R\u2032\u2032 t )\u2208b Ct\u22121,\u2113 o , (3) where for all \u2113\u2a7et \u2a7e1 we set b Ct,\u2113= [0, 1]t and b Ct,0 = {0}. Theorem 3. For all t and \u2113, b Ct,\u2113= Ct,\u2113. Proof: For the \ufb01rst direction, we prove by induction on t that for all \u2113\u2a7e1, if R = (R1, . . . , Rt) = b Ct,\u2113then R \u2208CEIA:DIA,\u03f5 t,\u2113 . Since CEIA:DIA,\u03f5 t,\u2113 = Ct,\u2113, we conclude that b Ct,\u2113\u2286Ct,\u2113. The base of the induction is t \u2a7d\u2113for all \u2113\u2a7e1. These cases are readily veri\ufb01ed. For the induction step, let R = (R1, R2, . . . , Rt) \u2208 b Ct,\u2113, 1 \u2a7d\u2113< t, such that R1 = h(p) for p \u2208[0, 0.5] and for 2 \u2a7dj \u2a7dt Rj = pR\u2032 j +(1\u2212p)R\u2032\u2032 j where (R\u2032 2, R\u2032 3, . . . , R\u2032 t) \u2208b Ct\u22121,\u2113\u22121 and (R\u2032\u2032 2, R\u2032\u2032 3, . . . , R\u2032\u2032 t ) \u2208b Ct\u22121,\u2113. By the induction hypothesis, (R\u2032 2, R\u2032 3, . . . , R\u2032 t) \u2208CEIA:DI t\u22121,\u2113\u22121 and (R\u2032\u2032 2, R\u2032\u2032 3, . . . , R\u2032\u2032 t ) \u2208CEIA:DI t\u22121,\u2113 . Thus, we have two codes: C1 an (\u2113\u22121)-change (t \u22121)-write ELM code which achieves the rate tuple (R\u2032 2, R\u2032 3, . . . , R\u2032 t) and C2 an \u2113-change (t \u22121)-write ELM code which achieves the rate tuple (R\u2032\u2032 2, R\u2032\u2032 3, . . . , R\u2032\u2032 t ). Then, we can design an \u2113-change t-write ELM code, such that on the \ufb01rst write the encoder programs a cell with probability p for p \u2208[0, 0.5], and then on the next writes it applies C1 for the cells that were programmed on the \ufb01rst write, and C2 for the other cells. Thus, the rate tuple R is achieved. \fThe second direction, Ct,\u2113\u2286b Ct,\u2113, is proved by induction on t, that is, for each t \u2a7e1 we prove that Ct,\u2113\u2286b Ct,\u2113for all 1 \u2a7d\u2113\u2a7dt. The base of the induction, t = 1 and \u2113= 1, is trivial. The induction assumption is that for each 1 \u2a7d\u2113\u2032 \u2a7dt\u22121, Ct\u22121,\u2113\u2032 \u2286b Ct\u22121,\u2113\u2032. For the induction step, let R = (R1, R2, . . . , Rt) \u2208Ct,\u2113which is achieved by the probabilities pj,i. Denote by R\u2032 = (R\u2032 2, R\u2032 3, . . . , R\u2032 t) \u2208 Ct\u22121,\u2113\u22121 the rate tuple which is attained by the probabilities p\u2032 j,i = pj+1,i+1, and by R\u2032\u2032 = (R\u2032\u2032 2, R\u2032\u2032 3, . . . , R\u2032\u2032 t ) \u2208Ct\u22121,\u2113the rate tuple which is attained by the probabilities p\u2032\u2032 j,i = pj+1,i. Recall that we de\ufb01ne b Ct\u22121,t = b Ct\u22121,t\u22121, and b Ct\u22121,0 = {0}. It can be easily veri\ufb01ed that for all j, 2 \u2a7dj \u2a7dt, Rj = p1,0R\u2032 j +(1\u2212p1,0)R\u2032\u2032 j . By the induction hypothesis, R\u2032 \u2208b Ct\u22121,\u2113\u22121 and R\u2032\u2032 \u2208b Ct\u22121,\u2113. Thus, by de\ufb01ning p = p1,0 we get a recursive form for R, and we can conclude that Ct,\u2113\u2286b Ct,\u2113. Next, using the result from Theorem 3, it is possible to \ufb01nd the maximum sum-rate of the EIA models, REIA t,\u2113 . Theorem 4. For all t and \u2113, REIA t,\u2113 = log \u2113 X i=0 \u0012t i \u0013 , and this value is achieved for p1,0 = p = P\u2113\u22121 i=0 \u0000t\u22121 i \u0001 P\u2113 i=0 \u0000t i \u0001 , where p1,0, p are de\ufb01ned in Equations (2), (3) in Ct,\u2113, b Ct,\u2113, respectively. For example, if \u2113= 2 the maximum sum-rate is achieved for p1,0 = p = 2t t2+t+2. Proof: First, we prove that REIA t,\u2113 \u2a7dlog P\u2113 i=0 \u0000t i \u0001 by counting all the possible sequences of t messages. We describe each possible sequence as a table of t rows and n columns, where n is the number of cells. Note that different sequences will be mapped to different matrices. Recall, that every cell can be programmed at most \u2113times. Thus, the number of different possible matrices is \u0010P\u2113 i=0 \u0000t i \u0001\u0011n , and the upper bound is proved. Next we assure that this upper bound is indeed tight. We prove this result by using the recursive formula for the capacity b Ct,\u2113 described in Equation (3). For \u2113= 1, W\u2113M is the binary WOM, and this upper bound, log(t+1) is known to be tight, and achieved for p = 1/(t+1) [8], [10], [23]. That is, the maximum sum-rate of one-change t-write W\u2113M is equal to log P\u2113 i=0 \u0000t i \u0001 = log(t+1). Let us denote Xt,\u2113= P\u2113 i=0 \u0000t i \u0001 and p = Xt\u22121,\u2113\u22121 Xt,\u2113 . Note that by the properties of the binomial coef\ufb01cients, Xt,\u2113= Xt\u22121,\u2113\u22121 + Xt\u22121,\u2113. Therefore, 1 \u2212p = Xt\u22121,\u2113 Xt,\u2113. By using the recursive formula for the capacity b Ct,\u2113described in Equation (3), we are only left to prove that for all 2 \u2a7d\u2113\u2a7dt \u22121, log Xt,\u2113= h(p) + p log Xt\u22121,\u2113\u22121 + (1 \u2212p) log Xt\u22121,\u2113. This relation holds since h(p) + p log Xt\u22121,\u2113\u22121 + (1 \u2212p) log Xt\u22121,\u2113 = p \u0010 log \u0010 Xt,\u2113 Xt\u22121,\u2113\u22121 \u0011 + log(Xt\u22121,\u2113\u22121) \u0011 +(1 \u2212p) \u0010 log \u0010 Xt,\u2113 Xt\u22121,\u2113 \u0011 + log(Xt\u22121,\u2113) \u0011 = p log Xt,\u2113+ (1 \u2212p) log(Xt,\u2113) = log(Xt,\u2113). A. The EIA:DU Model Constructions and Direct Part of Theorem 2 In this subsection, we study the EIA:DU model, that is, encoder informed all and decoder uninformed. Our main contribution is a construction of a capacity-achieving \u2113-change t-write EIA:DU-ELM code for the zero-error case, which assures that Ct,\u2113\u2286 CEIA:DU,z t,\u2113 . That is, the direct part of Theorem 2 is proved. Let us start with the \ufb01rst non-trivial case of t = 3 and \u2113= 2. Thus, we want to prove that C3,2 \u2286CEIA:DU,z 3,2 . Recall that, C3,2= n (R1, R2, R3)|R1 \u2a7dh(p1,0), R2\u2a7d1\u2212p1,0 + p1,0h(p2,1), R3\u2a7d1\u2212p1,0p2,1, and p1,0, p2,1\u2208[0, 0.5] o . which is achieved by setting p3,0 = p3,1 = p2,0 = 0.5 in Equation (2). The next theorem states the existence of a construction of ELM codes for this case. Theorem 5. For any \u03f5 > 0 and p1,0, p2,0, p2,1 \u2208[0, 0.5], there exists an explicit construction of a zero-error two-change three-write EIA:DU-ELM code satisfying R1 \u2a7eh(p1,0) \u2212\u03f5, R2 \u2a7e(1 \u2212p1,0)h(p2,0) + p1,0h(p2,1) \u2212\u03f5, and R3 \u2a7e(1 \u2212p1,0p2,1) \u2212\u03f5. \fBefore presenting our construction for two-change three-write EIA:DU-ELM codes, we introduce the following family of WOM codes. We then use these WOM codes as component codes in our construction of EIA:DU-ELM codes. Note that the WOM codes we use for our construction are given for n \u2192\u221e, and thus our constructions for ELM codes use such n. De\ufb01nition 6. An [n, 2; M1, M2]EI:DU,z q two-write q-ary EI:DU WOM code for the zero-error case is a coding scheme comprising of n q-ary cells. It consists of two pairs of encoding and decoding maps (Eq,1, Dq,1) and (Eq,2, Dq,2) which are de\ufb01ned as follows: (1) Eq,1 : [M1] \u2192[q]n and Dq,1 : Im(Eq,1) \u2192[M1] such that for all m1 \u2208[M1], Dq,1(Eq,1(m1)) = m1. (2) Eq,2 : [M2] \u00d7 Im(Eq,1) \u2192[q]n and Dq,2 : Im(Eq,2) \u2192[M2] such that for all (m2, c) \u2208[M2] \u00d7 Im(Eq,1), Eq,2(m2, c) \u2a7ec and Dq,2(Eq,2(m2, c)) = m2. We say that p = (p0, p1, . . . , pm\u22121) is a probability vector if Pm\u22121 i=0 pi = 1 and pi \u2a7e0 for all i \u2208[m]. We distinguish between an error-probability vector that is used in De\ufb01nition 1, and a probability vector. An error-probability vector is a vector of errorprobabilities, and not a probability vector, i.e., the sum of the elements of an error-probability vector does not need be 1. For two positive integers n, q and a probability vector p = (p0, p1, . . . , pq\u22121), we denote by B(n, p) the set of all length-n q-ary vectors of constant composition w = (w0, . . . , wq\u22121), where wi = pin for i \u2208[q] 1. Let pj,i\u2192k be the probability that on the j-th write, a cell in state i is programmed to state k, k \u2a7ei. A family of two-write q-ary capacity-achieving EI:DU WOM codes was constructed recently by Shpilka [21]. Particularly, given \u03f5 > 0 and probability vectors p1,0, p2,0, . . . , p2,q\u22122, Shpilka [21] constructed a family of two-write q-ary EI:DU WOM codes that match these probability vectors on the \ufb01rst and second writes. We state this result formally. Lemma 7. [21] For all (j, i) \u2208{(1, 0), (2, 0), (2, 1), . . . , (2, q \u22122)}, let pj,i = (pj,i\u2192i, pj,i\u2192i+1, . . . , pj,i\u2192q\u22121) be a probability vector. Then, for all \u03f5 > 0 there exists an [n, 2; M1, M2]EI:DU,z q two-write q-ary EI:DU WOM code satisfying: \u2022 Im(Eq,1) \u2286B(n, p1,0) and R1 = log M1 n \u2a7eh(p1,0) \u2212\u03f5. \u2022 For all c1 \u2208Im(Eq,1), m2 \u2208[M2], and c2 = Eq,2(m2, c1), the following condition holds. For i \u2208[q], let ci 2 be a length-w1,i, w1,i = np1,0\u2192i, substring of c2 at all locations k with value i before the second write, that is, c1,k = i. Then, ci 2 \u2208B(w1,i, p2,i). Furthermore, R2 = log M2 n \u2a7ePq\u22122 i=0 p1,0\u2192ih(p2,i) \u2212\u03f5. We refer to the family of WOM codes from Lemma 7 as an [n, 2; M1, M2]EI:DU q (\u03f5, p1,0, p2,0, . . . , p2,q\u22122) WOM code, where M1 = 2R1n and M2 = 2R2n are determined as the maximal possible values based on \u03f5, which tends to zero, and the probability vectors pj,i. For the case q = 2, for shorthand, given p1,0\u21921 = p we denote these codes by [n, 2; M1, M2]EI:DU,z(\u03f5, p) (where p2,0\u21921 = 0.5). Furthermore, using cooling codes, the work in [1] provides the following family of binary WOM codes. Lemma 8. For all p \u2208[0, 0.5] and \u03f5 > 0, there exists a two-write binary WOM code [n, 2; M1, M2]EI:DU,z(\u03f5, p) such that M1 = P\u03c4 i=0 \u0000n i \u0001 and M2 = 2n\u2212\u03c4\u22121, where \u03c4 = pn. Therefore, for any \u03f5 > 0, there exists n such that R1 = log M1 n \u2a7eh(p) \u2212\u03f5 and R2 = log M2 n \u2a7e1 \u2212p \u2212\u03f5. We are now ready to present a construction of two-change three-write EIA:DU-ELM codes which establishes the result in Theorem 5. Construction 9. Given p1,0, p2,0, p2,1 \u2208[0, 0.5] and \u03f5 > 0, we construct an [n, 3, 2; M1, M2, M3]EIA:DU,z ELM code where Mj = 2nRj for j = 1, 2, 3 such that R1 \u2a7eh(p1,0) \u2212\u03f5, R2 \u2a7e(1 \u2212p1,0)h(p2,0) + p1,0h(p2,1) \u2212\u03f5, and R3 \u2a7e(1 \u2212p1,0p2,1) \u2212\u03f5. We use the following two WOM codes. 1) Let p1,0 = (p1,0\u21920, p1,0\u21921, p1,0\u21922) = (1 \u2212p1,0, p1,0, 0), p2,0 = (p2,0\u21920, p2,0\u21921, p2,0\u21922) = (0, p2,0, 1 \u2212p2,0), and p2,1 = (p2,1\u21921, p2,1\u21922) = (1 \u2212p2,1, p2,1). Let C1 be an [n, 2; M1, M2]EI:DU,z 3 (\u03f5, p1,0, p2,0, p2,1) two-write ternary EI:DU WOM code from Lemma 7 with two pairs of encoder/decoder (E3,1, D3,1) and (E3,2, D3,2). 2) Let \u03c11 = p1,0p2,1, and C2 be an [n, 2; M \u2032 1, M3]EI:DU,z(\u03f5, \u03c11) two-write binary EI:DU WOM code from Lemma 8 with two pairs of encoder/decoder (E2,1, D2,1) and (E2,2, D2,2). The three pairs of encoder/decoder mappings (EEIA:DU j , DEIA:DU j ) for j = 1, 2, 3 are de\ufb01ned as follows. First write: EEIA:DU 1 (m1) = E3,1(m1) for all m1 \u2208[M1]. Similarly, DEIA:DU 1 (c1) = D3,1(c1). Note that since we chose the probability to program level 2 in the \ufb01rst write of C1 to be zero, the output of the encoder E3,1 is indeed a binary vector, so EEIA:DU 1 and DEIA:DU 1 are well de\ufb01ned. Second write: The idea is to use the second write encoder E3,2 of C1 with the probability vectors p2,0 and p2,1, and notice that here we write all cells to levels 1 or 2. Then, we can view this \u201cternary word\u201d as a binary word. Let c1 = (c1,1, . . . , c1,n) \u2208Im(EEIA:DU 1 ) be the cell-state vector after the \ufb01rst write, and note that this is a binary vector. The encoder/decoder (EEIA:DU 2 , DEIA:DU 2 ) are de\ufb01ned formally as follows. For all (m2, c1) \u2208[M2] \u00d7 Im(EEIA:DU 1 ), c2 = EEIA:DU 2 (m2, c1) = c\u2032 2(mod2), 1We assume here that pi is a rational number and n is large enough such that pin is an integer for i \u2208[q]. \fwhere c\u2032 2 = E3,2(m2, c1) \u2208[3]n. Furthermore, for all c2 \u2208Im(EEIA:DU 2 ), DEIA:DU 2 (c2) = D3,2(c\u2032 2) = m2, where c\u2032 2 = 2 \u00b7 1 \u2212c2, that is, c\u2032 2,i = 1 if c2,i = 1 and c\u2032 2,i = 2 if c2,i = 0. Third write: Let c2 be the cell-state vector after the second write. We note that the encoder on the third write knows the programcount vector v2 \u2208[3]n, but the decoder does not have this information. Among the n cells, there are \u03c11n cells which have been programmed twice, where \u03c11 = p1,0p2,1, and therefore (only) these cells cannot be programmed on this write. Hence, the encoder can interpret the vector v2 as a length-n binary vector indicating for each cell whether it can be programmed on this write. We denote this vector by c\u2032\u2032 2, so c\u2032\u2032 2,i = 1 if and only if v2,i = 2. We will use the code C2 to encode and decode on this write, and we denote by \u00af c the bitwise complement of a binary vector c. Speci\ufb01cally, the encoder/decoder mappings are de\ufb01ned as follows. For all m3 \u2208[M3] and v2 \u2208N2, EEIA:DU 3 (m3, v2) = E2,2(m3, c\u2032\u2032 2). Furthermore, for all c3 \u2208Im(EEIA:DU 3 ), DEIA:DU 3 (c3) = D2,2(c3). To illustrate Construction 9, we present the following example. Example 1. Let n = 7, p1,0 = 3/7, p2,0 = 1/2, and p2,1 = 1/3, we construct a [7, 3, 2; M1, M2, M3]EIA:DU,z three-change two-write ELM code as follows. In the \ufb01rst write, we encode a message m1 to obtain a binary vector of length 7, e.g., c1 = (1, 1, 1, 0, 0, 0, 0). In the second write, to encode a message m2, in the \ufb01rst step, we use the second write encoder E3,2 of the ternary code C1 in Lemma 7 with probability p2,0 = (0, 1/2, 1/2) and p2,1 = (2/3, 1/3) to obtain c\u2032 2 = E3,2(c1, m2), e.g., c\u2032 2 = (2, 1, 1, 1, 1, 2, 2). In the second step, we can replace symbol 2 by symbol 0 in the vector c\u2032 2 to obtain the binary vector c2 = (0, 1, 1, 1, 1, 0, 0). So, c2 is the output in the second write. We observe that it is not dif\ufb01cult to decode the vector c2 to obtain the message m2. In the last write, the encoder has all information and know that the \ufb01rst cell is programmed twice and the program-count vector is v2 = (2, 1, 1, 1, 1, 0, 0). The encoder also view the vector v2 as a binary vector c\u2032\u2032 2 = (1, 0, 0, 0, 0, 0, 0) where the \ufb01rst cell (=1) is not programmable. Using the second-write encoder E2,2 of the EIU:DI WOM code C2, we can encode the message m3 to obtain c\u2032 3 = E2,2(m3, c\u2032\u2032 2), e.g., c\u2032 3 = (1, 0, 0, 0, 1, 1, 1). We now take the bitwise complement of c\u2032 3 to obtain EEIA:DU 3 (m3, v2) = c3 = (0, 1, 1, 1, 0, 0, 0). So, all the three messages are c1 = (1, 1, 1, 0, 0, 0, 0), c2 = (0, 1, 1, 1, 1, 0, 0), and c3 = (0, 1, 1, 1, 0, 0, 0). 2 We now present the proof of Theorem 5. Proof of Theorem 5: Let Rj(Ci) be the rate of the WOM code Ci on the j-th write. For any \u03f5 > 0 and p1,0, p2,0, p2,1 \u2208 [0, 0.5], we choose the codes C1 and C2 in Construction 9 to satisfy R1(C1) \u2a7eh(p1,0) \u2212\u03f5 = h(p1,0) \u2212\u03f5, where p1,0 = (1 \u2212p1,0, p1,0, 0). R2(C1) \u2a7ep1,0\u21920h(p2,0) + p1,0\u21921h(p2,1) \u2212\u03f5 \u2a7e(1 \u2212p1,0)h(p2,0) + p1,0h(p2,1) \u2212\u03f5, and R2(C2) \u2a7e1 \u2212\u03c11 \u2212\u03f5 = 1 \u2212p1,0p2,1 \u2212\u03f5. The result follows from the fact that the rate tuple of the two-change three-write ELM code is (R1(C1), R2(C1), R2(C2)). The solution for the case t = 3, \u2113= 2 is generalized for any t and \u2113in the following theorem. Theorem 10. For all t and \u2113, Ct,\u2113\u2286CEIA:DU,z t,\u2113 , that is, for any \u03f5 > 0 and a rate t-tuple (R1, . . . , Rt) \u2208Ct,\u2113, there exists a zero-error \u2113-change t-write EIA:DU ELM code C such that its rate on the j-th write is at least Rj \u2212\u03f5 for all 1 \u2a7dj \u2a7dt, that is, Rj(C) \u2a7eRj \u2212\u03f5. To prove Theorem 10, we construct a zero-error \u2113-change t-write ELM code. The idea is to generalize Construction 9. Hence, for any given j, we use q-ary EI:DU WOM code from Lemma 7 to program all cells up to the two highest levels q \u22121 and q \u22122. So, the decoder can look at q \u22121 as 0 and q \u22122 as 1 to decode the original message. We now present the construction formally as follows. Construction 11. Given pj,i \u2208[0, 0.5], for all i \u2208[\u2113+ 1], 1 \u2a7dj \u2a7dt, and \u03f5 > 0, we construct an [n, t, \u2113; M1, . . . , Mt]EIA:DU,z ELM code where M1 = \u0000n p1,0n \u0001 and Mj = 2nRj for 2 \u2a7dj \u2a7dt such that Rj \u2a7ePm\u22121 i=0 Qj\u22121,ih(pj,i) \u2212\u03f5, where Qj\u22121,i is de\ufb01ned in Equation (1). The t pairs of encoder/decoder mappings (EEIA:DU j , DEIA:DU j ) are de\ufb01ned as follows. First write: Given p1,0, we program all words of length n, weight p1,0n as on the \ufb01rst write of Construction 9. Hence, M1 = \u0000n p1,0n \u0001 and the rate on the \ufb01rst write satis\ufb01es R1 \u2a7eh(p1,0) \u2212\u03f5. \fj-th write, 2 \u2a7dj \u2a7dt: Let m = min{j, \u2113}. We denote the cell-state vector and the cell-program-count vector after the j \u22121 writes by cj\u22121 = (cj\u22121,1, . . . , cj\u22121,n) \u2208Im(EEIA:DU j\u22121 ) and vj\u22121 = (vj\u22121,1, . . . , vj\u22121,n) \u2208B(n, qj\u22121) \u2282[m]n, respectively, where qj\u22121 = (Qj\u22121,0, Qj\u22121,1, . . . , Qj\u22121,m\u22121) and Qj,i are de\ufb01ned in Equation (1). To program on the j-th write, we use the two-write (2m + 1)ary WOM code from Lemma 7, [n, 2; M1,j, M2,j]EI:DU,z 2m+1 (\u03f5, p1,0, p2,0, p2,1, . . . , p2,2m\u22121) where p1,0 = (qj\u22121, 0, 0, . . . , 0) and for all i \u2208[m \u22121], p2,i = (p2,i\u2192i, . . . , p2,i\u21922m) = (0, . . . , 0, pj,i, 1 \u2212pj,i) if i is even and p2,i = (p2,i\u2192i, . . . , p2,i\u21922m) = (0, . . . , 0, 1 \u2212pj,i, pj,i) if i is odd. As in Lemma 7, M2,j = 2Rjn where Rj \u2a7ePm\u22121 i=0 Qj\u22121,ih(p2,i) \u2212\u03f5 = Pm\u22121 i=0 Qj\u22121,ih(pj,i) \u2212\u03f5 since p2,i = (p2,i\u2192i, . . . , p2,i\u21922m) = (0, . . . , 0, 1 \u2212pj,i, pj,i) or p2,i = (p2,i\u2192i, . . . , p2,i\u21922m) = (0, . . . , 0, pj,i, 1 \u2212pj,i). Hence, on the j-th write, we choose Mj = M2,j = 2Rjn. We denote the two pairs of the encoder/decoder of the used WOM code by (Em,1, Dm,1) and (Em,2, Dm,2). The idea is to push all cells to the two highest levels and view the obtained word as a binary word. Hence, to decode correctly, the decoder only needs to know the cell-state vector after the j-th write which is a binary word. We now de\ufb01ne the encoder/decoder (EEIA:DU j , DEIA:DU j ) formally as follows. For all each mj \u2208[Mj] and vj\u22121 \u2208Im(Em,1) cj = EEIA:DU j (mj, vj\u22121) = c\u2032 j(mod2), where c\u2032 j = Em,2(mj, vj\u22121) \u2208[2m + 1]n. Furthermore, for all cj \u2208Im(EEIA:DU j ), DEIA:DU j (cj) = Dm,2(c\u2032 j) = mj, where c\u2032 j,i = 2m \u22121 if cj,i = 1 and c\u2032 j,i = 2m if cj,i = 0. Proof of Theorem 10: Given all parameters as in Construction 11, the rate of this ELM code on the \ufb01rst write is R1 \u2a7e h(p1,0) \u2212\u03f5. Now, we consider the j-th write. Since we used the WOM code in Lemma 7 to program the j-th write of the ELM code, the rate on this write is exactly the rate on the second write of the used WOM code. Hence, the rate in the j-th write of the ELM code is Rj \u2a7ePm\u22121 i=0 Qj\u22121,ih(pj,i) \u2212\u03f5. Remark 1. In this section, we provide an explicit construction of zero-error two-change three-write EIA:DU ELM code and generalize the result to construct a zero-error \u2113-change t-write EIA:DU ELM code. Since Shpilka [21] provided a pair of polynomial time encoding/decoding algorithms of a family of two-write WOM codes, the encoder and decoder in Theorem 10 also run in polynomial time. As shown in Theorem 10, using these constructions, we can achieve any rate in the capacity region and thus achieve the maximum sum-rate when the length n tends to in\ufb01nity. However, for a \ufb01xed value of n, we can only achieve a high sum-rate but can not achieve the maximum sum-rate. Furthermore, Shpilka\u2019s technique only works for large block length [21]. Hence, for small value of block length n, we need other constructions to obtain a high sum-rate, for example, Construction 19 that will be presented later. B. The EIA:DIA Model Converse Part of Theorem 2 In this section, we prove the converse part of Theorem 2 for the EIA:DIA model \u03f5-error case. That is, we prove that CEIA:DIA,\u03f5 t,\u2113 \u2286 Ct,\u2113. For this direction we need to prove that if there exists an [n, t, \u2113; M1, . . . , Mt]EIA:DIA,pe ELM code where pe = (pe1, . . . , pet), then \u0012log M1 n \u2212\u03f51, log M2 n \u2212\u03f52, . . . , log Mt n \u2212\u03f5t \u0013 \u2208Ct,\u2113, where (\u03f51, \u03f52, . . . , \u03f5t) tends to 0 if pe tends to 0 and n tends to in\ufb01nity. In our proof \u03f5j = H(pej )+pej log(Mj) n , and therefore \u03f5j \u21920 when pej \u21920 and n \u2192\u221e. Let Xj be a length-n binary vector where Xj,k = 1 if and only if the k-th cell is intended to be programmed on the j-th write. Similarly, Yj, is a length-n binary vector, where Yj,k = 1 if and only if the value of the k-th cell was successfully changed on the j-th write, that is, Yj = cj \u2295cj\u22121. Note that the encoder knows the number of times each cell was programmed. Therefore, we can assume that a cell is not intended to be programmed more than \u2113times. Furthermore, the decoder also knows the number of times each cell was programmed. Thus we assume that Xj = Yj where Xj is the encoded word and Yj is the input of the decoder. Let S1, . . . , St be independent random variables, where Sj is uniformly distributed over the messages set [Mj], and \u02c6 Sj is the decoding result on the j-th write. Let Vj be an independent random variable on Nj, the set of all cell-programs-count vectors after the \ufb01rst j writes. The data processing yields the following Markov chain: Sj|Vj\u22121 \u2014 Xj|Vj\u22121 \u2014 Yj|Vj\u22121 \u2014 \u02c6 Sj|Vj\u22121 and therefore, I(Xj; Yj|Vj\u22121) \u2a7eI(Sj; \u02c6 Sj|Vj\u22121). Additionally, I(Sj; \u02c6 Sj|Vj\u22121) = H(Sj|Vj\u22121) \u2212H(Si| \u02c6 Sj, Vj\u22121) \u2a7eH(Sj) \u2212H(Sj| \u02c6 Sj) \u2a7elog(Mj) \u2212H(pej) \u2212pej log(Mj). \fThe \ufb01rst inequality follows from the independence of Vj\u22121 and Sj which implies that H(Sj|Vj\u22121) = H(Sj), and from the fact that conditioning does not increase the entropy. The second inequality follows from Fano\u2019s inequality [7, p. 38] H(Sj| \u02c6 Sj) \u2a7d H(pej) + pej log(Mj). Let L be an index random variable, which is uniformly distributed over the index set [n]. Since L is independent of all other random variables we get 1 nI(Xj; Yj|Vj\u22121) \u2a7d1 nH(Yj|Vj\u22121) (a) \u2a7d1 n Pn\u22121 k=0 H(Yj,k|Vj\u22121,k) (b) = H(Yj,L|Vj\u22121,L, L) (c) \u2a7dH(Yj,L|Vj\u22121,L) = P\u2113 i=0 Pr(Vj\u22121,L= i)H(Yj,L|Vj\u22121,L = i) (d) = P\u2113\u22121 i=0 Pr(Vj\u22121,L= i)H(Yj,L|Vj\u22121,L = i), where steps (a) and (c) follow from the fact that entropy of a vector is not greater than the sum of the entropies of its components, and conditioning does not increase the entropy. Step (b) follows from the fact that H(Yj,L|Vj\u22121,L, L) = Pn\u22121 k=0 Pr(L = k)H(Yj,k|Vj\u22121,L, L = k) = 1 n Pn\u22121 k=0 H(Yj,k|Vj\u22121,k), and step (d) follows from H(Yj,L|Vj\u22121,L = \u2113) = 0. Now, we set pj,i = Pr(Xj,L = 1|Nj\u22121,L = i) = Pr(Yj,L = 1|Nj\u22121,L = i), and thus we can conclude that Qj,i = Pr(Nj,L = i) where Qj,i is calculated in Equation (1), and then log(Mj) n \u2212\u03f5j \u2a7d1 nI(Xj; Yj|Nj\u22121) \u2a7dP\u2113\u22121 i=0 Pr(Nj\u22121,L = i)H(Yj,L|Nj\u22121,L = i) = P\u2113\u22121 i=0 Qj\u22121,ih (pj,i) , where \u03f5j = H(pej )+pej log(Mj) n , and the converse part is implied. By Theorem 10 in Subsection III-A and by the proof of the converse part in Subsection III-B we completed the proof of Theorem 2. Furthermore by Theorem 4 we conclude the following corollary. Corollary 12. For all t and \u2113, Ct,\u2113= b Ct,\u2113is the capacity region for all the EIA models for both the zero-error and the \u03f5-error cases and is denoted by CEIA t,\u2113 . The maximum sum-rate of all the EIA models is REIA t,\u2113 = log P\u2113 i=0 \u0000t i \u0001 . IV. THE CAPACITY OF THE EIP:DIA MODEL In this section we discuss the capacity region and the maximum sum-rate of the EIP:DIA model. Recall that if \u2113= 1 then by de\ufb01nition, EIP is equivalent to EIA and this model is equivalent to the known WOM model. Thus, in this section we assume that \u2113> 1. We focus on the \u03f5-error case and present the capacity region of this model. The zero-error case is harder to solve, and is left for future research. However the \u03f5-error case provides an upper bound for the zero-error case. Note that the EU:DI WOM model is simpler than the EIP:DIA ELM model, and even though its exact capacity for the zero-error case is still not known for general t. As done in the EIA models, let us denote by cj, j \u2208[t + 1], the length-n binary vector which represents the memory state after the j-th write, where c0 = 0. For 1 \u2a7dj \u2a7dt and i \u2208[\u2113+ 1], we de\ufb01ne the probabilities pj,0, pj,1, and Qj,i as follows. pj,k is the probability of programming a cell on the j-th write given that the value of this cell was k, k \u2208{0, 1}, and Qj,i is the probability of a cell to be programmed exactly i times after the \ufb01rst j writes. Additionally, let Qj,e, Qj,o be the probability of a cell to be programmed an even, odd number of times after the \ufb01rst j writes, respectively. Formally, Qj,i is de\ufb01ned recursively by using the probabilities pj\u2032,0 and pj\u2032,1 for j\u2032 \u2a7dj. We now assume that \u2113is even. The case of an odd \u2113is de\ufb01ned similarly. We de\ufb01ne Qj,i for j > 0 as follows. For even i \u2a7e0, Qj,i = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Qj\u22121,i\u22121pj,1 + Qj\u22121,i(1 \u2212pj,0), if 0 < i < \u2113, Qj\u22121,i\u22121pj,1 + Qj\u22121,i, if i = \u2113, Qj\u22121,i(1 \u2212pj,0), if i = 0, (4) and for odd i > 0, Qj,i = Qj\u22121,i\u22121pj,0 + Qj\u22121,i(1 \u2212pj,1). The base j = 0, is Q0,0 = 1 and Q0,i = 0 for i > 0. Furthermore, let Qj,e = P\u2113/2 i=0 Qj,2i and Qj,o = P\u2113/2 i=1 Qj,2i\u22121. \fNext, we de\ufb01ne the rates region e Ct,\u2113which will be proved to be the capacity region of the EIP:DIA model for the \u03f5-error case. We present here the de\ufb01nition for even \u2113, while the odd case is de\ufb01ned similarly. e Ct,\u2113= n (R1, R2, . . . , Rt)|\u22001 \u2a7dj \u2a7dt : Rj \u2a7dQj\u22121,oh(pj,1) + (Qj\u22121,e \u2212Qj\u22121,\u2113)h(pj,0), pj,0,pj,1\u2208[0, 0.5] and Qj,e, Qj,o, Qj,\u2113are de\ufb01ned above o . (5) For example, for t = 3, \u2113= 2, we have that e C3,2 = C3,2= n (R1, R2, R3)|R1 \u2a7dh(p1,0), R2\u2a7d1\u2212p1,0+p1,0h(p2,1), R3\u2a7d1\u2212p1,0p2,1, and p1,0, p2,1\u2208[0, 0.5] o , which is achieved by substituting p3,0 = p3,1 = p2,0 = 0.5 in Equations (2) and (5). Using the region e Ct,\u2113, the next theorem characterizes the capacity region of the EIP models for the \u03f5-error case. Theorem 13. The rates region e Ct,\u2113is the capacity region of t-write \u2113-change ELM EIP:DIA model for the \u03f5-error case. That is, e Ct,\u2113= CEIP :DIA,\u03f5 t,\u2113 . Proof: To show the achievable region, we should prove that for each \u03f5 > 0 and (R1, R2, . . . , Rt) \u2208e Ct,\u2113, there exists an [n, t; M1, . . . , Mt]EIP :DIA,pe t,\u2113 ELM code, where for all 1 \u2a7dj \u2a7dt, log Mj n \u2a7eRj \u2212\u03f5 and pe = (pe1, . . . , pet) \u2a7d(\u03f5, . . . , \u03f5). We use the well-known random channel-coding theorem [7, p. 200] on each write. We describe the encoding and decoding on each write. The j-th write presents a DMC with the input length-n binary vector Xj and the output is (Zj\u22121, Yj), where Zj\u22121 \u2208[\u2113+ 1]n represents the times each cell was programmed before the j-th write, and Yj \u2208[2]n represent the state of the memory after the j-th write. Let xj = Xj,k, zj = Zj\u22121,k, and yj = Yj,k for some index k. By the random coding theorem, for n large enough, the following region is achievable n (R1, . . . , Rt)| \u22001 \u2a7dj \u2a7dt, Rj \u2a7dI(xj; yj) o . By the de\ufb01nitions and notations of the probabilities pj\u2032,i\u2032 and Qj\u2032,i\u2032, I(xj; (zj\u22121, yj))=H(zj\u22121, yj) \u2212H(zj\u22121, yj|xj) = H(zj\u22121) + H(yj|zj\u22121) \u2212H(zj\u22121, yj|xj) (a) = H(zj\u22121) + H(yj|zj\u22121) \u2212H(zj\u22121) = H(yj|zj\u22121) = \u2113 X i=0 Pr(zj\u22121 = i)H(yj|zj\u22121 = i) (b) = \u2113\u22121 X i=0 Pr(zj\u22121 = i)H(yj|zj\u22121 = i) = \u2113/2 X i=1 (Qj\u22121,2i\u22121h (pj,1) + Qj\u22121,2i\u22122h (pj,0)) = Qj\u22121,oh(pj,1) + (Qj\u22121,e \u2212Qj\u22121,\u2113)h(pj,0). Step (a) follows from H((zj\u22121, yj)|xi) = H(zj\u22121|xj) since yj is a function of xj, zj\u22121, and H(zj\u22121|xj) = H(zj\u22121) because zj\u22121 is independent on xj. Step (b) is implied by H(yj|zj\u22121 = \u2113) = 0. Hence, we can achieve the region e Ct,\u2113for the \u2113-change t-write W\u2113M EIP:DIA model for the \u03f5-error case. The proof of the converse part, CEIP :DIA,\u03f5 t,\u2113 \u2286e Ct,\u2113, is similar to the proof of this part in Theorem 2, and hence is deferred to Appendix A. We could also present a family of capacity achieving codes using the binary erasure channel (BEC). Note that on the j-th write, both the encoder and the decoder know cj\u22121, the state of the memory before writing the new data, while the decoder also knows vj\u22121, the number of times each cell was programmed before the j-th write. Therefore, the encoder on the j-th write treats the one and the zero cells separately. On the cells with value one, the encoder writes zero with probability pj,1 (for example by using a constant weight code), while for the zero cells, the decoder knows which cells have been already programmed \u2113times before the j write. Thus, the encoding on the zero cells can be represented as encoding over BEC with erasure probability Q\u2113/Qe. The capacity of the BEC with erasure probability \u03c0 and probability \u03b1 for occurrence one in the encoded vector is (1 \u2212\u03c0)h(\u03b1) [7, p. 188]. By substituting pj,0 = \u03b1 and \u03c0 = Q\u2113/Qe, we get the rate on the j-th write Qj\u22121,oh(pj,1) + (Qj\u22121,e \u2212Qj\u22121,\u2113)h(pj,0). The following theorem is an immediate result deduced by the de\ufb01nitions of e Ct,\u2113and Ct,\u2113and Theorems 2 and 13. \fTheorem 14. For \u2113= 2 the capacity region of the EIP:DIA model for the epsilon error case is equal to the capacity region for the EIA models, i.e., CEIP :DIA,\u03f5 t,2 = CEIA t,2 . In Section V, we compare between the EIP:DIA model which was discussed in this section, and the EIA models, which were presented in Section III. V. A COMPARISON BETWEEN THE EIA MODELS AND THE EIP:DIA MODEL In this section we compare between the EIA models and the EIP:DIA model. The capacity of the EIA models, CEIA:DY,g t,\u2113 for g \u2208{z, \u03f5} and Y \u2208{IA, IP, U}, was stated in Section III to be equal to Ct,\u2113, while in Section IV we presented the capacity region of the EIP:DIA model for the \u03f5-error case, e Ct,\u2113= CEIP :DIA,\u03f5 t,\u2113 . The next theorem proves that for t > \u2113\u2a7e3 the maximum sum-rate of the EIP:DIA model for the epsilon-error case is smaller than the maximum sum-rate of the EIA models. Hence, the capacity region CEIP :DIA,\u03f5 t,\u2113 is a proper subset of the capacity region CEIA t,\u2113 for these parameters. Recall that for \u2113= 2 these regions were shown to be the same in Theorem 14, and therefore the maximum sum-rates of these models for \u2113= 2 are the same too. Theorem 15. For t > \u2113\u2a7e3, REIP :DIA,\u03f5 t,\u2113 < REIA t,\u2113 , and hence CEIP :DIA,\u03f5 t,\u2113 \u228aCEIA t,\u2113 . Proof: Let e R = ( e R1, e R2, . . . , e Rt) be a rate tuple which achieves the maximum sum-rate REIP :DIA,\u03f5 t,\u2113 , and we denote by e pj,0, e pj,1, and e Qj,i, 1 \u2a7dj \u2a7dt and i \u2208[\u2113+ 1], the probabilities which attain e R in e Ct,\u2113. Now we present a rate tuple R = (R1, R2, . . . , Rt) \u2208Ct,\u2113> e R. Then, we conclude that R \u2208CEIA t,\u2113 \\CEIP :DIA,\u03f5 t,\u2113 , which implies that REIP :DIA,\u03f5 t,\u2113 < REIA t,\u2113 and CEIP :DIA,\u03f5 t,\u2113 \u228aCEIA t,\u2113 . We assume now that \u2113is even, while the proof for the odd case is similar. Since e R is maximal rate tuple we have e pt\u22121,0 = e pt,0 = e pt,1 = 0.5. For all j and i, 1 \u2a7dj \u2a7dt \u22122 and i \u2208[\u2113], we de\ufb01ne pj,i = e pj,i\u2032 where i\u2032 = i mod 2. In addition, for all i \u2208[\u2113\u22121], pt\u22121,i = 0.5, pt\u22121,\u2113\u22121 = e pt\u22121,1, and for all i, pt,i = 0.5. Thus, for all j and i, 1 \u2a7dj \u2a7dt \u22122 and i \u2208[\u2113], Rj = e Rj and Qj,i = e Qj,i. For the (t \u22121)-th write we have, Rt\u22121 = 1 \u2212e Qt\u22122,\u2113\u22121 \u2212e Qt\u22122,\u2113+ e Qt\u22122,\u2113\u22121h(e pt\u22121,1) while e Rt\u22121 = e Qt\u22122,oh(e pt\u22121,1) + ( e Qt\u22122,e \u2212e Qt\u22122,\u2113), and for the last write Rt = e Rt = 1 \u2212e Qt\u22121,\u2113. Now we prove that e pt\u22121,1 < 0.5 which immediately implies that Rt\u22121 > e Rt\u22121 and thus completes the proof. Recall that e Rt = 1 \u2212e Qt\u22121,\u2113= 1 \u2212e Qt\u22122,\u2113\u2212e Qt\u22122,\u2113\u22121e pt\u22121,1. Thus, given the probabilities for the \ufb01rst t \u22122 writes, in order to achieve the maximal rate tuple e R, we have to maximize e Rt\u22121 + e Rt. That is, we choose e pt\u22121,1 which maximizes e Qt\u22122,oh(e pt\u22121,1) \u2212e Qt\u22122,\u2113\u22121e pt\u22121,1. The derivative is e Qt\u22122,0 log( 1\u2212e pt\u22121,1 e pt\u22121,1 ) \u2212e Qt\u22122,\u2113\u22121, and the maximum is obtained for e pt\u22121,1 = 1/(1 + 2 e Qt\u22122,\u2113\u22121/ e Qt\u22122,o). Since e R is maximal and t > \u2113\u2a7e3, we have e Qt\u22122,\u2113\u22121 > 0, and therefore e pt\u22121,1 \u0338= 0.5. We can summarize the results regrading the capacity region of the EIP:DIA model in the following corollary. Corollary 16 For all t > \u2113the following holds CEIP :DIA,\u03f5 t,\u2113 = e Ct,\u2113\u2286Ct,\u2113= CEIA t,\u2113 . Furthermore, \u2022 For t > \u2113= 2 all these regions are equal, in particular, CEIP :DIA,\u03f5 t,2 = CEIA t,2 . \u2022 For t > \u2113\u2a7e3, CEIP :DIA,\u03f5 t,\u2113 \u228aCEIA t,\u2113 and REIP :DIA,\u03f5 t,\u2113 < REIA t,\u2113 . VI. THE CAPACITY OF EU:DIA MODEL In this section we study the EU:DIA model for the \u03f5-error case, and provide the capacity region of this model. As in the EIP:DIA model, the capacity region for the zero-error case and the exact maximum sum-rate are left for future research. For 1 \u2a7dj \u2a7dt and i \u2208[\u2113+1], let pj be the probability of programming a cell on the j-th write, and Qj,i denotes the probability of a cell to be programmed exactly i times on the \ufb01rst j writes. Additionally, let Qj,e, Qj,o be the probabilities of a cell to be programmed even, odd number of times on the \ufb01rst j writes, respectively. Formally, Qj,i is de\ufb01ned recursively by using pj\u2032 probabilities for j\u2032 \u2a7dj. For j \u2a7e1, Qj,i = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Qj\u22121,i\u22121pj + Qj\u22121,i(1 \u2212pj), if 0 < i \u2a7d\u2113, Qj\u22121,i\u22121pj + Qj\u22121,i, if i = \u2113, Qj\u22121,i(1 \u2212pj), if i = 0, (6) where Q0,0 = 1 and Q0,i = 0 for i > 0. Then, we de\ufb01ne the region Ct,\u2113which is proved later in this section to be the capacity region CEU:DIA,\u03f5 t,\u2113 . Ct,\u2113= n (R1, R2, . . . , Rt)|\u22001 \u2a7dj \u2a7dt : Rj \u2a7dh(pj) \u2212Qj\u22121,\u2113h(pj), pj \u2208[0, 0.5], Qj,\u2113is de\ufb01ned above o . (7) \fThe next theorems establish the capacity region of the EU:DIA model for the \u03f5-error case and compare between this model and the EIP:DIA model. The techniques applied for the EU:DIA model are very similar to the proofs in Section IV. The proofs of Theorems 17 and 18 are similar to the proofs of Theorems 13 and 15, respectively. Therefore, these proofs are moved to Appendix A. Theorem 17. The rates region Ct,\u2113is the capacity region of t-write \u2113-change ELM EU:DIA model for the \u03f5-error case. That is, Ct,\u2113= CEU:DIA,\u03f5 t,\u2113 . Theorem 18. For t > \u2113\u2a7e2, REU:DIA,\u03f5 t,\u2113 < REIP :DIA,\u03f5 t,\u2113 , and hence CEU:DIA,\u03f5 t,\u2113 \u228aCEIP :DIA,\u03f5 t,\u2113 . VII. THE EIP:DU MODEL In this section, we study the EIP:DU model and its sum-rate. First, we note that CEIP :DU,\u03f5 t,\u2113 \u2286CEIP :DIA,\u03f5 t,\u2113 for all t, \u2113, and thus, REIP :DU,\u03f5 t,\u2113 \u2a7dREIP :DIA,\u03f5 t,\u2113 \u2a7dlog \u2113 X i=0 \u0012t i \u0013 . (8) That is, we obtain an upper bound of the maximum sum-rate REIP :DU,\u03f5 t,\u2113 . Note that for t > \u2113\u2a7e3 this upper bound is not tight (Theorem 15). We are now interested in some good lower bounds for the maximum sum-rate. Our goal is to provide several constructions with high sum-rate. We \ufb01rst present a general construction for the zero-error case and then show how to obtain higher sum-rate for the \u03f5-error case with t = 3, \u2113= 2. The following construction provides a family of \u2113-change t-write EIP:DU ELM codes for the zero-error case. Construction 19. Let (k1, . . . , k\u2113) be such that 1 \u2a7dki \u2a7dt for 1 \u2a7di \u2a7d\u2113and P\u2113 i=1 ki = t. Let [n, ki; Mji+1, . . . , Mji+ki]EI:DU,z be a binary ki-write EI:DU WOM code for 1 \u2a7di \u2a7d\u2113with sum-rate Ri where j1 = 0 and ji = Pi\u22121 r=1 kr. Each of which consists of n bits and ki pairs of encoding and decoding maps (EEI:DU ji+h , DEI:DU ji+h ) for 1 \u2a7dh \u2a7dki. We de\ufb01ne an [n, t, \u2113; M1, . . . , Mt]EIP :DU,z \u2113-change t-write ELM code consists of n bits and t pairs of encoders and decoders (EEIP :DU j , DEIP :DU j ) where EEIP :DU j = EEI:DU j and DEIP :DU j = DEI:DU j for 1 \u2a7dj \u2a7dt. The maximum sum-rate of the ELM codes from Construction 19 is Rsum \u2a7eP\u2113 i=1 log(ki + 1) \u2212\u03f5 since for 1 \u2a7di \u2a7d\u2113, Ri \u2a7e log(ki + 1) \u2212\u03f5/\u2113and Rsum = P\u2113 i=1 Ri. Hence, in order to maximize the sum-rate, our goal is to maximize the value of P\u2113 i=1 log(ki + 1) given that P\u2113 i=1 ki = t. Assume that t = k\u2113+ r, r \u2208[\u2113], then this maximum value will be achieved when choosing k1 = \u00b7 \u00b7 \u00b7 = kr = k + 1 and kr+1 = \u00b7 \u00b7 \u00b7 = k\u2113= k. The next corollary summarizes this result. Corollary 20. For all t and \u2113, where t = k\u2113+ r, r \u2208[\u2113], REIP :DU,z t,\u2113 \u2a7er log(k + 2) + (\u2113\u2212r) log(k + 1) =\u2113log \u0012\u0016t \u2113 \u0017 +1 \u0013 +(t mod \u2113) log   1+ 1 \u0004 t \u2113 \u0005 + 1 ! . Proof: We choose (k1, . . . , k\u2113) such that k1 = \u00b7 \u00b7 \u00b7 = kr = k + 1 and kr+1 = \u00b7 \u00b7 \u00b7 = k\u2113= k and thus P\u2113 i=1 ki = t. We note that k = \u0004 t \u2113 \u0005 and r = t mod \u2113. Since we presented in Construction 19 an [n, t, \u2113; M1, . . . , Mt]EIP :DU,z \u2113-change t-write ELM code with sum-rate Rsum = P\u2113 i=1 Ri \u2a7er log(k + 2) + (\u2113\u2212r) log(k + 1) \u2212\u03f5 for any \u03f5 > 0, we obtain the result in Corrollary 20. From the above corollary, we have a lower bound of the maximum sum-rate of the EIP:DU model. Recall that REIP :DU,z t,\u2113 \u2a7d REIA t,\u2113 = log P\u2113 i=0 \u0000t i \u0001 , that is the exact maximum sum-rate of the EIA model is an upper bound of the maximum sum-rate of the EIP:DU model. Hence, we obtain a lower bound and an upper bound of the maximum sum-rate of the EIP:DU ELM model. We note that when t \u2a7d\u2113, we always achieve the full capacity, that is, the maximum sum-rate is t. When t > \u2113, the maximum sum-rate is dif\ufb01cult to compute exactly and there is a gap between the above lower and upper bounds. We illustrate the results for \u2113= 2, t \u2208[3, 25] in the following \ufb01gure. The following result shows that for \u2113= 2 the sum-rate of the ELM code from Construction 19 is already close to the upper bound when t is large and n \u2192\u221e. Proposition 21. For \u2113= 2 and t \u2a7e3, REIP :DU,z t,2 \u2a7eREIA t,2 \u22121. Proof: Recall that REIA t,2 = log P2 i=0 \u0000t i \u0001 = log t2+t+2 2 . When t is even, there exists a positive integer t1 such that t = 2t1. In this case, REIP :DU,z t,2 \u2a7e2 log(t1 + 1) = log(t2 1 + 2t1 + 1) \fFig. 1: The upper and lower bounds of the maximum sum-rates of the EIP:DU ELM codes when \u2113= 2, t \u2208[3, 25]. and, REIA t,2 = log 4t2 1 + 2t1 + 1 2 . Hence, REIA t,2 \u2212REIP :DU,z t,2 \u2a7dlog 4t2 1 + 2t1 + 1 2(t2 1 + 2t1 + 1) \u2a7dlog 2 = 1. When t is odd, there exists a positive integer t2 such that t = 2t2 + 1. In this case, REIP :DU,z t,2 \u2a7elog(t2 + 1) + log(t2 + 2) = log(t2 2 + 3t2 + 2) and, REIA t,2 = log 4t2 2 + 6t2 + 4 2 . Hence, REIA t,2 \u2212REIP :DU,z t,2 \u2a7dlog 4t2 2 + 6t2 + 4 2(t2 2 + 3t2 + 2) \u2a7dlog 2 = 1. In conclusion, the proposition is proven. We note that when t = 3 and \u2113= 2, the maximum achievable sum-rate of the codes in Construction 19 is log 6 \u22482.585, while the upper bound is log 7 \u22482.807. Lastly, we show how to improve this result for the \u03f5-error case. The main ideas of the following construction are as follows. On the \ufb01rst two writes, we follow exactly the \ufb01rst two writes of Construction 9 which is a construction for a two-change three-write EIA:DU ELM code. After the second write, there are \u03c11n cells which were programmed twice, where \u03c11 = p1,0p2,1. However, while the encoder in the EIA:DU model knows these positions, the encoder in the third write in the EIP:DU model does not know these positions. In order to overcome this dif\ufb01culty, we use the following family of binary EU:DU WOM codes. De\ufb01nition 22. An [n, 2; M1, M2] EU:DU,(pe1,pe2) 2 (p1, p2) two-write binary EU:DU WOM code is a coding scheme comprising of n bits. It consists of two pairs of encoding and decoding maps (EEU:DU j , DEU:DU j ) for j = 1, 2. For the map EEU:DU j , Im(EEU:DU j ) is its image and Im\u2217(EEU:DU j ) is the set of all the cell-state vectors which can be obtained after the j-th write. We note that Im(EEU:DU 0 ) = Im\u2217(EEU:DU 0 ) = {(0, . . . , 0)} and Im\u2217(EEU:DU 2 ) = {max{c1, c2} where ci \u2208Im(EEU:DU i ) : i = 1, 2}. The encoding and decoding maps are de\ufb01ned as follows. For j = 1, 2, EEU:DU j : [Mj] \u2192B(n, (1 \u2212pj, pj)) and DEU:DU j : Im\u2217(EEU:DU j ) \u2192[Mj] such that for all m \u2208[Mj], X (m,c)\u2208[Mj]\u00d7Im\u2217(EEU:DU j\u22121 ) Pr(m)Pr(c)Im \u0000DEU:DU j ( max{c, EEU:DI j (m)}) \u0001 \u2a7dpei. Two-write binary EU:DU WOM codes have been studied for a long time [23]. Recently, in [11] several constructions of EU:DU WOM codes were presented. Assume that there exists a capacity achieving code for the Z channel, then the following result for EU:DU WOM codes can be received based upon the constructions from [11]. Lemma 23. [11] For all 0 \u2a7dp1, p2 \u2a7d0.5 and \u03f5 > 0 there exists an [n, 2; M1, M2]EU:DU,(0,\u03f5) two-write binary EU:DU WOM code satisfying: \u2022 c1 \u2208B(n, (1 \u2212p1, p1)), and R1 = log M1 n \u2a7eh(p1) \u2212\u03f5. \f\u2022 c2 \u2208B(n, (1 \u2212p2, p2)), and R2 = log M2 n \u2a7eh(p1p2) \u2212p2h(p1) \u2212\u03f5, where ci \u2208Im(EEU:DU i ) for i = 1, 2. We refer to the family of WOM codes from Lemma 23 as an [n, 2; M1, M2]EU:DU,(0,\u03f5) q (\u03f5, p1, p2) WOM code, where M1 = 2R1n and M2 = 2R2n are determined as the maximal possible values based on \u03f5, which tends to zero, and the probabilities p1 and p2. We are now ready to present a construction of two-change three-write EIP:DU ELM code. Construction 24. Given p1,0, p2,0, p2,1, p3 \u2208[0, 0.5], we use the following two codes: \u2022 An [n, 3, 2; M1, M2, M \u2032 3]EIA:DU,z code from Construction 9 with the \ufb01rst two pairs of encoder/decoder (EEIA:DU i , DEIA:DU i ) for i = 1, 2. \u2022 An [n, 2; M \u2032 1, M3]EU:DU,(0,\u03f5))(\u03f5, \u03c11, p3) two-write binary EU:DU WOM code from Lemma 23, \u03c11 = p1,0p2,1, with the pair of encoder/decoder in the second write (EEU:DU 2 , DEU:DU 2 ). We construct an [n, 3, 2; M1, M2, M3]EIP :DU,(0,0,\u03f5) two-change three-write EIP:DU ELM code where its 3 pairs of encoding/decoding maps (EEIP :DU j , DEIP :DU j ) for j = 1, 2, 3 are de\ufb01ned as follows. (1) For i = 1, 2, EEIP :DU i = EEIA:DU i and DEIP :DU i = DEIA:DU i . That is, the \ufb01rst two writes of this EIP:DU ELM code are exactly the same as the \ufb01rst two writes of the EIA:DU ELM code from Construction 9. (2) After the \ufb01rst two writes, we note that \u03c11n cells are already programmed twice, and thus can not be programmed this time. Hence, we use the pair of encoder/decoder (EEU:DU 2 , DEU:DU 2 ) to encode/decode information. The pair of encoder/decoder in the third write is de\ufb01ned formally as follows: EEIP :DU 3 : [M3] \u00d7 Im\u2217(EEIP :DU 2 ) \u2192[2]n such that for all m3 \u2208[M3] and c2 \u2208Im\u2217(EEIP :DU 2 ), EEIP :DU 3 (m3, c2) = EEU:DU 2 (m3). Furthermore, DEIP :DU 3 : Im\u2217(EEIP :DU 3 ) \u2192[M3] such that for all c\u2217 3 \u2208Im\u2217(EEU:DU 3 ), DEIP :DU 3 (c\u2217 3) = DEU:DU 2 (c\u2217 3) = m3. On the \ufb01rst two writes, it is clear that R1 \u2a7eh(p1,0) \u2212\u03f5 and R2 \u2a7e(1 \u2212p1,0)h(p2,0) + p1,0h(p2,1) \u2212\u03f5. In the third write, R3 \u2a7e h(p1,0p2,1p3) \u2212p3h(p1,0p2,1) \u2212\u03f5. In conclusion, we constructed a two-change three-write EIP:DU ELM code satisfying R1 \u2a7eh(p1,0)\u2212\u03f5, R2 \u2a7e(1\u2212p1,0)h(p2,0)+ p1,0h(p2,1) \u2212\u03f5 and R3 \u2a7eh(p1,0p2,1p3) \u2212p3h(p1,0p2,1) \u2212\u03f5 for all \u03f5 > 0. Therefore, the following region is achievable for the \u03f5-error case: CEIP :DU 3,2 = {(R1, R2, R3) : R1 \u2a7dh(p1,0), R2 \u2a7d(1 \u2212p1,0)h(p2,0) + p1,0h(p2,1), R3 \u2a7dh(p1,0p2,1p3) \u2212p3h(p1,0p2,1), p1,0, p2,0, p2,1, p3 \u2208[0, 1]}. The sum-rate of the above code is Rsum = R1 + R2 + R3 \u2a7eh(p1,0) + (1 \u2212p1,0)h(p2,0) + p1,0h(p2,1) + h(p1,0p2,1p3) \u2212 p3h(p1,0p2,1) \u2212\u03f5 for any \u03f5 > 0. By choosing p1,0 = 3/7, p2,0 = 1/2, p2,1 = 2/3, and p3 = 1/2, we obtain the sum-rate Rsum = R1 + R2 + R3 \u22482.64. Remark 2. In this section, we construct a family of zero-error \u2113-change t-write EIP:DU ELM codes for any \u2113and t. Using some ef\ufb01cient encoding/decoding algorithms of the well-known binary t-write EI:DU WOM codes, we can encode/decode our EIP:DU ELM codes ef\ufb01ciently in polynomial time. When n tends to in\ufb01nity, we can obtain some codes with high sum-rate and thus get a lower bound on the maximal sum-rate of the EIP:DU model. We note that the lower bound is not tight even though it is close to the upper bound. We actually improve the lower bound for the \u03f5-error case when \u2113= 2 and t = 3 in Construction 24. Using some known polynomial time encoding/decoding algorithms of a two-write EU:DU WOM code in Lemma 23 [11], the encoding and decoding algorithms in Construction 24 also run in polynomial time. Since the exact capacity region and the maximum sum-rate of the EIP:DU model are not known yet, we expect to have better constructions in near future. VIII."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2007.11626v1",
+                    "title": "Explicit Baranyai Partitions for Quadruples, Part I: Quadrupling Constructions",
+                    "abstract": "It is well known that, whenever $k$ divides $n$, the complete $k$-uniform\nhypergraph on $n$ vertices can be partitioned into disjoint perfect matchings.\nEquivalently, the set of $k$-subsets of an $n$-set can be partitioned into\nparallel classes so that each parallel class is a partition of the $n$-set.\nThis result is known as Baranyai's theorem, which guarantees the existence of\n\\emph{Baranyai partitions}. Unfortunately, the proof of Baranyai's theorem uses\nnetwork flow arguments, making this result non-explicit. In particular, there\nis no known method to produce Baranyai partitions in time and space that scale\nlinearly with the number of hyperedges in the hypergraph. It is desirable for\ncertain applications to have an explicit construction that generates Baranyai\npartitions in linear time. Such an efficient construction is known for $k=2$\nand $k=3$. In this paper, we present an explicit recursive quadrupling\nconstruction for $k=4$ and $n=4t$, where $t \\equiv 0,3,4,6,8,9\n~(\\text{mod}~12)$. In a follow-up paper (Part II), the other values of~$t$,\nnamely $t \\equiv 1,2,5,7,10,11 ~(\\text{mod}~12)$, will be considered.",
+                    "authors": "Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Alexander Vardy, Chengmin Wang",
+                    "published": "2020-07-22",
+                    "updated": "2020-07-22",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "Introduction A hypergraph is a pair G = (V, E), where V is a set of elements, called vertices, and E is a set of nonempty subsets of V, called hyperedges or edges. The size of the vertex set is called the order of the hypergraph. A k-uniform hypergraph is a hypergraph such that all edges in E have size k. A 2-uniform hypergraph is also called a graph. Let (V k) denote the set of all k-subsets of V. A hypergraph (V, (V k)) is called a complete k-uniform hypergraph. A one-factor or a parallel class in a k-uniform hypergraph is a set of disjoint edges that partition the set of vertices V. A one-factorization of a hypergraph G = (V, E) is a pair (V, F), where F is a set of parallel classes that partition the set of edges E. Given a complete k-uniform hypergraph of order n, all of its its edges can be partitioned into parallel classes. Of course, this requires that k divides n, and the number of parallel classes is (n k)/ n k = (n\u22121 k\u22121). This result is well known as Baranyai\u2019s theorem; the resulting partitions are called Baranyai partitions. More precisely, given a set V of cardinality n, an (n, k)-Baranyai partition of V, denoted BP(n, k), is a one-factorization of the complete k-uniform hypergraph G = (V, (V k)). The existence of BP(n, 2) is known since the 19th century; extensive study of related one-factorizations can be found in [21]. Explicit and ef\ufb01cient recursive constructions of BP(n, 3), for every n divisible by 3, were given by Peltesohn [18] in her doctoral thesis from 1936. Bermond proved the existence of BP(n, 4) in the early 1970s, using coloring techniques from graph theory, but his result was never published [19, p. 537]. In fact, after almost \ufb01fty years, it does not appear possible to reconstruct the proof in its entirety [2], and it is not completely clear which values of n have been solved. Moreover, it seems that the techniques used by Bermond would not lead to a construction whose complexity scales linearly in the number of hyperedges. The general case was resolved by Baranyai [1] in 1975, using network \ufb02ow methods. Speci\ufb01cally, Baranyai [1] established the following existence result. Theorem 1. If k and n are positive integers, then BP(n, k) exists whenever k | n. The proof of the foregoing theorem of Baranyai involves several iterations of a network \ufb02ow algorithm. Unfortunately, even today, the best-known algorithm [17] for computing the maximum \ufb02ow in a network G = (V, E) requires O(|V|\u00b7|E|) time. The complexity of producing Baranyai partitions is even higher. For example, the iterative implementation given by Brouwer and Schrijver [3], based on network \ufb02ow techniques, has complexity O(n3k\u22122) for a \ufb01xed k. However, for certain applications, it would be desirable to generate Baranyai partitions in time and space that scale linearly with the number of hyperedges. Hence, we say that a construction of BP(n, k) is ef\ufb01cient if its complexity is O \u0000(n k) \u0001 . Our goal herein is to provide an explicit and ef\ufb01cient construction of Baranyai partitions for k = 4. Our results also lead, via enumerative coding, to ef\ufb01cient algorithms for generating a speci\ufb01c parallel class in BP(n, 4) or a speci\ufb01c hyperedge in the parallel class. The complexity of these algorithms is O(n). To this end, we introduce ef\ufb01cient recursive quadrupling constructions for BP(4t, 4). Speci\ufb01cally, we will show how to construct BP(4t, 4), given BP(t + \u03b41, 4) and BP(t + \u03b42, 3), where 0 \u2264\u03b41 \u22643 and 0 \u2264\u03b42 \u22642. In this paper, we provide such recursive constructions for t \u22610, 3, 4, 6, 8, 9 (mod 12). In other words, we prove the following theorem. Theorem 2. If t \u22610, 3, 4, 6, 8, 9 (mod 12) and there exists an ef\ufb01cient construction for BP(t + \u03b4, 4), where 0 \u2264\u03b4 \u22643, then there exists an ef\ufb01cient construction for BP(4t, 4). In a follow-up paper [4] we consider the remaining cases, namely t \u22611, 2, 5, 7, 10, 11 (mod 12). Together with Theorem 2, this makes it possible to ef\ufb01ciently construct BP(n, 4) for every n divisible by 4. 2 \fThe rest of this paper is organized as follows. We begin in Section 2 with some background on combinatorial designs, which is necessary for our constructions. The discussion in Section 2 brie\ufb02y covers Steiner systems, large sets of Steiner systems, resolvable Steiner systems, one-factorizations, near-onefactorizations, and Latin squares. In Section 3, a doubling construction for BP(2t, 4) from BP(t, 4) is presented for the case where t \u22614 or 8 (mod 12). This construction is based on resolvable Steiner quadruple systems and one-factorizations of the complete graph Kt. In Section 4, the ingredients for our quadrupling constructions are discussed. First, the set of quadruples is partitioned into \ufb01ve groups. Then the parallel classes of the Baranyai partitions we construct are classi\ufb01ed into \ufb01ve types, depending upon which groups take part in the parallel class. In Section 5, we present constructions for each of the \ufb01ve types of parallel classes introduced in Section 4. For three of the \ufb01ve types, the construction depends on whether t is even or odd. For one type, the construction depends on the value of t modulo 4. Finally, for one of the types we will restrict ourselves, in this section, to values of t that are divisible by 3. In Section 6, we summarize the results from Section 5 to present a quadrupling construction of BP(4t, 4) for t \u22610, 3, 4, 6, 8, 9 (mod 12). The case t \u22619 (mod 12) is more involved and requires further modi\ufb01cation of the constructions given in Section 4. Section 7 is devoted to complexity analysis of several problems associated with Baranyai partitions and constructions thereof. As an example of our methods, we present in Section 7 an ef\ufb01cient enumerative-coding algorithm for constructing BP(2n, 4) for all n \u22652. We conclude with a brief discussion in Section 8. Note on terminology. For the sake of brevity, in what follows, whenever we say there exists a BP(n, k) we mean that there also exists an ef\ufb01cient construction for it with complexity O \u0000(n k) \u0001 . 2. Background on Combinatorial Designs Our constructions will make use of some well-known combinatorial designs which will be described in the following subsections. 2.1. Steiner Systems, Large Sets, and Resolvable Designs A Steiner system S(t, k, v) is a pair (Q, B), where Q is a v-set of points and B is a set of k-subsets (called blocks) of Q, such that each t-subset of Q is contained in exactly one block of B. The literature on Steiner systems is very rich (many book chapters and surveys, see for example [5]). Our interest is only in Steiner systems S(k \u22121, k, v), Steiner systems S(1, k, v), and Steiner systems S(k, k, v). Note, that S(k, k, v) is equivalent to the complete k-uniform hypergraph on v vertices. A large set of Steiner systems S(t, k, v) is a partition of all the (v k) k-subsets of Q into Steiner systems S(t, k, v). In our context, a BP(n, k) is a large set of Steiner systems S(1, k, v). Large sets were extensively studied during the years. For example, a large set of S(2, 3, v) is known for each v \u22611 or 3 (mod 6) [12, 15, 16, 20]. No large set of S(3, 4, v) is known, but many pairwise disjoint sets of S(3, 4, v) were found, e.g. [7, 8]. A Steiner system S(k \u22121, k, v) is resolvable if its block set B can be partitioned into subsets B1, B2, . . ., such that each subset Bi is a Steiner system S(1, k, v), i.e. each Bi is a subset of disjoint blocks whose union is the point set Q. It is important to note that BP(n, k) is a resolvable Steiner system S(k, k, n). Of special interest in our exposition are two types of resolvable Steiner systems. A Steiner system S(3, 4, v) is called a Steiner quadruple system and it is denoted by SQS(v). Such a system exists if and only if 3 \fv \u22612 or 4 (mod 6) [10]. It is known that there exists an ef\ufb01cient construction for resolvable SQS(v) if and only if v \u22614 or 8 (mod 12) [11, 13]. The number of parallel classes in a BP(n, k) will be used in our exposition to prove the correctness of the constructions which follow. The total number of k-subsets of an n-set is (n k) and a parallel class contains n k k-subsets. This implies the following theorem. Theorem 3. The number of parallel classes in a BP(n, k) is (n\u22121 k\u22121). Finally, we will use a BP(n, 2) which is a resolvable Steiner system S(2, 2, n), also called onefactorization of the complete graph Kv. This structure is discussed in the next subsection. 2.2. One-Factorization and Near-One-Factorization A one-factorization of the complete graph Km, m even, is a partition of the edges of Km into perfect matchings. In other words, the set F = {F0, F1, . . . , Fm\u22122} is a one-factorization of Km if each Fi, 0 \u2264i \u2264m \u22122, is a perfect matching (called a one-factor), and the Fi\u2019s are pairwise disjoint. The m 2 pairs of Fi are ordered arbitrarily, Fi,0, Fi,1, . . . , Fi,(m\u22122)/2. If m is odd, then there is no perfect matching in Km and we de\ufb01ne a near-one-factorization F = {F0, F1, . . . , Fm\u22121} to be a partition of the edges in Km into sets of m\u22121 2 pairwise disjoint edges, where each Fi has one isolated vertex. Each Fi is called a near-one-factor. In the sequel we assume that the set of vertices of Km is Zm and in the near-one-factor Fi, 0 \u2264i \u2264m \u22121, the vertex i is isolated. The m\u22121 2 pairs of Fi are ordered arbitrarily, Fi,0, Fi,1, . . . , Fi,(m\u22123)/2. As was noted before, there has been done extensive study on one-factorizations and near-one-factorization and an excellent book for this is [21]. Example 1. For m = 3 there are three near-one-factors in a near-one-factorization F = {F0, F1, F2}, where F0 = {{1, 2}}, F1 = {{0, 2}}, and F2 = {{0, 1}}. For m = 4, this yields a one-factorization F \u2032 = {F \u2032 0, F \u2032 1, F \u2032 2}, where F \u2032 0 = {{1, 2}, {0, 3}}, F \u2032 1 = {{0, 2}, {1, 3}}, and F \u2032 2 = {{0, 1}, {2, 3}}. The examples for m = 3 and m = 4 are special cases of a general construction. Let m > 1 be a positive odd integer. We de\ufb01ne the near-one-factor Fk, 0 \u2264k \u2264m \u22121, as follows. Fk \u225c{{x, y} : x, y \u2208Zm, x \u0338= y, x + y \u22612k (mod m)} . The one-factorization F \u2032 of Km+1 is de\ufb01ned from the near-one-factorization F of Km as follows F \u2032 k \u225cFk \u222a{{k, m}}. 4 \f2.3. Latin Squares All the designs which were previously discussed consist of blocks and hence they are also called block designs. The last combinatorial design which will be used in our constructions is not a block design, but it is heavily used in constructions of various types of block designs including Steiner systems. A Latin square of order n is an n \u00d7 n matrix, say M, with entries from a given n-set Q such that each row and each column of M is a permutation of the elements of Q. An important result in our discussion, which can be proved for example by using the well-known Hall\u2019s marriage Theorem [9] is given next. Theorem 4. If a k \u00d7 n matrix \u02c6 M has rows which are permutations of an n-set Q and columns with distinct elements of Q, then \u02c6 M can be completed to an n \u00d7 n Latin square. 3. A Doubling Construction for t \u22614 or 8 (mod 12) Let n = 2t, where t \u22614 or 8 (mod 12), and let S be a BP(t, 4) on the point set Zt. Furthermore, let T be a resolvable SQS(t) and let F = {F0, F1, . . . , Ft\u22122} be a one-factorization of Kt on the point set Zt. We form a BP(n, 4) P on the point set Zt \u00d7 Z2. The parallel classes of P will be of three types, Type S, Type T , and Type F. A quadruple {(x1, i1), (x2, i2), (x3, i3), (x4, i4)} is in a parallel class of Type S if |{x1, x2, x3, x4}| = 4, in a parallel class of Type T if |{x1, x2, x3, x4}| = 3, and in a parallel class of Type F if |{x1, x2, x3, x4}| = 2. Each type does not have any quadruple with a different structure. 3.1. Parallel Classes of Type S For a given parallel class R of S form eight sets Ri, i \u2208Z8, in P. For each i \u2208Z8, given a block {x1, x2, x3, x4} \u2208R we form the following two blocks in Ri. {(x1, 0), (x2, j2), (x3, j3), (x4, j4)}, {(x1, 1), (x2, j2 + 1), (x3, j3 + 1), (x4, j4 + 1)}, where (j2j3j4) is the binary representation of i. The following result can be easily veri\ufb01ed. Lemma 1. Given a parallel class R of S, each set Ri, i \u2208Z8, is a parallel class, of Type S, on the point set Zt \u00d7 Z2. Each two parallel classes Ri and R\u2032 j, 0 \u2264i, j \u22647, where R \u0338= R\u2032 or i \u0338= j, are disjoint. The number of parallel classes of Type S in P is 8(t\u22121 3 ). Proof. Since R is a parallel class of S on the point set Zt, it follows that Ri forms a parallel class on the point set Zt \u00d7 Z2 for each i \u2208Z8. If B = {(x1, \u21131), (x2, \u21132), (x3, \u21133), (x4, \u21134)} is a block of a de\ufb01ned parallel class Ri, then {x1, x2, x3, x4} is a block of the parallel class R. Hence, |{x1, x2, x3, x4}| = 4 which implies that B is a block in a parallel class of Type S. Suppose that Ri and R\u2032 j are two parallel classes which are not disjoint and let B = {(x1, \u21131), (x2, \u21132), (x3, \u21133), (x4, \u21134)} \u2208Ri \u2229R\u2032 j. By the de\ufb01nition of Ri and R\u2032 j, we have that {x1, x2, x3, x4} \u2208R \u2229R\u2032. Since both R and R\u2032 are parallel classes of S, in which each two different parallel classes are disjoint, it follows that R = R\u2032. Moreover, the value of \u21131 implies that either \u21132\u21133\u21134 or (\u21132 + 1)(\u21133 + 1)(\u21134 + 1) is the binary representation of i and j which implies that i = j, that is Ri = R\u2032 j, a contradiction. Thus, Ri \u2229R\u2032 j = \u2205if R \u0338= R\u2032 or i \u0338= j. 5 \fBy Theorem 3, the number of parallel classes in S is (t\u22121 3 ). Each parallel class of S induces eight parallel classes of Type S in P. Hence, the number of parallel classes of Type S in P is 8(t\u22121 3 ). Example 2. For t = 8 there are 35 parallel classes in a unique BP(8, 4), where a parallel class has two quadruples of the form {X, Z8 \\ X}, where X is a quadruple of Z8. For the parallel class R = {{0, 1, 2, 3}, {4, 5, 6, 7}} form the eight parallel classes R0, R1, R2, R3, R4, R5, R6, R7 for P as follows. R0 = {{(0, 0), (1, 0), (2, 0), (3, 0)}, {(0, 1), (1, 1), (2, 1), (3, 1)}, {(4, 0), (5, 0), (6, 0), (7, 0)}, {(4, 1), (5, 1), (6, 1), (7, 1)}}, R1 = {{(0, 0), (1, 0), (2, 0), (3, 1)}, {(0, 1), (1, 1), (2, 1), (3, 0)}, {(4, 0), (5, 0), (6, 0), (7, 1)}, {(4, 1), (5, 1), (6, 1), (7, 0)}}, R2 = {{(0, 0), (1, 0), (2, 1), (3, 0)}, {(0, 1), (1, 1), (2, 0), (3, 1)}, {(4, 0), (5, 0), (6, 1), (7, 0)}, {(4, 1), (5, 1), (6, 0), (7, 1)}}, R3 = {{(0, 0), (1, 0), (2, 1), (3, 1)}, {(0, 1), (1, 1), (2, 0), (3, 0)}, {(4, 0), (5, 0), (6, 1), (7, 1)}, {(4, 1), (5, 1), (6, 0), (7, 0)}}, R4 = {{(0, 0), (1, 1), (2, 0), (3, 0)}, {(0, 1), (1, 0), (2, 1), (3, 1)}, {(4, 0), (5, 1), (6, 0), (7, 0)}, {(4, 1), (5, 0), (6, 1), (7, 1)}}, R5 = {{(0, 0), (1, 1), (2, 0), (3, 1)}, {(0, 1), (1, 0), (2, 1), (3, 0)}, {(4, 0), (5, 1), (6, 0), (7, 1)}, {(4, 1), (5, 0), (6, 1), (7, 0)}}, R6 = {{(0, 0), (1, 1), (2, 1), (3, 0)}, {(0, 1), (1, 0), (2, 0), (3, 1)}, {(4, 0), (5, 1), (6, 1), (7, 0)}, {(4, 1), (5, 0), (6, 0), (7, 1)}}, R7 = {{(0, 0), (1, 1), (2, 1), (3, 1)}, {(0, 1), (1, 0), (2, 0), (3, 0)}, {(4, 0), (5, 1), (6, 1), (7, 1)}, {(4, 1), (5, 0), (6, 0), (7, 0)}}. 3.2. Parallel Classes of Type T For a given parallel class R of T form 24 sets Ri,j,k, i \u2208Z6, j, k \u2208Z2, in P. Given a block {x1, x2, x3, x4} \u2208R we form the following two blocks in R0,j,k: {(x1, 0), (x1, 1), (x2, j), (x3, k)}, {(x2, j + 1), (x3, k + 1), (x4, 0), (x4, 1)}. The following two blocks are constructed in R1,j,k, j, k \u2208Z2: {(x1, 0), (x1, 1), (x2, j), (x4, k)}, {(x2, j + 1), (x3, 0), (x3, 1), (x4, k + 1)}. The following two blocks are constructed in R2,j,k, j, k \u2208Z2: {(x1, 0), (x1, 1), (x3, j), (x4, k)}, {(x2, 0), (x2, 1), (x3, j + 1), (x4, k + 1)}. The following two blocks are constructed in R3,j,k, j, k \u2208Z2: {(x1, j), (x2, 0), (x2, 1), (x3, k)}, {(x1, j + 1), (x3, k + 1), (x4, 0), (x4, 1)}. The following two blocks are constructed in R4,j,k, j, k \u2208Z2: {(x1, j), (x2, 0), (x2, 1), (x4, k)}, {(x1, j + 1), (x3, 0), (x3, 1), (x4, k + 1)}. The following two blocks are constructed in R5,j,k, j, k \u2208Z2: {(x1, j), (x2, k), (x3, 0), (x3, 1)}, {(x1, j + 1), (x2, k + 1), (x4, 0), (x4, 1)}. The following result can be easily veri\ufb01ed. 6 \fLemma 2. Given a parallel class R of T, each set Ri,j,k, i \u2208Z6, j, k \u2208Z2, is a parallel class, of Type T , on the point set Zt \u00d7 Z2. Each two parallel classes Ri,j,k and R\u2032 m,r,s, i, m \u2208Z6, j, k, r, s \u2208Z2, where R \u0338= R\u2032 or (i, j, k) \u0338= (m, r, s), are disjoint. The number of parallel classes of Type T in P is 4(t \u22121)(t \u22122). Proof. It is easy to verify that since R is a parallel class of T on the point set Zt, it follows that Ri,j,k forms a parallel class on the point set Zt \u00d7 Z2 for each i \u2208Z6, j, k \u2208Z2. Moreover, if B = {(x1, \u21131), (x2, \u21132), (x3, \u21133), (x4, \u21134)} is a block of a de\ufb01ned parallel class, one can easily veri\ufb01ed that |{x1, x2, x3, x4}| = 3 which implies that B is a block in a parallel class of Type T . Suppose that Ri,j,k and R\u2032 m,r,s are two parallel classes which are not disjoint and B = {(x1, \u21131), (x2, \u21132), (x3, \u21133), (x4, \u21134)} \u2208Ri,j,k \u2229R\u2032 m,r,s. Since |{x1, x2, x3, x4}| = 3 and T is an SQS(t), one can easily verify from the construction that B \u2208Ri,j,k \u2229R\u2032 m,r,s can only happen if i = m. Assume, for example, that i = 0, {y1, y2, y3, y4} \u2208T, and B = {(y1, 0), (y1, 1), (y2, \u03b1), (y3, \u03b2)} \u2208 R0,j,k \u2229R\u2032 0,r,s. It implies that (\u03b1, \u03b2) = (j, k) = (r, s). Moreover, it implies that {y1, y2, y3, y4} \u2208 R \u2229R\u2032 and hence R = R\u2032, that is Ri,j,k = R\u2032 m,r,s, a contradiction. Thus, Ri,j,k \u2229R\u2032 m,r,s = \u2205if R \u0338= R\u2032 or (i, j, k) \u0338= (m, r, s). Simple counting arguments show that the number of parallel classes in an SQS(t) is (t\u22121)(t\u22122) 6 . Each parallel class of T in an SQS(t) induces 24 parallel classes of Type T in P. Hence, the number of parallel classes of Type T in P is 4(t \u22121)(t \u22122). 3.3. Parallel Classes of Type F For each one-factor Fi, 0 \u2264i \u2264t \u22122, form the following set \u02c6 Ri in P. \u02c6 Ri \u225c{{(x, 0), (y, 0), (x, 1), (y, 1)} : {x, y} \u2208Fi} The following result can be easily veri\ufb01ed. Lemma 3. For each one-factor Fi, 0 \u2264i \u2264t \u22122, the set \u02c6 Ri is a parallel class of size t/2 on the point set Zt \u00d7 Z2. Each two parallel classes \u02c6 Ri and \u02c6 Rj, 0 \u2264i < j \u2264t \u22122, are disjoint. Each de\ufb01ned block {(x1, i1), (x2, i2), (x3, i3), (x4, i4)} of a parallel class \u02c6 R of Type F satis\ufb01es |{x1, x2, x3, x4}| = 2. The number of parallel classes of Type F in P is t \u22121. 3.4. Conclusion for t \u22614 or 8 (mod 12) The constructions of the blocks of Type S, Type T , and Type F imply the following theorem. Theorem 5. If there exists a BP(t, 4), where t \u22614 or 8 (mod 12), then there exists a BP(2t, 4). Proof. By Lemmas 1, 2, and 3, all the de\ufb01ned parallel classes of Type S, Type T , and Type F, are pairwise disjoint and there are 8 \u0012t \u22121 3 \u0013 + 4(t \u22121)(t \u22122) + (t \u22121) = \u00122t \u22121 3 \u0013 such parallel classes. This implies by Theorem 3 that each quadruple of Zt \u00d7 Z4 is contained in exactly one of the parallel classes. 7 \f4. Ingredients for the Quadrupling Construction Let n = 4t, where t is a positive integer, t + \u03b41 is divisible by 4, and t + \u03b42 is divisible by 3, where 0 \u2264\u03b41 \u22643, and 0 \u2264\u03b42 \u22642. Assume that S is a BP(t + \u03b41, 4) on the point set Zt \u222a{\u21261, . . . , \u2126\u03b41}. Assume T is a BP(t + \u03b42, 3) on the point set Zt \u222a{\u21261, . . . , \u2126\u03b42}. The constructed BP(4t, 4), P, in the following sections is on the point set Zt \u00d7 Z4. In a quadruple {y0, y1, y2, y3} \u2208S and a triple {x0, x1, x2} \u2208T, where y0, y1, y2, y3, x0, x1, x2 \u2208Zt we assume that the elements are ordered in increasing order, i.e. y0 < y1 < y2 < y3 and x0 < x1 < x2. The number of quadruples in Zt \u00d7 Z4 is (4t 4), the number of quadruples in a parallel class of P is t, and the number of parallel classes in P is (4t\u22121 3 ). 4.1. Con\ufb01gurations, Groups, and Types of Parallel Classes Each quadruple X of the point set Zt \u00d7 Z4 is written as X = {(x0, i0), (x1, i1), (x2, i2), (x3, i3)}. Let j0 = |X \u2229Zt \u00d7 {0}|, j1 = |X \u2229Zt \u00d7 {1}|, j2 = |X \u2229Zt \u00d7 {2}|, and j3 = |X \u2229Zt \u00d7 {3}|. We say that X is a quadruple from con\ufb01guration (j0, j1, j2, j3). There is a total of 35 possible con\ufb01gurations. For ji, i \u2208Z4, there is a con\ufb01guration (j0, j1, j2, j3) if j0 + j1 + j2 + j3 = 4. There are \u220f3 i=0 ( t ji) distinct quadruples in con\ufb01guration (j0, j1, j2, j3). The 35 con\ufb01gurations are partitioned into \ufb01ve groups as follows: Group 1: In this group there are four con\ufb01gurations (4, 0, 0, 0), (0, 4, 0, 0), (0, 0, 4, 0), and (0, 0, 0, 4). Group 2: In this group there are twelve con\ufb01gurations (3, 1, 0, 0), (1, 3, 0, 0), (3, 0, 1, 0), (1, 0, 3, 0), (3, 0, 0, 1), (1, 0, 0, 3), (0, 3, 1, 0), (0, 1, 3, 0), (0, 3, 0, 1), (0, 1, 0, 3), (0, 0, 3, 1), and (0, 0, 1, 3). Group 3: In this group there are six con\ufb01gurations (2, 2, 0, 0), (2, 0, 2, 0), (2, 0, 0, 2), (0, 2, 2, 0), (0, 2, 0, 2), and (0, 0, 2, 2). Group 4: In this group there are twelve con\ufb01gurations (2, 1, 1, 0), (2, 1, 0, 1), (2, 0, 1, 1), (1, 2, 1, 0), (1, 2, 0, 1), (0, 2, 1, 1), (1, 1, 2, 0), (1, 0, 2, 1), (0, 1, 2, 1), (1, 1, 0, 2), (1, 0, 1, 2), and (0, 1, 1, 2). Group 5: The only con\ufb01guration contained in this group is (1, 1, 1, 1). The construction which will be described in the sequel has \ufb01ve types of parallel classes. \u2022 In Type 1, most of the quadruples in each parallel class are from Group 1. There might be some quadruples from Group 2, Group 3, or Group 5. \u2022 In Type 2, if t is divisible by 3 then all the quadruples in each parallel class are from Group 2. If t is not divisible by 3, then there might be some quadruples from Group 3 or Group 4. \u2022 In Type 3, if t is even then all the quadruples in each parallel class are from Group 3. If t is odd, then t \u22121 of the quadruples in each parallel class are from Group 3 and one quadruple in each parallel class is from Group 5, i.e. from con\ufb01guration (1,1,1,1). \u2022 In Type 4, if t is even then all the quadruples in each parallel class are from Group 4. If t is odd then t \u22121 of the quadruples in each parallel class are from Group 4 and one quadruple in each parallel class is from Group 5, i.e. from con\ufb01guration (1,1,1,1). \u2022 In Type 5, all the quadruples in each parallel class are from Group 5. 8 \f5. Parallel Classes for the Quadrupling Construction 5.1. Parallel Classes of Type 1 Given any parallel class R in S, construct a parallel class R\u2032 in P. For any given B \u2208R we form a subset of quadruples in R\u2032 which will be de\ufb01ned in the sequel. We distinguish between the four different residues of t modulo 4. 5.1.1. Type 1 for t divisible by 4 The set S is a BP(t, 4) on Zt and R a parallel class in S. If B = {x0, x1, x2, x3} \u2208R, then construct the following set with four quadruples in R\u2032. B\u2032 i \u225c{{(x0, i), (x1, i), (x2, i), (x3, i)} : i \u2208Z4}. Lemma 4. 1. The subset R\u2032 is a parallel class on the point set Zt \u00d7 Z4. 2. Each quadruple of Group 1 is contained in exactly one such parallel class. 3. The number of parallel classes of Type 1 when t is divisible by 4 is (t\u22121 3 ). Each such parallel class contains t quadruples from Group 1. Proof. Since R is a parallel class in S, it follows that R\u2032 = [ B\u2208R,i\u2208Z4 B\u2032 i forms a parallel class on the point set Zt \u00d7 Z4. For each i \u2208Z4, each quadruple {(x0, i), (x1, i), (x2, i), (x3, i)} is contained in exactly one parallel class in P since any block {x0, x1, x2, x3} is contained in exactly one parallel class in S. By Theorem 3, the number of parallel classes in a BP(t, 4) is (t\u22121 3 ). Each parallel class of S corresponds to exactly one parallel class of Type 1 in P, and hence the number of parallel classes of Type 1 in P is also (t\u22121 3 ). Clearly, each such parallel class contains t quadruples from Group 1. 5.1.2. Type 1 for t \u22613 (mod 4) The set S is a BP(t + 1, 4) on the point set Zt \u222a{\u21261} and R is a parallel class in S. For each block B \u2208R construct a set of quadruples R\u2032 as follows. 1. If \u21261 / \u2208B = {x0, x1, x2, x3} then construct the following four quadruples in R\u2032: {(x0, i), (x1, i), (x2, i), (x3, i)}, i \u2208Z4. 2. If B = {x0, x1, x2, \u21261}, then construct the following three quadruples in R\u2032: {(x0, i), (x1, i), (x2, i), (xi, 3)}, i \u2208{0, 1, 2}. 9 \fLemma 5. The subset of quadruples from Group 2 contained in the parallel classes of Type 1 when t \u22613 (mod 4) is {{(x, 0), (y, 0), (z, 0), (x, 3)}, {(x, 1), (y, 1), (z, 1), (y, 3)}, {(x, 2), (y, 2), (z, 2), (z, 3)} : x, y, z \u2208Zt}. This subset contains exactly (t \u22122)(t 2) quadruples from Group 2. Proof. In a BP(t + 1, 4), any four different points from Zt \u222a{\u21261} occurs as a block exactly once. Hence, the point \u21261 occurs with any three points, of Zt, in some block exactly once. Therefore, the number of quadruples containing \u21261 is (t 3). Each such quadruple induces three quadruples from Group 2 de\ufb01ned above. Thus, this set contains exactly 3(t 3) = (t \u22122)(t 2) quadruples from Group 2. Lemma 6. 1. The subset R\u2032 is a parallel class on the point set Zt \u00d7 Z4. 2. Each quadruple of Group 1 is contained in exactly one such parallel class. 3. The number of parallel classes of Type 1 when t \u22613 (mod 4) is (t 3). Each such parallel class contains t \u22123 quadruples from Group 1 and three quadruples from Group 2. Proof. The \ufb01rst two claims of the lemma are readily veri\ufb01ed. For the third claim, note that each parallel class of S corresponds to a unique parallel class of P. By Theorem 3 there are (t 3) parallel classes in a BP(t + 1, 4) and hence the number of parallel classes of Type 1, in P when t \u22613 (mod 4), is (t 3). Clearly, each such parallel class contains t \u22123 quadruples from Group 1 and three quadruples from Group 2. 5.1.3. Type 1 for t \u22612 (mod 4) The set S is a BP(t + 2, 4) on the point set Zt \u222a{\u21261, \u21262} and R is a parallel class in S. For each block B \u2208R de\ufb01ne the following blocks in R\u2032. 1. If \u21261, \u21262 / \u2208B = {x0, x1, x2, x3} then construct the following four quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (x3, i)}, i \u2208Z4. 2. If \u21262 / \u2208B = {x0, x1, x2, \u21261}, then construct the following three quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (xi, 3)}, i \u2208Z3. 3. If \u21261 / \u2208B = {x0, x1, x2, \u21262}, then construct the following three quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (xi+1, 3)}, i \u2208Z3. 4. If B = {x0, x1, \u21261, \u21262}, then construct the two quadruples in R\u2032. {(x0, 0), (x1, 0), (x0, 1), (x1, 1)}}. {(x0, 2), (x1, 2), (x0, 3), (x1, 3)}}. 10 \fRecall, that there is an order between the elements of each block of S which implies that there is no ambiguity in the de\ufb01nitions of the blocks of P. Lemma 7. 1. The subset of quadruples from Group 2 contained in the parallel classes of Type 1 when t \u2261 2 (mod 4) is {{(x, 0), (y, 0), (z, 0), (x, 3)}, {(x, 1), (y, 1), (z, 1), (y, 3)}, {(x, 2), (y, 2), (z, 2), (z, 3)} : x, y, z \u2208Zt} \u222a{{(x, 0), (y, 0), (z, 0), (y, 3)}, {(x, 1), (y, 1), (z, 1), (z, 3)}, {(x, 2), (y, 2), (z, 2), (x, 3)} : x, y, z \u2208Zt}. This subset contains exactly t(t \u22121)(t \u22122) quadruples from Group 2. 2. The subset of quadruples from Group 3 contained in the parallel classes of Type 1 when t \u2261 2 (mod 4) is {{(x, i), (y, i), (x, i + 1), (y, i + 1)} : x, y \u2208Zt, i = 0, 2} . This subset contains exactly (t \u22121)t quadruples from Group 3. Proof. The structure of the quadruples in P when t \u22612 (mod 4) is an immediate consequence from the de\ufb01nition of the blocks in the construction. It is easily to verify that \u21261 (without \u21262) is contained (t 3) quadruples in S and the same is true for \u21262 (without \u21261). Hence, there are exactly 2 \u00b7 3 \u00b7 (t 3) = t(t \u22121)(t \u22122) quadruples from Group 2 in P. Since the pair {\u21261, \u21262} is contained in (t 2) quadruples of S, it follows that there are exactly 2(t 2) = (t \u22121)t quadruples from Group 3 in P. Lemma 8. 1. The subset R\u2032 is a parallel class, of Type 1, on the point set Zt \u00d7 Z4. 2. Each quadruple of Group 1 is contained in exactly one such parallel class. 3. The number of parallel classes of Type 1 when t \u22612 (mod 4) is (t+1 3 ). There are (t 2) parallel classes, in which each parallel class contains t \u22122 quadruples from Group 1 and two quadruples from Group 3. There are (t 3) parallel classes, in which each parallel class contains t \u22126 quadruples from Group 1 and six quadruples from Group 2. Proof. The \ufb01rst two claims of the lemma are readily veri\ufb01ed. For the third claim, note that \u21261 (\u21262, respectively) without \u21262 (\u21261, respectively) occurs in exactly (t 3) parallel classes of S. Each such parallel class of S (which contains one block with \u21261 and another block with \u21262) induces one parallel class of Type 1 in P, which contains t \u22126 quadruples from Group 1 and six quadruples from Group 2. Similarly, the pair {\u21261, \u21262} occurs in exactly (t 2) parallel classes of S. Each such parallel class induces one parallel class of Type 1 in P, which contains t \u22122 quadruples from Group 1 and two quadruples from Group 3. Thus, the total number of parallel classes of Type 1 is (t 3) + (t 2) = (t+1 3 ). 11 \f5.1.4. Type 1 for t \u22611 (mod 4) The set S is a BP(t + 3, 4) on the point set Zt \u222a{\u21261, \u21262, \u21263} and R is a parallel class in S. For each block B \u2208R construct the following blocks in R\u2032 \u2208P. 1. If \u21261, \u21262, \u21263 / \u2208B = {x0, x1, x2, x3} then construct the following four quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (x3, i)}, i \u2208Z4. 2. If \u21261, \u21262 / \u2208B = {x0, x1, x2, \u21263}, then construct the following three quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (xi, 3)}, i \u2208Z3. 3. If \u21261, \u21263 / \u2208B = {x0, x1, x2, \u21262}, then construct the following three quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (xi+1, 3)}, i \u2208Z3. 4. If \u21262, \u21263 / \u2208B = {x0, x1, x2, \u21261}, then construct the following three quadruples in R\u2032. {(x0, i), (x1, i), (x2, i), (xi+2, 3)}, i \u2208Z3. 5. If \u21261 / \u2208B = {x0, x1, \u21262, \u21263}, then construct the following two quadruples in R\u2032. {(x0, 0), (x1, 0), (x0, 1), (x1, 1)}, {(x0, 2), (x1, 2), (x0, 3), (x1, 3)}. 6. If \u21262 / \u2208B = {x0, x1, \u21261, \u21263}, then construct the following two quadruples in R\u2032. {(x0, 0), (x1, 0), (x0, 3), (x1, 3)}, {(x0, 1), (x1, 1), (x0, 2), (x1, 2)}. 7. If \u21263 / \u2208B = {x0, x1, \u21261, \u21262}, then construct the following two quadruples in R\u2032. {(x0, 0), (x1, 0), (x0, 2), (x1, 2)}, {(x0, 1), (x1, 1), (x0, 3), (x1, 3)}. 8. If B = {x0, \u21261, \u21262, \u21263}, then construct the following quadruple in R\u2032. {(x0, 0), (x0, 1), (x0, 2), (x0, 3)}}. Lemma 9. 1. The subset of quadruples from Group 2 contained in the parallel classes of Type 1 when t \u2261 1 (mod 4) is {{(x, i), (y, i), (z, i), (x, 3)}, {(x, i), (y, i), (z, i), (y, 3)}, {(x, i), (y, i), (z, i), (z, 3)} : x, y, z \u2208Zt, i = 0, 1, 2}. This subset contains exactly 9(t 3) quadruples from Group 2. 12 \f2. The subset of quadruples from Group 3 contained in parallel classes of Type 1 when t \u22611 (mod 4) is {{(x, 0), (y, 0), (x, 1), (y, 1)} : x, y \u2208Zt} \u222a{{(x, 2), (y, 2), (x, 3), (y, 3)} : x, y \u2208Zt} \u222a{{(x, 0), (y, 0), (x, 2), (y, 2)} : x, y \u2208Zt} \u222a{{(x, 1), (y, 1), (x, 3), (y, 3)} : x, y \u2208Zt} \u222a{{(x, 0), (y, 0), (x, 3), (y, 3)} : x, y \u2208Zt} \u222a{{(x, 1), (y, 1), (x, 2), (y, 2)} : x, y \u2208Zt} . This subset contains exactly 3(t \u22121)t quadruples from Group 3. 3. The subset of quadruples from Group 5 contained in the parallel classes of Type 1 when t \u2261 1 (mod 4) is {{(i, 0), (i, 1), (i, 2), (i, 3)} : i \u2208Zt} . This subset contains exactly t quadruples from Group 5 which form a parallel class. Proof. 1. When t \u22611 (mod 4), for each i \u2208Z3 and B = {x, y, z, \u2126i} \u2208R, the subset of quadruples from Group 2 contained in the parallel classes of Type 1 contains the following set with three quadruples {{(x, i), (y, i), (z, i), (x, 3)}, {(x, i), (y, i), (z, i), (y, 3)}, {(x, i), (y, i), (z, i), (z, 3)}} In addition, for each i \u2208Z3, \u21261 (without \u21262 or \u21263) is contained in (t 3) quadruples of S. The same is true for \u21262 (without \u21261 or \u21263) and \u21263 (without \u21261 or \u21262). Hence, there are exactly 3 \u00b7 3 \u00b7 (t 3) = 9(t 3) quadruples from Group 2 in P. 2. For any 1 \u2264i < j \u22643 and B = {x, y, \u2126i, \u2126j} \u2208R, the subset of quadruples from Group 3 which are used in Type 1 contains two of the quadruples de\ufb01ned above. Since the pair {\u2126i, \u2126j} is contained in (t 2) quadruples of S, there are exactly 3 \u00b7 2 \u00b7 (t 2) = 3(t \u22121)t quadruples from Group 3 in P. 3. For any quadruple {x, \u21261, \u21262, \u21263} \u2208R in S, the corresponding quadruple from Group 5 contained in the parallel classes of Type 1 is {(x, 0), (x, 1), (x, 2), (x, 3)}. Since the triple {\u21261, \u21262, \u21263} is contained in t quadruples of S, it follows that there are exactly t quadruples from Group 5 in P. The claims in the following lemma are proved similarly to the ones in previous lemmas and hence we omit the proof and leave it to the interested reader. Lemma 10. 1. The subset R\u2032 is a parallel class on the point set Zt \u00d7 Z4. 2. Each quadruple of Group 1 is contained in exactly one such parallel class. 3. The number of parallel classes of Type 1 when t \u22611 (mod 4) is (t+2 3 ). There are t parallel classes, where each parallel class contains t \u22121 quadruples from Group 1 and one quadruple from Group 5. There are 3(t 2) parallel classes, in which each parallel class contains t \u22125 quadruples from Group 1, three quadruples of Group 2, and two quadruples from Group 3. There are (t 3) \u2212(t 2) parallel classes, in which each parallel class contains t \u22129 quadruples from Group 1 and nine quadruples from Group 2. 13 \f5.2. Parallel Classes of Type 2 for t divisible by 3 Given any parallel class R in T, a few parallel classes are de\ufb01ned for P. The construction heavily depends on the residue of t modulo 4. Let M be a t \u00d7 t Latin square on the point set Zt, where the rows and the columns are indexed by Zt. Given any parallel class R in T, construct 4t, 4t \u22121, 4t \u22122, or 4t \u22123, parallel classes in P, depending if t (mod 4) equals 0, 3, 2, or 1, respectively. The \ufb01rst 3t parallel classes will be denoted by Ri,j, i \u2208Zt, j \u2208{0, 1, 2}. For a given j \u2208{0, 1, 2}, let \u03c6 : {0, 1, 2, 3} \\ {j} \u2212 \u2192{0, 1, 2} be any bijection. Let B0, B1, . . . , B(t\u22123)/3, be the blocks of R. For a given i \u2208Zt, j \u2208{0, 1, 2}, de\ufb01ne the following t quadruples in Ri,j. {(x0, s), (x1, s), (x2, s), (M(i, 3m + \u03c6(s)), j)}, {x0, x1, x2} \u2208Bm, 0 \u2264m \u2264t \u22123 3 , s \u2208{0, 1, 2, 3} \\ {j}. For the last t, t \u22121, t \u22122, t \u22123 parallel classes, we distinguish now between four cases which have similar solutions, where we de\ufb01ne 0, 1, 2, or 3, respectively, rows of the following matrix M. If t \u22611 or 2 or 3 (mod 4) and if Bm = {xm0, xm1, xm2}, 0 \u2264m \u2264t\u22123 3 , then M(0, 3m + i) = xmi, for i \u2208Z3. If t \u22611 or 2 (mod 4) and if Bm = {xm0, xm1, xm2}, 0 \u2264m \u2264t\u22123 3 , then M(1, 3m + i) = xmi+1, for i \u2208Z3. If t \u22611 (mod 4) and if Bm = {xm0, xm1, xm2}, 0 \u2264m \u2264t\u22123 3 , then M(2, 3m + i) = xmi+2, for i \u2208Z3. For each congruence, we complete M to be a Latin square (see Theorem 4). Each one of the rows of M, which was not predetermined corresponds to a parallel class denoted by Ri, i \u2208Zt, i \u22650 if t is divisible by 4; i \u22651 if t \u22613 (mod 4); i \u22652 if t \u22612 (mod 4); and i \u22653 if t \u22611 (mod 4). Each parallel class has t quadruples as follows. {(x0, s), (x1, s), (x2, s), (M(i, 3m + s), 3)}, {x0, x1, x2} \u2208Bm, 0 \u2264m \u2264t \u22123 3 , s \u2208{0, 1, 2}. Remark 1. We can avoid Theorem 4 and de\ufb01ne the completion of M to a t \u00d7 t Latin square. We have not done this to restrict the length of this section. Lemma 11. Each set (Ri,j or Ri) of Type 2 de\ufb01ned for t divisible by 3 is a parallel class on the point set Zt \u00d7 Z4 in P. Moreover, 1. If t is divisible by 4, then there are 4t parallel classes of Type 2. These parallel classes contain all the quadruples of Group 2. There are 2t(t \u22121)(t \u22122) = 4t(t\u22121 2 ) such parallel classes, in which each parallel class contains t quadruples from Group 2. 2. If t \u22611 (mod 4), then there are 4t \u22123 parallel classes of Type 2. Each quadruple of Group 2 is contained in either one of these parallel classes or in the parallel classes of Type 1. There are (4t \u22123)(t\u22121 2 ) such parallel classes, in which each parallel class contains t quadruples from Group 2. 3. If t \u22612 (mod 4), then there are 4t \u22122 parallel classes of Type 2. Each quadruple of Group 2 is contained in either one of these parallel classes or in the parallel classes of Type 1. There are (2t \u22121)(t \u22121)(t \u22122) = (4t \u22122)(t\u22121 2 ) such parallel classes, in which each parallel class contains t quadruples from Group 2. 14 \f4. If t \u22613 (mod 4), then there are 4t \u22121 parallel classes of Type 2. Each quadruple of Group 2 is contained in either one of these parallel classes or in the parallel classes of Type 1. There are (4t \u22121)(t\u22121 2 ) such parallel classes, in which each parallel class contains t quadruples from Group 2. Each such parallel class of Type 2 contains only quadruples from Group 2. Proof. If t is divisible by 4, then by Lemma 4, no quadruples of Group 2 are contained in the parallel classes of Type 1. Hence, for each parallel class of BP(t, 3), there are 4t parallel classes of Type 2 in P. By Theorem 3, there exist (t\u22121 2 ) parallel classes in a BP(t, 3). Thus, there are 4t(t\u22121 2 ) such parallel classes in P, where each parallel class contains t quadruples from Group 2. Claims 2, 3, and 4 are proved in a similar way. 5.3. Parallel Classes of Type 3 For the de\ufb01nition of the parallel classes of Type 3 in P we distinguish between even t and odd t. 5.3.1. Type 3 for even t For even t a one-factorization F = {F0, F1, . . . , Ft\u22122} of Kt is required. For each 0 \u2264i, s \u2264t \u22122, 0 \u2264r \u2264t\u22122 2 , and a, b, c, d \u2208Z4, where |{a, b, c, d}| = 4, the following set is de\ufb01ned in P: {{(x, a), (y, a), (z, b), (v, b)}, {(x, c), (y, c), (z, d), (v, d)} : {x, y} = Fi,j, {z, v} = Fs,j+r, 0 \u2264j \u2264(t \u22122)/2} . Note that there are three options for a, b, i.e., {a, b} = {0, 1}, {a, b} = {0, 2} and {a, b} = {0, 3} and {c, d} = {0, 1, 2, 3} \\ {a, b}. Lemma 12. 1. For each 0 \u2264i, s \u2264t \u22122, 0 \u2264r \u2264t\u22122 2 , and a, b, c, d \u2208Z4, where |{a, b, c, d}| = 4, the t de\ufb01ned quadruples form a parallel class with t quadruples from Group 3. 2. If t is divisible by 4, then each quadruple from the con\ufb01gurations of Group 3 appears only in one of the de\ufb01ned parallel classes. 3. If t \u22612 (mod 4), then each quadruple from the con\ufb01gurations of Group 3 appears either in one of the de\ufb01ned parallel classes or in the parallel classes of Type 1. 4. The number of de\ufb01ned parallel classes of Type 3 for even t is 3(t \u22121)(t 2). Proof. Combining Lemma 4 and Lemma 8, the \ufb01rst three claims are readily veri\ufb01ed from the de\ufb01nition of the blocks in the construction. Since, in the de\ufb01nition of the construction, there are three options for a, b, c, d and for each such option the triple (i, s, r) has (t \u22121)2 t 2 different choices, it follows that there are 3(t \u22121)(t 2) de\ufb01ned parallel classes of Type 3 in the construction of P. Lemma 13. The set of quadruples from Group 3 which are used in Type 1 when t \u22612 (mod 4), {{(x, i), (y, i), (x, i + 1), (y, i + 1)} : x, y \u2208Zt, i = 0, 2} , form t \u22121 parallel classes of Type 3. 15 \fProof. The number of all the pairs {x, y} with x, y \u2208Zt, x \u0338= y, is (t 2). Consider a one-factorization F of Kt and a given one-factor Fj, 0 \u2264j \u2264t \u22122, one can form one parallel class {{(x, i), (y, i), (x, i + 1), (y, i + 1)} : {x, y} \u2208Fj, i = 0, 2} , Hence, the t \u22121 one-factors of F induce t \u22121 parallel classes of Type 3. 5.3.2. Type 3 for odd t For odd t a near-one-factorization F = {F0, F1, . . . , Ft\u22121} of Kt is required. Let G1, G2, G3 be three pairwise disjoint sets, each one of size t\u22121 2 t2, of quadruples from con\ufb01guration (1,1,1,1) with the following properties. Each pair {(x, 0), (y, 1)} and each pair {(x, 2), (y, 3)}, x, y \u2208Zt, is contained in exactly t\u22121 2 quadruples of G1. Each pair {(x, 0), (y, 3)} and each pair {(x, 1), (y, 2)}, x, y \u2208Zt, is contained in exactly t\u22121 2 quadruples of G2. Each pair {(x, 0), (y, 2)} and each pair {(x, 1), (y, 3)}, x, y \u2208Zt, is contained in exactly t\u22121 2 quadruples of G3. For a quadruple B = {(i, 0), (j, 1), (k, 2), (m, 3)} of G1, assume that B contains the r-th appearance of the pair {(i, 0), (j, 1)} in G1 and the s-th appearance of the pair {(k, 2), (m, 3)} in G1. De\ufb01ne the set of quadruples which contains B and the quadruples of the following sets. {{(x, 0), (y, 0), (z, 1), (v, 1)} : {x, y} = Fi,p, {z, v} = Fj,p+r, 0 \u2264p \u2264t \u22123 2 } . {{(x, 2), (y, 2), (z, 3), (v, 3)} : {x, y} = Fk,p, {z, v} = Fm,p+s, 0 \u2264p \u2264t \u22123 2 } . For a quadruple B = {(i, 0), (j, 1), (k, 2), (m, 3)} of G2, assume that B contains the r-th appearance of the pair {(i, 0), (m, 3)} in G2 and the s-th appearance of the pair {(j, 1), (k, 2)} in G2. De\ufb01ne the set of quadruples which contains B and the quadruples of the following sets. {{(x, 0), (y, 0), (z, 3), (v, 3)} : {x, y} = Fi,p, {z, v} = Fm,p+r, 0 \u2264p \u2264t \u22123 2 } . {{(x, 1), (y, 1), (z, 2), (v, 2)} : {x, y} = Fj,p, {z, v} = Fk,p+s, 0 \u2264p \u2264t \u22123 2 } . For a quadruple B = {(i, 0), (j, 1), (k, 2), (m, 3)} of G3, assume that B contains the r-th appearance of the pair {(i, 0), (k, 2)} in G3 and the s-th appearance of the pair {(j, 1), (m, 3)} in G3. De\ufb01ne the set of quadruples which contains B and the quadruples of the following sets. {{(x, 0), (y, 0), (z, 2), (v, 2)} : {x, y} = Fi,p, {z, v} = Fk,p+r, 0 \u2264p \u2264t \u22123 2 } . {{(x, 1), (y, 1), (z, 3), (v, 3)} : {x, y} = Fj,p, {z, v} = Fm,p+s, 0 \u2264p \u2264t \u22123 2 } . The following result can be readily veri\ufb01ed. Lemma 14. 1. For each block B = {(i, 0), (j, 1), (k, 2), (m, 3)} in G1, G2 or G3, the t de\ufb01ned quadruples form a parallel class with t \u22121 quadruples from Group 3 and one quadruple from Group 5. 16 \f2. If t is odd, then each quadruple from the con\ufb01gurations of Group 3 appears either in one of the de\ufb01ned parallel classes or in the parallel classes of Type 1. 3. The number of de\ufb01ned parallel classes of Type 3 for odd t is 3t(t 2). Lemma 15. The set of quadruples from Group 3 contained in the parallel classes of Type 1 when t \u2261 1(mod 4), {{(x, i), (y, i), (x, i + 1), (y, i + 1)} : x, y \u2208Zt, i \u2208Z4} , {{(x, i), (y, i), (x, i + 2), (y, i + 2)} : x, y \u2208Zt, i \u2208{0, 1}} , form 3t parallel classes of Type 3. Each one of these 3t parallel classes contains exactly one quadruple from con\ufb01guration (1, 1, 1, 1). Proof. By Lemma 9, when t \u22611(mod 4), the quadruples from Group 3 contained in the parallel classes of Type 1 are {{(x, i), (y, i), (x, i + 1), (y, i + 1)} : x, y \u2208Zt, i \u2208Z4} , {{(x, i), (y, i), (x, i + 2), (y, i + 2)} : x, y \u2208Zt, i \u2208{0, 1}} . Consider a near-one-factorization F of Kt and a given near-one-factor Fj of F, 0 \u2264j \u2264t \u22121, we can form three parallel classes {{(x, 0), (y, 0), (x, 1), (y, 1)}, {(x, 2), (y, 2), (x, 3), (y, 3)} : {x, y} \u2208Fj, } \u222a{{(j, 0), (j, 1), (j, 2), (j, 3)}} {{(x, 0), (y, 0), (x, 2), (y, 2)}, {(x, 1), (y, 1), (x, 3), (y, 3)} : {x, y} \u2208Fj, } \u222a{{(j, 0), (j, 1), (j, 2), (j, 3)}} {{(x, 0), (y, 0), (x, 3), (y, 3)}, {(x, 1), (y, 1), (x, 2), (y, 2)} : {x, y} \u2208Fj, } \u222a{{(j, 0), (j, 1), (j, 2), (j, 3)}} Hence, the related t near-one-factors induce 3t parallel classes of Type 3 and each one of these 3t parallel classes contains exactly one quadruple from con\ufb01guration (1, 1, 1, 1). 5.4. Parallel Classes of Type 4 For Type 4, the con\ufb01gurations of Group 4 are partitioned into six subsets, M1, M2, M3, M4, M5, and M6, where M1 = {(2, 1, 1, 0), (0, 1, 1, 2)}, M2 = {(2, 1, 0, 1), (0, 1, 2, 1)}, M3 = {(2, 0, 1, 1), (0, 2, 1, 1)}, M4 = {(1, 2, 0, 1), (1, 0, 2, 1)}, M5 = {(1, 2, 1, 0), (1, 0, 1, 2)}, and M6 = {(1, 1, 2, 0), (1, 1, 0, 2)}. In the construction we distinguish between even t and odd t. 5.4.1. Type 4 for even t Let F = {F0, F1, . . . , Ft\u22122} be a one-factorization of Kt. For M1, let U1 be a set with (t \u22121)t2 quadruples de\ufb01ned by U1 \u225c{{(x, 0), (y, 1), (z, 2), (x, 3)} : x \u2208Zt\u22121, y, z \u2208Zt}. For each x \u2208Zt\u22121 and each pair {(y, 1), (z, 2)} \u2208Zt \u00d7 {1, 2} there is exactly one quadruple {(x, 0), (y, 1), (z, 2), (x, 3)} \u2208U1 which contains the triple {(x, 0), (y, 1), (z, 2)} and exactly one quadruple {(x, 0), (y, 1), (z, 2), (x, 3)} \u2208U1 which contains the triple {(y, 1), (z, 2), (x, 3)}. 17 \fGiven a quadruple X = {(i, 0), (j, 1), (k, 2), (i, 3)} of the set U1 we form the following set Ri,j,k in P, i \u2208Zt\u22121, j, k \u2208Zt, which contains t quadruples {{(x, 0), (y, 0), (j+ r, 1), (k + r, 2)}, {(x, 3), (y, 3), (j\u2212r \u22121, 1), (k \u2212r \u22121, 2)} : {x, y} = Fi,r, 0 \u2264r \u2264t \u22122 2 }. For Mi, 2 \u2264i \u22646 a set Ui is de\ufb01ned similarly as follows. U2 \u225c{{(x, 0), (y, 1), (x, 2), (z, 3)} : x \u2208Zt\u22121, y, z \u2208Zt}. For each x \u2208Zt\u22121 and each pair {(y, 1), (z, 3)} \u2208Zt \u00d7 {1, 3} there is exactly one quadruple {(x, 0), (y, 1), (x, 2), (z, 3)} \u2208U2 which contains the triple {(x, 0), (y, 1), (z, 3)} and exactly one quadruple {(x, 0), (y, 1), (x, 2), (z, 3)} \u2208U2 which contains the triple {(y, 1), (x, 2), (z, 3)}. U3 \u225c{{(x, 0), (x, 1), (y, 2), (z, 3)} : x \u2208Zt\u22121, y, z \u2208Zt}. For each x \u2208Zt\u22121 and each pair {(y, 2), (z, 3)} \u2208Zt \u00d7 {2, 3} there is exactly one quadruple {(x, 0), (x, 1), (y, 2), (z, 3)} \u2208U3 which contains the triple {(x, 0), (y, 2), (z, 3)} and exactly one quadruple {(x, 0), (x, 1), (y, 2), (z, 3)} \u2208U3 which contains the triple {(x, 1), (y, 2), (z, 3)}. U4 \u225c{{(y, 0), (x, 1), (x, 2), (z, 3)} : x \u2208Zt\u22121, y, z \u2208Zt}. For each x \u2208Zt\u22121 and each pair {(y, 0), (z, 3)} \u2208Zt \u00d7 {0, 3} there is exactly one quadruple {(y, 0), (x, 1), (x, 2), (z, 3)} \u2208U4 which contains the triple {(y, 0), (x, 1), (z, 3)} and exactly one quadruple {(y, 0), (x, 1), (x, 2), (z, 3)} \u2208U4 which contains the triple {(y, 0), (x, 2), (z, 3)}. U5 \u225c{{(y, 0), (x, 1), (z, 2), (x, 3)} : x \u2208Zt\u22121, y, z \u2208Zt}. For each x \u2208Zt\u22121 and each pair {(y, 0), (z, 2)} \u2208Zt \u00d7 {0, 2} there is exactly one quadruple {(y, 0), (x, 1), (z, 2), (x, 3)} \u2208U5 which contains the triple {(y, 0), (x, 1), (z, 2)} and exactly one quadruple {(y, 0), (x, 1), (z, 2), (x, 3)} \u2208U5 which contains the triple {(y, 0), (z, 2), (x, 3)}. U6 \u225c{{(y, 0), (z, 1), (x, 2), (x, 3)} : x \u2208Zt\u22121, y, z \u2208Zt}. For each x \u2208Zt\u22121 and each pair {(y, 0), (z, 1)} \u2208Zt \u00d7 {0, 1} there is exactly one quadruple {(y, 0), (z, 1), (x, 2), (x, 3)} \u2208U6 which contains the triple {(y, 0), (z, 1), (x, 2)} and exactly one quadruple {(y, 0), (z, 1), (x, 2), (x, 3)} \u2208U6 which contains the triple {(y, 0), (z, 1), (x, 3)}. Similarly to the de\ufb01nition of Ri,j,k for a quadruple {(i, 0), (j, 1), (k, 2), (i, 3)} \u2208U1, a related set of quadruple is de\ufb01ned for each element of U2 (also U3, U4, U5, and U6), where the de\ufb01nition is changed related to the positions of the 2\u2019s in the con\ufb01gurations of M2 (M3, M4, M5, and M6, respectively). Lemma 16. 1. Each set Ri,j,k de\ufb01ned by U1 is a parallel class of Type 4. 2. Each quadruple from con\ufb01gurations (2, 1, 1, 0) and (0, 1, 1, 2) is contained in exactly one of the de\ufb01ned parallel classes. 3. The total number of parallel classes of Type 4 (for S6 i=1 Ui) is 6(t \u22121)t2. Each such parallel class contains only quadruples from Group 4. It contains t quadruples from Group 4. 18 \fProof. Claim 1 and 2 can be easily veri\ufb01ed. For Claim 3, given any quadruples X = {(i, 0), (j, 1), (k, 2), (i, 3)} of the set U1 where i \u2208Zt\u22121 and j, k \u2208Zt, one-factor of Fi of Kt is used to form a set Ri,j,k in P. It is easy to verify that Ri,j,k is a parallel class of Type 4 in P. Hence, there are (t \u22121)t2 parallel classes of Type 4 obtained from the set U1. Similarly, for each Ui, 2 \u2264i \u22646, there are (t \u22121)t2 parallel classes of Type 4. Thus, there are 6(t \u22121)t2 parallel classes of Type 4. Example 3. For t = 4 the set U1 contains the following 48 quadruples {(0, 0), (0, 1), (0, 2), (0, 3)}, {(0, 0), (0, 1), (1, 2), (0, 3)}, {(0, 0), (0, 1), (2, 2), (0, 3)}{(0, 0), (0, 1), (3, 2), (0, 3)}, {(0, 0), (1, 1), (0, 2), (0, 3)}, {(0, 0), (1, 1), (1, 2), (0, 3)}, {(0, 0), (1, 1), (2, 2), (0, 3)}{(0, 0), (1, 1), (3, 2), (0, 3)}, {(0, 0), (2, 1), (0, 2), (0, 3)}, {(0, 0), (2, 1), (1, 2), (0, 3)}, {(0, 0), (2, 1), (2, 2), (0, 3)}{(0, 0), (2, 1), (3, 2), (0, 3)}, {(0, 0), (3, 1), (0, 2), (0, 3)}, {(0, 0), (3, 1), (1, 2), (0, 3)}, {(0, 0), (3, 1), (2, 2), (0, 3)}{(0, 0), (3, 1), (3, 2), (0, 3)}, {(1, 0), (0, 1), (0, 2), (1, 3)}, {(1, 0), (0, 1), (1, 2), (1, 3)}, {(1, 0), (0, 1), (2, 2), (1, 3)}{(1, 0), (0, 1), (3, 2), (1, 3)}, {(1, 0), (1, 1), (0, 2), (1, 3)}, {(1, 0), (1, 1), (1, 2), (1, 3)}, {(1, 0), (1, 1), (2, 2), (1, 3)}{(1, 0), (1, 1), (3, 2), (1, 3)}, {(1, 0), (2, 1), (0, 2), (1, 3)}, {(1, 0), (2, 1), (1, 2), (1, 3)}, {(1, 0), (2, 1), (2, 2), (1, 3)}{(1, 0), (2, 1), (3, 2), (1, 3)}, {(1, 0), (3, 1), (0, 2), (1, 3)}, {(1, 0), (3, 1), (1, 2), (1, 3)}, {(1, 0), (3, 1), (2, 2), (1, 3)}{(1, 0), (3, 1), (3, 2), (1, 3)}, {(2, 0), (0, 1), (0, 2), (2, 3)}, {(2, 0), (0, 1), (1, 2), (2, 3)}, {(2, 0), (0, 1), (2, 2), (2, 3)}{(2, 0), (0, 1), (3, 2), (2, 3)}, {(2, 0), (1, 1), (0, 2), (2, 3)}, {(2, 0), (1, 1), (1, 2), (2, 3)}, {(2, 0), (1, 1), (2, 2), (2, 3)}{(2, 0), (1, 1), (3, 2), (2, 3)}, {(2, 0), (2, 1), (0, 2), (2, 3)}, {(2, 0), (2, 1), (1, 2), (2, 3)}, {(2, 0), (2, 1), (2, 2), (2, 3)}{(2, 0), (2, 1), (3, 2), (2, 3)}, {(2, 0), (3, 1), (0, 2), (2, 3)}, {(2, 0), (3, 1), (1, 2), (2, 3)}, {(2, 0), (3, 1), (2, 2), (2, 3)}{(2, 0), (3, 1), (3, 2), (2, 3)}. The \ufb01rst 8 quadruples of U1 with the one-factorization F = {F0, F1, F2}, where F0 = {F0,0 = {1, 2}, F0,1 = {0, 3}}, F1 = {F1,0 = {0, 2}, F1,1 = {1, 3}}, F2 = {F2,0 = {0, 1}, F2,1 = {2, 3}}, yield the following 8 parallel classes. From the quadruple {(0, 0), (0, 1), (0, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (0, 1), (0, 2)}, {(0, 0), (3, 0), (1, 1), (1, 2)}, {(1, 3), (2, 3), (3, 1), (3, 2)}, {(0, 3), (3, 3), (2, 1), (2, 2)}}. From the quadruple {(0, 0), (0, 1), (1, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (0, 1), (1, 2)}, {(0, 0), (3, 0), (1, 1), (2, 2)}, {(1, 3), (2, 3), (3, 1), (0, 2)}, {(0, 3), (3, 3), (2, 1), (3, 2)}}. From the quadruple {(0, 0), (0, 1), (2, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (0, 1), (2, 2)}, {(0, 0), (3, 0), (1, 1), (3, 2)}, {(1, 3), (2, 3), (3, 1), (1, 2)}, {(0, 3), (3, 3), (2, 1), (0, 2)}}. From the quadruple {(0, 0), (0, 1), (3, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (0, 1), (3, 2)}, {(0, 0), (3, 0), (1, 1), (0, 2)}, {(1, 3), (2, 3), (3, 1), (2, 2)}, {(0, 3), (3, 3), (2, 1), (1, 2)}}. From the quadruple {(0, 0), (1, 1), (0, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (1, 1), (0, 2)}, {(0, 0), (3, 0), (2, 1), (1, 2)}, {(1, 3), (2, 3), (0, 1), (3, 2)}, {(0, 3), (3, 3), (3, 1), (2, 2)}}. From the quadruple {(0, 0), (1, 1), (1, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (1, 1), (1, 2)}, {(0, 0), (3, 0), (2, 1), (2, 2)}, {(1, 3), (2, 3), (0, 1), (0, 2)}, {(0, 3), (3, 3), (3, 1), (3, 2)}}. From the quadruple {(0, 0), (1, 1), (2, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (1, 1), (2, 2)}, {(0, 0), (3, 0), (2, 1), (3, 2)}, {(1, 3), (2, 3), (0, 1), (1, 2)}, {(0, 3), (3, 3), (3, 1), (0, 2)}}. From the quadruple {(0, 0), (1, 1), (3, 2), (0, 3)} we form the parallel class {{(1, 0), (2, 0), (1, 1), (3, 2)}, {(0, 0), (3, 0), (2, 1), (0, 2)}, {(1, 3), (2, 3), (0, 1), (2, 2)}, {(0, 3), (3, 3), (3, 1), (1, 2)}}. 19 \f5.4.2. Type 4 for odd t Let F = {F0, F1, . . . , Ft\u22121} be a near-one-factorization of Kt and let {Hi : 1 \u2264i \u22646} be a set with six pairwise disjoint subsets from Group 5, i.e. con\ufb01guration (1,1,1,1), where the subset Hi is associated with the subset Mi, for 1 \u2264i \u22646. For example H1 is a set with t3 quadruples de\ufb01ned by H1 \u225c{{(i, 0), (j, 1), (k, 2), (i + j + k, 3)} : i, j, k \u2208Zt}. For each i, j, k \u2208Zt there is exactly one quadruple {(i, 0), (j, 1), (k, 2), (\u2113, 3)} \u2208H1 which contains the triple {(i, 0), (j, 1), (k, 2)} and exactly one quadruple {(i, 0), (j, 1), (j, 2), (\u2113, 3)} \u2208H1 which contains the triple {(j, 1), (k, 2), (\u2113, 3)}. Related subsets Hi\u2019s of quadruples are de\ufb01ned for each Mi, for each 2 \u2264i \u22646. Remark 2. The Hi\u2019s have similar properties as the Ui\u2019s. But, the Hi\u2019s are pairwise disjoint, while the Ui\u2019s might not be disjoint as they were de\ufb01ned. Given a quadruple X = {(i, 0), (j, 1), (k, 2), (\u2113, 3)} of the subset H1 we form a set Ri,j,k in P which contains X and the following set with t \u22121 quadruples {{(x, 0), (y, 0), (j + r + 1, 1), (k + r + 1, 2)} : {x, y} = Fi,r, 0 \u2264r \u2264t \u22123 2 } [ {{(j \u2212r \u22121, 1), (k \u2212r \u22121, 2), (x, 3), (y, 3)} : {x, y} = F\u2113,r, 0 \u2264r \u2264t \u22123 2 }. Related subsets of quadruples are de\ufb01ned for each Hi and Mi, for each 2 \u2264i \u22646. The following lemma has a proof similar to the one of Lemma 16. Lemma 17. 1. Each set Ri,j,k de\ufb01ned by H1 is a parallel class of Type 4. 2. Each quadruple from con\ufb01gurations (2, 1, 1, 0) and (0, 1, 1, 2) is contained in exactly one of the de\ufb01ned parallel classes. 3. The total number of parallel classes of Type 4 (for S6 i=1 Hi) is 6t3. Each such parallel class contains t \u22121 quadruples from Group 4 and one quadruple from Group 5. 5.5. Parallel Classes of Type 5 Type 5 contains quadruples only from Group 5 which has the unique con\ufb01guration (1, 1, 1, 1). There are t4 quadruples in this con\ufb01guration. Each parallel class contains t quadruples and hence there are t3 possible pairwise disjoint parallel classes which contain all the quadruples from Group 5. They can be easily constructed in a few ways. But, some of them might have been used with quadruples from other con\ufb01gurations in Type 1, Type 3, or Type 4. Hence, our construction will take into account the quadruples which are used with the con\ufb01gurations in other types. We start with the following four subsets of quadruples from con\ufb01guration (1, 1, 1, 1). A \u225c{{(i, 0), (i, 1), (i, 2), (i, 3)} : i \u2208Zt}, 20 \fB \u225c{{(i, 0), (0, 1), (i, 2), (0, 3)} : i \u2208Zt}, C \u225c{{(i, 0), (i, 1), (0, 2), (0, 3)} : i \u2208Zt}, D \u225c{{(i, 0), (0, 1), (0, 2), (0, 3)} : i \u2208Zt}. For X = {(x0, 0), (x1, 1), (x2, 2), (x3, 3)} and Y = {(y0, 0), (y1, 1), (y2, 2), (y3, 3)} the addition X + Y is de\ufb01ned by X + Y \u225c{(x0 + y0, 0), (x1 + y1, 1), (x2 + y2, 2), (x3 + y3, 3)}. Recall that the computation in the \ufb01rst coordinate is performed modulo t. For two subsets of quadruples S1 and S2 from con\ufb01guration (1, 1, 1, 1), the addition S1 + S2 is de\ufb01ned by S1 + S2 \u225c{X + Y : X \u2208S1, Y \u2208S2}. Lemma 18. The set of quadruples A + B + C + D = {X + Y + Z + V : X \u2208A, Y \u2208B, Z \u2208C, V \u2208D} consists of all the t4 quadruples of Zt \u00d7 Z4 from con\ufb01guration (1, 1, 1, 1). Proof. First, assume for the contrary that two quadruples of Zt \u00d7 Z4 are generated in two different ways in A + B + C + D. Assume that there exist eight parameters i1, i2, i3, i4, and j1, j2, j3, j4 such that (i1, i2, i3, i4) \u0338= (j1, j2, j3, j4) and the two quadruples {(i1, 0), (i1, 1), (i1, 2), (i1, 3)} + {(i2, 0), (0, 1), (i2, 2), (0, 3)} + {(i3, 0), (i3, 1), (0, 2), (0, 3)} + {(i4, 0), (0, 1), (0, 2), (0, 3)} and {(j1, 0), (j1, 1), (j1, 2), (j1, 3)} + {(j2, 0), (0, 1), (j2, 2), (0, 3)} + {(j3, 0), (j3, 1), (0, 2), (0, 3)} + {(j4, 0), (0, 1), (0, 2), (0, 3)}, are equal. This implies that i1 + i2 + i3 + i4 \u2261j1 + j2 + j3 + j4 (mod t), i1 + i3 \u2261j1 + j3 (mod t), i1 + i2 \u2261j1 + j2 (mod t), i1 \u2261j1 (mod t). This set of equations has exactly one solution which is (i1, i2, i3, i4) = (j1, j2, j3, j4), a contradiction. Hence, all the t4 quadruples formed by X + Y + Z + V, where X \u2208A, Y \u2208B, Z \u2208C, and V \u2208D are distinct, which implies the claim of the lemma. The following lemma can be easily veri\ufb01ed. Lemma 19. The set of quadruples in A forms a parallel class and for any quadruple X from con\ufb01guration (1, 1, 1, 1) the set X + A \u225c{X + Y : Y \u2208A} of t quadruples forms a parallel class from con\ufb01guration (1, 1, 1, 1). 21 \fWe are now in position to de\ufb01ne the parallel classes of Type 5, i.e. the parallel classes with quadruples only from con\ufb01guration (1, 1, 1, 1). For even t, this is rather simple since there are no quadruples from con\ufb01guration (1, 1, 1, 1) in Types 1, 2, 3, and 4. Since by Lemma 18, A + B + C + D contains all the quadruples from con\ufb01guration (1, 1, 1, 1), it follows from Lemma 19 that for each quadruples X \u2208 B + C + D, the set X + A is a parallel class. This implies that Lemma 20. For even t, the set of quadruples from con\ufb01guration (1, 1, 1, 1) can be partitioned into t3 parallel classes, each one of size t. 5.5.1. Type 5 for odd t The situation is more complicated in the de\ufb01nition of the quadruples of Type 5 for odd t. We partition the t4 quadruples from con\ufb01guration (1, 1, 1, 1) into \ufb01ve subsets, L1, L2, L3, L4, L5, as follows. L1 \u225c{{(i, 0), (0, 1), (0, 2), (0, 3)} : 1 \u2264i \u22646} + A + B + C and it is partitioned into six subsets H1, H2, H3, H4, H5, H6, where Hi \u225c{(i, 0), (0, 1), (0, 2), (0, 3)} + A + B + C, 1 \u2264i \u22646. L2 \u225c{{(i, 0), (i, 1), (0, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A + B, L3 \u225c{{(i, 0), (i, 1), (0, 2), (0, 3)} : t + 1 2 \u2264i \u2264t \u22121} + A + B, L4 \u225c{{(i, 0), (0, 1), (0, 2), (0, 3)} : 7 \u2264i \u2264t + 11 2 } + A + C, L5 \u225c(A + B + C + D) \\ 4 [ i=1 Li. Lemma 21. 1. The subsets L1, L2, L3, L4, and L5 are pairwise disjoint. 2. The subset L5 can be partitioned into parallel classes. Proof. By Lemma 18 we have that Zt \u00d7 Z4 = A + B + C + D = {{(i1, 0), (i2, 1), (i3, 2), (i4, 3)} : 0 \u2264i1, i2, i3, i4 \u2264t \u22121}. (1) It is easy to verify that each one of quadruples in Zt \u00d7 Z4 is contained in exactly one of the four subsets of quadruples A + B + C, L1, A + B + C + D1, A + B + C + D2, (2) where D1 \u225c{{(i, 0), (0, 1), (0, 2), (0, 3)} : 7 \u2264i \u2264t+11 2 }, D2 \u225c{{(i, 0), (0, 1), (0, 2), (0, 3)} : t+13 2 \u2264 i \u2264t \u22121}, and as de\ufb01ned before L1 = {{(i, 0), (0, 1), (0, 2), (0, 3)} : 1 \u2264i \u22646} + A + B + C. 22 \fSimilarly by the de\ufb01nition of L2 and L3 we have that A + B + C = (A + B) \u222aL2 \u222aL3 . (3) Similar arguments and the de\ufb01nitions of D1 and L4 imply that A + B + C + D1 = L4 \u222a(A + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121} + C + D1). (4) Therefore, by Equations (1), (2), (3), and (4), we have L5 = Zt \u00d7 Z4 \\ (L1 \u222aL2 \u222aL3 \u222aL4) = (A + B) \u222a(A + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121} + C + D1) \u222a(A + B + C + D2). Hence L5 = S + A for some subset S of quadruples from con\ufb01guration (1, 1, 1, 1) which implies by Lemma 19 that L5 can be partitioned into pairwise disjoint parallel classes. Lemma 22. For any integer r, the subset S \u225c{(r, 0), (0, 1), (0, 2), (0, 3)} + A + B + C of con\ufb01guration (1, 1, 1, 1) contains t3 quadruples and each triple {(x1, i), (x2, j), (x3, k)}, where |{i, j, k}| = 3 is contained in exactly one quadruple of S. Proof. Consider the matrix G = \uf8ee \uf8f0 1 1 1 1 1 0 1 0 1 1 0 0 \uf8f9 \uf8fb, and the set of codewords in the code \u02c6 C \u225c{(i1, i2, i3)G : i1, i2, i3 \u2208Zt}. Note that A + B + C = {{(j0, 0), (j1, 1), (j2, 2), (j3, 3)} : (j0, j1, j2, j3) \u2208\u02c6 C}. The claim follows now from the observation that the matrix de\ufb01ned by any three columns of G is nonsingular. Corollary 1. The subsets H1, H2, H3, H4, H5, H6 satisfy the properties required for Type 4 when t is odd. These subsets of quadruples from Group 5 contained in the parallel classes of Type 4 form 6t2 parallel classes de\ufb01ned for Type 5 (con\ufb01guration (1, 1, 1, 1)). Lemma 23. 1. Each pair {(y, 0), (z, 1)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L2. Each pair {(y, 2), (z, 3)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L2. 2. Each pair {(y, 0), (z, 3)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L3. Each pair {(y, 1), (z, 2)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L3. 3. Each pair {(y, 0), (z, 2)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L4. Each pair {(y, 1), (z, 3)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L4. Proof. First note that in A there are t quadruples and for any given x \u2208Zt and i \u2208Z4, the element (x, i) is contained in exactly one of the quadruples of A. In B each pair {(y, 0), (0, 1)} is contained in exactly one of the quadruples. Hence, each pair {(y, 0), (z, 1)}, y, z \u2208Zt, is contained in exactly one quadruple of A + B. Therefore, each pair {(y, 0), (z, 1)}, y, z \u2208Zt, is also contained in exactly one quadruple of {(j, 0), (j, 1), (0, 2), (0, 3)} + A + B, for any j \u2208Zt. Thus, each pair {(y, 0), (z, 1)}, y, z \u2208Zt, is contained in exactly t\u22121 2 times in the quadruples of L2. The other claims in the lemma are proved in exactly the same way. 23 \fCorollary 2. The subset L2 has the properties required for G1 in Type 3 for odd t, the subset L3 has the properties required for G2 in Type 3 for odd t, and the subset L4 has the properties required for G3 in Type 3 for odd t. These subsets of quadruples from Group 5 contained in the parallel classes of Type 3 form 3(t 2) parallel classes of con\ufb01guration (1, 1, 1, 1). Remark 3. The de\ufb01nitions of D1 and D2 given in Lemma 21 imply that the construction for odd t works only if (t + 13)/2 \u2264t \u22121 which implies that t \u226515. 6. Proof of the Main Result In this section we will prove Theorem 2, i.e., the existence of a quadrupling construction whenever t \u22610, 3, 4, 6, 8, 9 (mod 12). For each residue modulo 12 a separate proof will be given. Case 1: t \u22610 (mod 12). In this case the parallel classes are taken exactly as de\ufb01ned in Section 5. By Lemma 4 there are (t\u22121 3 ) parallel classes of Type 1. By Lemma 11 there are 2t(t \u22121)(t \u22122) parallel classes of Type 2. By Lemma 12 there are 3(t \u22121)(t 2) parallel classes of Type 3. By Lemma 16 there are 6(t \u22121)t2 parallel classes of Type 4. By Lemma 20 there are t3 parallel classes of Type 5. All these parallel classes together imply that there are a total of \u0012t \u22121 3 \u0013 + 2t(t \u22121)(t \u22122) + 3(t \u22121) \u0012t 2 \u0013 + 6(t \u22121)t2 + t3 = \u00124t \u22121 3 \u0013 parallel classes as required for BP(4t, 4). Thus, if there exist a BP(t, 4) and a BP(t, 3), then there exists a BP(4t, 4). Case 2: t \u22613 (mod 12). In this case the parallel classes are taken exactly as de\ufb01ned in Section 5, except for the parallel classes of Type 5. By Lemma 6 there are (t 3) parallel classes of Type 1. By Lemma 11 there are (4t \u22121)(t\u22121 2 ) parallel classes of Type 2. Lemma 5, Lemma 6, and Lemma 11, imply that each quadruple from Group 1 and Group 2 is contained in exactly one of the parallel classes of Type 1 or Type 2. By Lemma 14 there are 3t(t 2) parallel classes of Type 3. In these 3t(t 2) parallel classes there are 3t(t 2) quadruples from Group 5. By Lemma 17 there are 6t3 parallel classes of Type 4. In these 6t3 parallel classes there are 6t3 quadruples from Group 5. By Lemma 14, Lemma 17, Lemma 21, Corollary 1, and Corollary 2, there are t3 \u22123(t 2) \u22126t2 parallel classes of Type 5. All these parallel classes together imply that there are a total of \u0012t 3 \u0013 + (4t \u22121) \u0012t \u22121 2 \u0013 + 3t \u0012t 2 \u0013 + 6t3 + t3 \u22123 \u0012t 2 \u0013 \u22126t2 = \u00124t \u22121 3 \u0013 parallel classes as required for BP(4t, 4). Thus, if there exist a BP(t + 1, 4) and a BP(t, 3), then there exists a BP(4t, 4). Case 3: t \u22614 (mod 12). By Theorem 5, if there exists a BP(t, 4) then there exists a BP(2t, 4), and since 2t \u22618 (mod 12), Theorem 5 also implies that there exists a BP(4t, 4). Case 4: t \u22616 (mod 12). In this case the parallel classes are taken exactly as de\ufb01ned in Section 5, except for the parallel classes of Type 3. By Lemma 8 there are (t+1 3 ) parallel classes of Type 1. By Lemma 11 there are (2t \u22121)(t \u22121)(t \u22122) parallel classes of Type 2. Lemma 7, Lemma 8, and Lemma 11, imply that each 24 \fquadruple from Group 1 and Group 2 is contained in exactly one of the parallel classes of Type 1 or Type 2. By Lemma 12 there are 3(t \u22121)(t 2) parallel classes of Type 3, from which by Lemma 13, the quadruples of t \u22121 parallel classes are already contained in the parallel classes of Type 1. By Lemma 16 there are 6(t \u22121)t2 parallel classes of Type 4. By Lemma 20 there are t3 parallel classes of Type 5. All these parallel classes together imply that there are a total of \u0012t + 1 3 \u0013 + (2t \u22121)(t \u22121)(t \u22122) + 3(t \u22121) \u0012t 2 \u0013 \u2212(t \u22121) + 6(t \u22121)t2 + t3 = \u00124t \u22121 3 \u0013 parallel classes as required for BP(4t, 4). Thus, if there exist a BP(t + 2, 4) and a BP(t, 3), then there exists a BP(4t, 4). Case 5: t \u22618 (mod 12). By Theorem 5, if there exists a BP(t, 4) then there exists a BP(2t, 4), and since 2t \u22614 (mod 12), Theorem 5 also implies that there exists a BP(4t, 4). Case 6: t \u22619 (mod 12). This case is the most complicated one. For this case we will have to change the de\ufb01nitions of the parallel classes of Type 5 for odd t. More precisely, we will have to change the de\ufb01nition of the partition of the t4 quadruples from con\ufb01guration (1, 1, 1, 1) into \ufb01ve subsets. We de\ufb01ne the following \ufb01ve subsets, where A, B, C, and D are de\ufb01ned as in Section 5.5. Remark 4. The following de\ufb01nition can also be used for other values of odd t. But, the previous de\ufb01nition is much simpler to verify and can also be seen as an introduction for the more complicated de\ufb01ntion considered in the next few paragraphs. Partition the t4 quadruples from con\ufb01guration (1, 1, 1, 1) into \ufb01ve subsets, L\u2032 1, L\u2032 2, L\u2032 3, L\u2032 4, L\u2032 5, as follows. L\u2032 1 \u225c{{(i, 0), (0, 1), (0, 2), (0, 3)} : 1 \u2264i \u22646} + A + B + C and it is partitioned into six subsets H1, H2, H3, H4, H5, H6, where Hi \u225c{(i, 0), (0, 1), (0, 2), (0, 3)} + A + B + C, 1 \u2264i \u22646. L\u2032 2 \u225c{{(i, 0), (i, 1), (0, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A + B, Remark 5. These de\ufb01nitions are the same one as were de\ufb01ned in Section 5.5.1. L\u2032 3 \u225c{(\u22121, 0), (0, 1), (0, 2), (0, 3)} + {{(i, 0), (i, 1), (0, 2), (0, 3)} : t + 1 2 \u2264i \u2264t \u22121} + A + B, L\u2032 4 \u225c({{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A) \u222a{(7, 0), (0, 1), (0, 2), (0, 3)} + {{(i, 0), (i, 1), (0, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 and i \u0338= \u22127} +{{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A 25 \f\u222a{(7, 0), (\u22127, 1), (0, 2), (0, 3)} + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A L\u2032 5 \u225c(A + B + C + D) \\ 4 [ i=1 L\u2032 i. Lemma 24. 1. The subsets L\u2032 1, L\u2032 2, L\u2032 3, L\u2032 4, and L\u2032 5 are pairwise disjoint. 2. The subset L\u2032 5 can be partitioned into parallel classes. Proof. By Lemma 18 we have that Zt \u00d7 Z4 = A + B + C + D = {{(i1, 0), (i2, 1), (i3, 2), (i4, 3)} : 0 \u2264i1, i2, i3, i4 \u2264t \u22121}. (5) First, we have to show that the four subsets L\u2032 1, L\u2032 2, L\u2032 3, and L\u2032 4 are pairwise disjoint. To see that note that L\u2032 1 = {{(i, 0), (0, 1), (0, 2), (0, 3)} : 1 \u2264i \u22646} + A + B + C, L\u2032 2 = {(0, 0), (0, 1), (0, 2), (0, 3)} + A + B + {{(i, 0), (i, 1), (0, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 }, L\u2032 3 = {(\u22121, 0), (0, 1), (0, 2), (0, 3)} + A + B + {{(i, 0), (i, 1), (0, 2), (0, 3)} : t + 1 2 \u2264i \u2264t \u22121}, L\u2032 4 = D1 \u222aD2 \u222aD3, where D1 \u2286{(0, 0), (0, 1), (0, 2), (0, 3)} + A + B + {(0, 0), (0, 1), (0, 2), (0, 3)}, D2 \u2286{(7, 0), (0, 1), (0, 2), (0, 3)} + A + B + {{(i, 0), (i, 1), (0, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 and i \u0338= \u22127}, D3 \u2286{(14, 0), (0, 1), (0, 2), (0, 3)} + A + B + {(\u22127, 0), (\u22127, 1), (0, 2), (0, 3)}. With this representation it is readily veri\ufb01ed that these four subsets are pairwise disjoint. It follows that the \ufb01ve subsets L\u2032 1, L\u2032 2, L\u2032 3, L\u2032 4, and L\u2032 5 are pairwise disjoint. Now, it is readily veri\ufb01ed that each subset L\u2032 1, L\u2032 2, L\u2032 3, L\u2032 4, and L\u2032 5 can be written as S + A for some subset S of quadruples from con\ufb01guration (1, 1, 1, 1) which implies by Lemma 19 that L\u2032 5 can be partitioned into pairwise disjoint parallel classes. The structure of the quadruples in L\u2032 1, L\u2032 2, L\u2032 3, and L\u2032 4 implies the following result. Corollary 3. The quadruples of the subset A are contained in L\u2032 5 and form the quadruples from con\ufb01guration (1, 1, 1, 1) which are used in Type 1 when t \u22611 (mod 4). Lemma 25. 1. Each pair {(y, 0), (z, 1)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L\u2032 2. Each pair {(y, 2), (z, 3)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L\u2032 2. 2. Each pair {(y, 0), (z, 3)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L\u2032 3. Each pair {(y, 1), (z, 2)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L\u2032 3. 26 \f3. Each pair {(y, 0), (z, 2)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L\u2032 4. Each pair {(y, 1), (z, 3)}, y, z \u2208Zt, is contained exactly t\u22121 2 times in the quadruples of L\u2032 4. Proof. The proof for L\u2032 2 and L\u2032 3 is exactly as in the proof of Lemma 23. Hence, we concentrate in the proof of the claim for L\u2032 4. First note that in A there are t quadruples and for any given x \u2208Zt and i \u2208Z4, the element (x, i) is contained in exactly one of the quadruples of A. Next, we consider the subset {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A. Each pair {(i, 0), (i, 2)} and each pair {(j, 1), (j, 3)}, where i, j \u2208Zt, is contained in exactly t\u22121 2 quadruples of this subset. Consider now the subset {{(j, 0), (j, 1), (0, 2), (0, 3)} : j \u2208Zt} + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A. In this subset each pair {(i, 0), (j, 2)}, i, j \u2208Zt, and each pair {(k, 1), (m, 3)}, k, m \u2208Zt, is contained in exactly t\u22121 2 quadruples. The same is true for the subset {(7, 0), (0, 1), (0, 2), (0, 3)} + {{(j, 0), (j, 1), (0, 2), (0, 3)} : j \u2208Zt} +{{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A. Consider now the two subsets {(7, 0), (0, 1), (0, 2), (0, 3)} + {(0, 0), (0, 1), (0, 2), (0, 3)} + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A and {(7, 0), (0, 1), (0, 2), (0, 3)} + {(\u22127, 0), (\u22127, 1), (0, 2), (0, 3)} + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A. These two subsets contain the same pairs {(i, 0), (j, 2)}, {(k, 1), (m, 3)}, i, j, k, m \u2208Zt, and their multiplicity as in the two subsets {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A and {(7, 0), (\u22127, 1), (0, 2), (0, 3)} + {{(i, 0), (0, 1), (i, 2), (0, 3)} : 1 \u2264i \u2264t \u22121 2 } + A. Thus, each pair {(y, 0), (z, 2)}, y, z \u2208Zt, is contained in exactly t\u22121 2 times in the quadruples of L\u2032 4. Similarly, each pair {(y, 1), (z, 3)}, y, z \u2208Zt, is contained in exactly t\u22121 2 times in the quadruples of L\u2032 4. Corollary 4. The subset L\u2032 2 has the properties required for G1 in Type 3 for odd t, the subset L\u2032 3 has the properties required for G2 in Type 3 for odd t, and the subset L\u2032 4 has the properties required for G3 in Type 3 for odd t. These subsets form 3(t 2) parallel classes of con\ufb01guration (1, 1, 1, 1). 27 \fLemma 26. 1. In L\u2032 2 there exists a subset S with exactly t quadruples and the following properties. The quadruples of S have the form {(i, 0), (i, 1), (j, 2), (j, 3)}. For each i \u2208Zt, S has exactly one quadruple of the form {(i, 0), (i, 1), (j, 2), (j, 3)}. For each j \u2208Zt, S has exactly one quadruple of the form {(i, 0), (i, 1), (j, 2), (j, 3)}. 2. In L\u2032 3 there exists a subset S with exactly t quadruples and the following properties. The quadruples of S have the form {(i, 0), (j, 1), (j, 2), (i, 3)}. For each i \u2208Zt, S has exactly one quadruple of the form {(i, 0), (j, 1), (j, 2), (i, 3)}. For each j \u2208Zt, S has exactly one quadruple of the form {(i, 0), (j, 1), (j, 2), (i, 3)}. 3. In L\u2032 4 there exists a subset S with exactly t quadruples and the following properties. The quadruples of S have the form {(i, 0), (j, 1), (i, 2), (j, 3)}. For each i \u2208Zt, S has exactly one quadruple of the form {(i, 0), (j, 1), (i, 2), (j, 3)}. For each j \u2208Zt, S has exactly one quadruple of the form {(i, 0), (j, 1), (i, 2), (j, 3)}. Proof. We de\ufb01ne the subset S for each one of the subsets L\u2032 2, L\u2032 3, and L\u2032 4. For L\u2032 2 let S \u225c{(1, 0), (1, 1), (0, 2), (0, 3)} + A. For L\u2032 3 let S \u225c{(\u22121, 0), (0, 1), (0, 2), (0, 3)} + {(t + 1 2 , 0), (t + 1 2 , 1), (0, 2), (0, 3)} +{(t + 1 2 , 0), (0, 1), (t + 1 2 , 2), (0, 3)} + A = {(0, 0), (t + 1 2 , 1), (t + 1 2 , 2), (0, 3)} + A. For L\u2032 4 let S \u225c{(1, 0), (0, 1), (1, 2), (0, 3)} + A. It is readily veri\ufb01ed that each subset S is contained in the related set L\u2032 2, L\u2032 3, and L\u2032 4, and also S satisfy the requirements of the claim. Corollary 5. The subsets L\u2032 2, L\u2032 3, and L\u2032 4, without the subset S (of each one separately), satisfy the properties required for Type 3 when t \u22611 (mod 4). These subsets of quadruples from Group 5 which are used in Type 3 form 3(t 2) \u22123 parallel classes of con\ufb01guration (1, 1, 1, 1) de\ufb01ned in Type 5. We continue with this case, where the parallel classes are taken as de\ufb01ned in Section 5, with several exceptions. These exceptions occur since quadruples from Groups 3 and 5 are used in Type 1. Furthermore, quadruples of Group 5 are also used in Type 3 and in Type 4. Hence, our enumeration should be made carefully. By Lemma 10 there are (t+2 3 ) parallel classes of Type 1. By Lemma 9 these parallel classes contain 3(t \u22121)t quadruples from Group 3 and t quadruples from Group 5. By Lemma 11 we have (4t \u22123)(t\u22121 2 ) parallel classes of Type 2. Lemma 9, Lemma 10, and Lemma 11, imply that each quadruple from Group 1 and Group 2 is contained in exactly one of the parallel classes of Type 1 or Type 2. By Lemma 14 there are 3t(t 2) parallel classes of Type 3. By Lemma 15, the 3(t \u22121)t quadruples of 3t parallel classes were already used in Type 1. By Lemma 14 each one of these 3t(t 2) \u22123t parallel classes contains one quadruple from Group 5. By Lemma 17 there are 6t3 parallel classes of Type 4 and 28 \feach one contains one quadruple from Group 5. There are t3 parallel classes of Type 5, from which by Corollary 3 implies that one parallel class is used in Type 1. Combining this with Lemma 24, Corollary 1, and Corollary 5, imply that there are t3 \u22121 \u22123(t 2) + 3 \u22126t2 parallel classes of Type 5. Taking all these parallel classes together, there are a total of \u0012t + 2 3 \u0013 + (4t \u22123) \u0012t \u22121 2 \u0013 + 3t \u0012t 2 \u0013 \u22123t + 6t3 + t3 \u22121 \u22123 \u0012t 2 \u0013 + 3 \u22126t2 = \u00124t \u22121 3 \u0013 parallel classes as required. Thus, if there exist a BP(t + 3, 4) and a BP(t, 3), then there exists a BP(4t, 4). The last part of the proof is to give some required initial conditions for our recursive quadrupling constructions. The only part of our proof that required some initial values for t is for t \u22613 which works only from t \u226515. Hence, an initial condition is required are for BP(12, 4). There are several ways to construct BP(12, 4). These constructions are left as an exercise for the interested reader. Thus, we have completed the proof of Theorem 2. 7. Ef\ufb01ciency of the Constructions Recall the general de\ufb01nition of the Baranyai partition problem. Let X be a \ufb01nite set of size n and k be an integer, where 0 \u2264k \u2264|X|. We are interested in a BP(n, k), P1 \u2294P2 \u2294\u00b7 \u00b7 \u00b7 \u2294P(n\u22121 k\u22121), where Pi is a parallel class and \u2294is used to denote disjoint union. Such a partition is represented by an (n/k) \u00d7 (n\u22121 k\u22121) matrix M, where the i-th column of M contains the n/k blocks in Pi. We call M an (n, k)-Baranyai matrix. There are several algorithmic problems which can be considered for Baranyai partitions. LISTING PROBLEM: Given n and k, output an (n, k)-Baranyai matrix. COLUMN GENERATION PROBLEM: Given n, k, and 1 \u2264i \u2264(n\u22121 k\u22121), output the i-th column, COL(n, k, i), of an (n, k)-Baranyai matrix. ENTRY PROBLEM: Given n, k, 1 \u2264i \u2264(n\u22121 k\u22121), and 1 \u2264j \u2264n/k, output the (i, j)-th entry, ENT(n, k, i, j), of an (n, k)-Baranyai matrix. It is clear that the computational complexity is highest for the LISTING PROBLEM, followed by the COLUMN GENERATION PROBLEM, and then the ENTRY PROBLEM. But we also have LISTING PROBLEM \u2264P COLUMN GENERATION PROBLEM \u2264P ENTRY PROBLEM, where \u2264P denotes polynomial-time Turing reduction. For general \ufb01xed k, linear-time algorithms (in the number of k-subsets) for the LISTING PROBLEM is not known. The known proofs of Baranyai\u2019s theorem requires solving network \ufb02ow problems and the implementation for Baranyai partition is of a high degree polynomial (O(n3k\u22122) for the implementation presented in [3]; see also [19, pp. 536 \u2013 542]). The COLUMN GENERATION PROBLEM and the ENTRY PROBLEM were not considered in previous works and it is unclear how the network \ufb02ow algorithm can provide more ef\ufb01cient solutions these problems. Our methods are more ef\ufb01cient than the solutions via the network \ufb02ow techniques for these problems. Our solution to the LISTING PROBLEM problem is linear in the number of 4-subsets, 29 \fi.e. its complexity is O(n4) compared to O(n10) using network \ufb02ow. Our solution to the COLUMN GENERATION PROBLEM has complexity O(n) and the same for the ENTRY PROBLEM. The ef\ufb01ciency of our constructions is based on the enumerative coding of Cover [6]. We demonstrate the enumerative coding on a BP(2n, 4) for any n \u22652. This Baranyai\u2019s partition has (2n\u22121 3 ) parallel classes, each one with 2n\u22122 quadruples. The construction of this Baranyai\u2019s partition is presented and analysed in Section 3. The construction has three types of parallel classes, Type S, Type T , and Type F. In Type S there are 8(2n\u22121\u22121 3 ) parallel classess, in Type T there are 4(2n\u22121 \u22121)(2n\u22121 \u22122) parallel classes, and in Type F there are 2n\u22121 \u22121 parallel classes. These numbers of parallel classes will be taken into account in the enumerative coding. We distinguish now between the three algorithmic problems. The COLUMN GENERATION PROBLEM: Assume we look for COL(2n, 4, \u03b2). We distinguish between three cases. 1. 1 \u2264\u03b2 \u22648(2n\u22121\u22121 3 ). In this case the parallel class is of Type S (see Section 3.1). For this type the recursion generates eight parallel classes for each parallel class of BP(2n\u22121, 4). Hence, we compute COL(2n\u22121, 4, \u2308\u03b2 8\u2309) and assume that with this recursion we obtain the quadruples X1, X2, . . . , X2n\u22123. If \u03b2 = i + 8(\u2308\u03b2 8 \u2309\u22121) + 1, then COL(2n, 4, \u03b2) is obtained by applying the i-th parallel class (i \u2208Z8 see Section 3.1), in the construction for Type S, on the quadruples X1, X2, . . . , X2n\u22123 as presented in Section 3.1. 2. 8(2n\u22121\u22121 3 ) < \u03b2 \u22648(2n\u22121\u22121 3 ) + 4(2n\u22121 \u22121)(2n\u22121 \u22122). In this case the parallel class is of Type T (see Section 3.2). For this an ef\ufb01cient construction for resolvable SQS(2n\u22121) is required. Such a construction is not dif\ufb01cult to \ufb01nd and in general it can be obtained by considering the constructions in [11, 13]. In general the complexity for the construction of the parallel class of the resolution for SQS(t) is O(t). The number of parallel classes in such a resolution of SQS(2n\u22121) is (2n\u22121\u22121)(2n\u22122\u22121) 3 . For this type the recursion generates twenty four parallel classes for each parallel class of SQS(2n\u22121). The parallel class which is required is number \u2308\u03b2\u22128(2n\u22121\u22121 3 ) 24 \u2309. Assume that this parallel class contains the quadruples X1, X2, . . . , X2n\u22123. If \u03b2 = i + 24(\u2308\u03b2\u22128(2n\u22121\u22121 3 ) 24 \u2309\u22121) + 1, then COL(2n, 4, \u03b2) is obtained by applying the i-th parallel class (i \u2208Z24, where a translation from R\u03b1,\u03b2,\u03b3 should be made see Section 3.2), in the construction for Type T , on the quadruples X1, X2, . . . , X2n\u22123 as presented in Section 3.2. 3. 8(2n\u22121\u22121 3 ) + 4(2n\u22121 \u22121)(2n\u22121 \u22122) < \u03b2. In this case the the parallel class is of Type F (see Section 3.3). If i = \u03b2 \u22128(2n\u22121\u22121 3 ) \u22124(2n\u22121 \u2212 1)(2n\u22121 \u22122) \u22121, then COL(2n, 4, \u03b2) is the following set of blocks. {{(a, 0), (b, 0), (a, 1), (b, 1)} : {a, b} \u2208Fi}, where {F0, F1, . . . , F2n\u22121\u22122} is any one-factorization of K2n\u22121. The complexity of this enumerative coding of the COLUMN GENERATION PROBLEM for BP(t, 4) is O(t). This immediately implies that an algorithm for the ENTRY PROBLEM can be given with the 30 \fsame complexity or even a better one. The algorithm can be simpler for the LISTING PROBLEM, but by applying the COLUMN GENERATION PROBLEM for all the columns of Baranyai matrix to obtain a solution for the LISTING PROBLEM implies complexity O(n4) which cannot be improved since this is the size of the list (matrix). 8."
+                },
+                {
+                    "url": "http://arxiv.org/abs/2005.03102v1",
+                    "title": "Constrained de Bruijn Codes: Properties, Enumeration, Constructions, and Applications",
+                    "abstract": "The de Bruijn graph, its sequences, and their various generalizations, have\nfound many applications in information theory, including many new ones in the\nlast decade. In this paper, motivated by a coding problem for emerging memory\ntechnologies, a set of sequences which generalize sequences in the de Bruijn\ngraph are defined. These sequences can be also defined and viewed as\nconstrained sequences. Hence, they will be called constrained de Bruijn\nsequences and a set of such sequences will be called a constrained de Bruijn\ncode. Several properties and alternative definitions for such codes are\nexamined and they are analyzed as generalized sequences in the de Bruijn graph\n(and its generalization) and as constrained sequences. Various enumeration\ntechniques are used to compute the total number of sequences for any given set\nof parameters. A construction method of such codes from the theory of\nshift-register sequences is proposed. Finally, we show how these constrained de\nBruijn sequences and codes can be applied in constructions of codes for\ncorrecting synchronization errors in the $\\ell$-symbol read channel and in the\nracetrack memory channel. For this purpose, these codes are superior in their\nsize on previously known codes.",
+                    "authors": "Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Alexander Vardy, Van Khu Vu, Eitan yaakobi",
+                    "published": "2020-05-06",
+                    "updated": "2020-05-06",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION The de Bruijn graph of order m, Gm, was introduced in 1946 by de Bruijn [7]. His target in introducing this graph was to \ufb01nd a recursive method to enumerate the number of cyclic binary sequences of length 2k such that each binary k-tuple appears as a window of length k exactly once in each sequence. It should be mentioned that in parallel also Good [31] de\ufb01ned the same graph and hence it is sometimes called the de Bruijn-Good graph. Moreover, it did not take long for de Bruijn himself to \ufb01nd out that his discovery was not novel. In 1894 Flye-Sainte Marie [24] has proved the enumeration result of de Bruijn without the de\ufb01nition of the graph. Nevertheless, the graph continues to carry the name of de Bruijn as well as the related sequences (cycles in the graph) which he enumerated. Later in 1951 van Aardenne-Ehrenfest and de Bruijn [1] generalized the enumeration result for any arbitrary alphabet of \ufb01nite size \u03c3 greater than one, using a generalized graph for an alphabet \u03a3 of size \u03c3. Formally, the de Bruijn graph G\u03c3,k has \u03c3k vertices, each one is represented by a word of length k over an alphabet \u03a3 with \u03c3 letters. The in-degree and the out-degree of each vertex in the graph is \u03c3. There is a directed edge from the vertex (x0, x1, . . . , xk\u22121) to the vertex (y1, y2, . . . , yk), where xi, yj \u2208\u03a3, if and only if yi = xi, 1 \u2264i \u2264k \u22121. This edge is represented by the (k + 1)-tuple (x0, x1, . . . , xk\u22121, yk). The sequences enumerated by de Bruijn are those whose length is \u03c3k and each k-tuple over \u03a3 appears in a window of k consecutive (cyclically) symbols in the sequence. Such a sequence enumerated by de Bruijn is represented by an Eulerian cycle in G\u03c3,k\u22121, where each k consecutive symbols represent an edge in the graph. This sequence is also an Hamiltonian cycle in G\u03c3,k, where each consecutive symbols represent a vertex in the graph. Henceforth, we assume that the cycles are Hamiltonian, i.e., with no repeated vertices. Throughout the years since de Bruijn introduced his graph and the related sequences, there have been many generalizations for the graph and for the sequences enumerated by de Bruijn. Such generalizations include enumeration of sequences of length \u2113, where \u2113< \u03c3k, in G\u03c3,k [48] or coding for two-dimensional arrays in which all the n \u00d7 m sub-arrays appear as windows in exactly one position of the large array [19]. The interest in the de Bruijn graph, its sequences, and their generalizations, is due to their diverse important applications. One of the \ufb01rst applications of this graph was in the introduction of shift-register sequences in general and linear feedback shift registers in particular [28]. These will have an important role also in our research. Throughout the years de Bruijn sequences, the de Bruijn graph, and their generalizations, e.g. for larger dimensions, have found variety of applications. These applications include cryptography [25], [41], and in particular linear complexity of sequences, e.g. [10], [20], [22], [27], [38], interconnection networks, e.g. [4], [23], [55], [58], [61], VLSI testing, e.g. [3], [42], two-dimensional generalizations, e.g. [6], [19], [45] with applications to self-locating patterns, range-\ufb01nding, data scrambling, mask con\ufb01gurations, robust undetectable digital watermarking of two-dimensional test images, and structured light, e.g. [35], [49], [50], [54], [57]. Some interesting modern applications are combined with biology, like the genome assembly as part of DNA sequencing, e.g. [9], [16], [36], [44], [52], [65] and coding for DNA storage, e.g. [11], [26], [39], [53]. This is a small sample of examples for the wide range of applications in which the de Bruijn graph, its sequences, and their generalizations, were used. The current work is not different. Motivated by applications to certain coding problems for storage, such as the \u2113-symbol read channel and the racetrack memory channel, we introduce a new type of generalization for de Bruijn sequences, the constrained de Bruijn sequences. In these sequences, a k-tuple cannot repeat within a segment starting in any of b consecutive positions. This generalization is quite natural and as the name hints, it can be viewed as a type of a constrained sequence. The goal of this paper is to study this type of sequences \f3 in all natural directions: enumeration, constructions, and applications. Our generalization is motivated by the need to combat synchronization errors (which are shift errors known also as deletions and sticky insertions) in certain memories. These types of synchronization errors occur in some new memory technologies, mainly in racetrack memories [12], [13], and in other technologies which can be viewed as an \u2113-symbol read channel [8], [14], [63]. By using constrained de Bruijn sequences to construct such codes we will be able to increase the rate of codes which correct such synchronization errors. But, we believe that de Bruijn constrained sequences and codes (sets of sequences) are of interest for their own right from both practical and theoretical points of view. The new de\ufb01ned sequences can be viewed as constrained codes and as such they pose some interesting problems. This is the reason that the new sequences and the related codes will be called constrained de Bruijn sequences and constrained de Bruijn codes, respectively. The rest of this paper is organized as follows. In Section II we introduce some necessary concepts which are important in our study, such as the length and the period of sequences in general and of de Bruijn sequences in particular. We will also introduce some elementary concepts related to shift-register sequences. In Section III we de\ufb01ne the new type of sequences, the constrained de Bruijn sequences and their related codes. There will be a distinction between cyclic and acyclic sequences. We consider the concept of periodicity for acyclic and cyclic sequences and de\ufb01ne the concept of forbidden patterns. The main result in this section will be a theorem which reveals that by the given de\ufb01nitions, three types of codes de\ufb01ned differently, form exactly the same set of sequences for an appropriate set of parameters for each de\ufb01nition. In Section IV the enumeration for the number of constrained de Bruijn sequences with a given set of parameters will be considered. We start with a few very simple enumeration results and continue to show that for some parameters, the rates of the new de\ufb01ned codes approach 1, and there are other parameters with one symbol of redundancy. Most of the section will be devoted to a consideration of the new de\ufb01ned codes as constrained codes. Enumeration based on the theory of constrained codes will be applied. These considerations yield also ef\ufb01cient encoding and decoding algorithms for the new de\ufb01ned codes. In Section V, a construction based on shift-register sequences will be given. This construction will be our main construction for codes with large segments in which no repeated k-tuples appear. The next two sections are devoted to applications of constrained de Bruijn sequences in storage memories. In Section VI, the application to the \u2113-symbol read channel is discussed. This application yields another application for another new type of storage channel, the racetrack channel which is considered in Section VII. We conclude in Section VIII, where we also present a few directions for future research. II. PRELIMINARIES In this section we will give some necessary de\ufb01nitions and notions concerning, cyclic and acyclic sequences, paths and cycles in the de Bruijn graph, shift-register sequences and in particular those which are related to primitive polynomials and are known as maximum length shift-register sequences. In Section II-A we discuss the length and period of sequences as well as cyclic and acyclic sequences. Shift-register sequences are discussed in Section II-B. A. Paths and Cycles in the de Bruijn Graph A sequence s = (s1, s2, . . . , sn) over an alphabet \u03a3 is a sequence of symbols from \u03a3, i.e., si \u2208\u03a3 for 1 \u2264i \u2264n. The length of such a sequence is the number of its symbols n and the sequence is considered to be acyclic. A cyclic sequence s = [s1, s2, . . . , sn] is a sequence of length n for which the symbols can be read \f4 from any position in a sequential order, where the element s1 follows the element sn. This means that s can be written also as [si, si+1, . . . , sn, s1, . . . , si\u22121] for each 2 \u2264i \u2264n. De\ufb01nition 1. A cyclic sequence s = [s1, . . . , sn], where si \u2208\u03a3 and \u03c3 = |\u03a3|, is called a weak de Bruijn sequence (cycle) of order k, if all the n windows of consecutive k symbols of s are distinct. A cyclic sequence s = [s1, . . . , sn] is called a de Bruijn sequence of order k, if n = \u03c3k and s is a weak de Bruijn sequence of order k. The connection between a weak de Bruijn cycle of length n (as a cycle in the de Bruijn graph) and a weak de Bruijn sequence of length n in the graph is very simple. The sequence is generated from the cycle by considering the \ufb01rst digit of the consecutive vertices in the cycle. The cycle is generated from the sequence by considering the consecutive windows of length k in the sequence. This is the place to de\ufb01ne some common concepts and some notation for sequences. An acyclic sequence s = (s1, . . . , sn) has n digits read from the \ufb01rst to the last with no wrap around. If all the windows of length k, starting at position i, 1 \u2264i \u2264n \u2212k + 1, are distinct, then the sequence corresponds to a simple path in G\u03c3,k. The period of a cyclic sequence s = [s0, s1, . . . , sn\u22121] is the least integer p, such that si = si+p, where indices are taken modulo n, for each i, 0 \u2264i \u2264n \u22121. It is well known that the period p divides the length of a sequence n. If n = rp then the periodic sequence s is formed by a concatenation of r copies of the \ufb01rst p entries of s. It is quite convenient to have the length n and the period p equal if possible. There is a similar de\ufb01nition for a period of acyclic sequence with slightly different properties, which is given in Section III. In this paper we will assume (if possible) that for a given cyclic sequence s = [s1, . . . , sn] the period is n as the length. Hence, usually we won\u2019t distinguish between the length and period for cyclic sequences. We will elaborate more on this point in Section II-B. Finally, the substring (window) (si, si+1, . . . , sj) will be denoted by s[i, j] and this substring will be always considered as an acyclic sequence, no matter if s is cyclic or acyclic. In addition, s[i] will be sometimes used instead of si and the set {1, 2, . . . , n} is denoted by [n]. B. Feedback Shift Registers and their Cycle Structure The theory on the sequences in the de Bruijn graph cannot be separated from the theory of shift-register sequences developed mainly by Golomb [28]. This theory, developed \ufb01fty years ago, was very in\ufb02uential in various applications related to digital communication [29], [30]. A short summary on the theory of shift-registers taken from [28] which is related to our work, is given next. A characteristic polynomial (for a linear feedback shift register de\ufb01ned in the sequel) c(x) of degree k over Fq, the \ufb01nite \ufb01eld with q elements, is a polynomial given by c(x) = 1 \u2212 k X i=1 cixi , where ci \u2208Fq. For such a polynomial a function f on k variables from Fq is de\ufb01ned by f(x1, x2, . . . , xk) = k X i=1 cixk+1\u2212i . For the function f we de\ufb01ne a state diagram with the set of vertices Qk \u225c{(x1, x2, . . . , xk) : xi \u2208Fq} . \f5 If xk+1 = f(x1, x2, . . . , xk) then an edge from (x1, x2, . . . , xk) to (x2, . . . , xk, xk+1) is de\ufb01ned for the related state diagram. A feedback shift register of length k has qk states corresponding to the set Qk of all qk k-tuples over Fq. The feedback function xk+1 = f(x1, x2, . . . , xk) de\ufb01nes a mapping from Qk to Qk. The feedback function can be linear or nonlinear. The shift register is called nonsingular if its state diagram consists of disjoint cycles. Any such state diagram is called a factor in Gq,k, where a factor in a graph is a set of vertex-disjoint cycles which contains all the vertices of the graph. There is a one-to-one correspondence between the set of factors in the de Bruijn graph Gq,k and the set of state diagrams for the nonsingular shift registers of order k over Fq. It is well-known [28] that a binary feedback shift-register is nonsingular if and only if its feedback function has the form f(x1, x2, . . . , xk) = x1 + g(x2, . . . , xk) , where g(x2, . . . , xk) is any binary function on k\u22121 variables. A similar representation also exists for nonsingular feedback shift-registers over Fq. An Hamiltonian cycle in Gq,k is a de Bruijn cycle which forms a de Bruijn sequence. There are (q!)qk\u22121/qk distinct such cycles in Gq,k and there are many methods to generate such cycles [21], [25]. One important class of sequences in the graph, related to de Bruijn sequences, are the so called m-sequences, or maximal length linear shift-register sequences. A shift-register is called linear if its feedback function f(x1, x2, . . . , xk) is linear. An m-sequence is a sequence of length qk \u22121 generated by a linear shift-register associated with a primitive polynomial of degree k over Fq. Each primitive polynomial is associated with such a sequence. In such a sequence all windows of length k are distinct and the only one which does not appear is the all-zero window. The following theorem is well known [28]. Theorem 1. The number of distinct m-sequences of order k over Fq (the same as the number of primitive polynomials of order k over Fq) is \u03c6(qk \u22121) k , where \u03c6 is the Euler function. The exponent of a polynomial f(x) is the smallest integer e such that f(x) divides xe \u22121. The length of the longest cycle, of the state diagram, formed by the shift-register associated with the characteristic polynomial f(x) is the exponent of f(x). The length of the cycles associated with a multiplication of several irreducible polynomials can be derived using the exponents of these polynomials and some related algebraic and combinatorial methods. This theory leads immediately to the following important result. Theorem 2. The state diagram associated with a multiplication of two distinct characteristic primitive polynomials f(x) and g(x) of order k over Fq contains qk + 1 cycles of length qk \u22121 and the all-zero cycle. Each possible (2k)-tuple over Fq appears exatly once as a window in one of these cycles. The m-sequences related to f(x) and g(x) are two of these cycles. III. CONSTRAINED DE BRUIJN CODES, PERIODS, AND FORBIDDEN PATTERNS In this section we give the formal de\ufb01nitions for constrained de Bruijn sequences and codes. We present a de\ufb01nition for a family of sequences with a certain period and a de\ufb01nition for a family of sequences avoiding certain substrings. Three different de\ufb01nitions will be given and it will be proved that with the appropriate \f6 parameters the related three families of sequences contain the same sequences. Each de\ufb01nition will have later some role in the enumeration of constrained de Bruijn sequences which will be given in Section IV. A. Constrained de Bruijn Codes De\ufb01nition 2. \u2022 A sequence s = (s1, . . . , sn), over some alphabet \u03a3, is called a (b, k)-constrained de Bruijn sequence if s[i, i + k \u22121] \u0338= s[j, j + k \u22121] for all i, j \u2208[n \u2212k + 1] such that 0 < |i \u2212j| \u2264b \u22121. In other words, in each substring of s whose length is b+k \u22121 there is no repeated k-tuple, i.e. each subset of b consecutive windows of length k in s contains b distinct k-tuples. \u2022 A set of distinct (b, k)-constrained de Bruijn sequences of length n is called a (b, k)-constrained de Bruijn code. The set of all (b, k)-constrained de Bruijn sequences of length n will be denoted by CDB(n, b, k). The alphabet \u03a3 and its size sigma should be understood from the context throught our discussion. \u2022 A cyclic sequence s = [s1, . . . , sn], over \u03a3, is called a cyclic (b, k)-constrained de Bruijn sequence if (s1, . . . , sn) is a (b, k)-constrained de Bruijn sequence and s[i, i + k \u22121] \u0338= s[j, j + k \u22121] for all i, j \u2208[n] such that i < j and i \u2212j + n \u2264b \u22121. In other words, in each cyclic substring of s whose length is b + k \u22121 there is no repeated k-tuple. Note, that two sequences which differ only in a cyclic shift are considered to be the same sequence. \u2022 A set of distinct cyclic (b, k)-constrained de Bruijn sequences of length n is called a cyclic (b, k)constrained de Bruijn code. The set of all cyclic (b, k)-constrained de Bruijn sequences of length n will be denoted by C\u2217 DB(n, b, k). We note that by the de\ufb01nition of a cyclic constrained de Bruijn sequence two codewords in a cyclic (b, k)-constrained de Bruijn code cannot differ only in a cyclic shift, but in some applications one might consider these cyclic shifts of codewords as distinct codewords. de Bruijn sequences form a special case of constrained de Bruijn sequences as asserted in the following theorem which is readily veri\ufb01ed by de\ufb01nition. Theorem 3. \u2022 The cyclic sequence s of length qk over Fq is a de Bruijn sequence if and only if s is a cyclic (qk, k)constrained de Bruijn sequence. \u2022 The (acyclic) de Bruijn sequence s of length qk + k \u22121 over Fq is a de Bruijn sequence if and only if s is a (qk, k)-constrained de Bruijn sequence. Theorem 3 introduce the connection between de Bruijn sequences and constrained de Bruijn sequences. But, there are a few main differences between de Bruijn sequences of order k and (b, k)-constrained de Bruijn sequences. \u2022 A de Bruijn sequence is usually considered and is used as a cyclic sequence while a constrained de Bruijn sequence will be generally used as an acyclic sequence. \u2022 A de Bruijn sequence contains each possible k-tuple exactly once as a window of length k, while in a (b, k)-constrained de Bruijn sequence, each possible k-tuple can be repeated several times or might not appear at all. \u2022 A de Bruijn sequence contains each possible k-tuple exactly once in one period of the sequence, while in a (b, k)-constrained de Bruijn sequence, each possible k-tuple can appear at most once among any b consecutive k-tuples in the sequence. \f7 \u2022 The length (or equivalently period in this case) of a de Bruijn sequence is strictly qk, while there is no constraint on the length and the period of a (b, k)-constrained de Bruijn sequence. These differences between de Bruijn sequences and constrained de Bruijn sequences are important in the characterization, enumeration, constructions, and applications of the two types of sequences. B. Periods and Forbidden Subsequences There are two concepts which are closely related to constrained de Bruijn sequences, the period of an acyclic sequence and avoided patterns. Let u and s be two sequences over \u03a3. The sequence s avoids u (or s is u-avoiding) if u is not a substring of s. Let F be a set of sequences over \u03a3 and s a sequence over \u03a3. The sequence s avoids F (or s is F-avoiding) if no sequence in F is a substring of s. Let A(n; F) denote the set of all \u03c3-ary sequences of length n which avoid F. A subset of A(n; F), i.e. a set of F-avoiding sequences of length n, is called an F-avoiding code of length n. The second concept is the period of a sequence. For cyclic sequences the length and the period of the sequence either coincide or have a very strict relation, where the period of the sequence divides its length. This is not the case for acyclic sequences. De\ufb01nition 3. \u2022 A sequence s = (s1, s2, . . . , sn) \u2208\u03a3n is a period p sequence if it satis\ufb01es si = si+p for all 1 \u2264i \u2264n\u2212p. Note, that the de\ufb01nition for periods of a cyclic sequence coincides with this de\ufb01nition). \u2022 A sequence s \u2208\u03a3n is called an m-limited length for period p substrings if any substring with period p of s has length at most m. \u2022 A set of m-limited length, period p sequences from \u03a3n is called a code of m-limited length for period p substrings. The set of all such m-limited length for period p substrings is denoted by CLP (n, m, p). The \ufb01rst lemma is a straightforward observation. Lemma 1. If F is the set of all sequences of length m + 1 and period p, then for each i, 1 \u2264i \u2264m, the set CLP (n, i, p) is an F-avoiding code. Proof. Recall, that for each i, the set of sequences in CLP (n, i, p), have length n. If s is a sequence in CLP (n, i, p), 1 \u2264i \u2264m, then each substring of s with period p has length at most i, where i is a positive integer strictly smaller than m + 1. The set F contains the sequences of length m + 1 and period p and hence a sequence of F cannot be a substring of s. Thus, CLP (n, i, p) is an F-avoiding code. C. Equivalence between the Three Types of Codes Let Fp,p+k be the set of all period p sequences of length p + k for any given 1 \u2264p \u2264b \u22121 and let F = \u222ab\u22121 p=1Fp,p+k. The following result implies a strong relation between constrained de Bruijn codes, code of limited length for period p substrings, and F-avoiding codes. By the related de\ufb01nitions of these concepts we have the following theorem. Theorem 4. For all given admissible n, b, k, and any code C \u2282\u03a3n, A(n; F) = CDB(n, b, k) = b\u22121 \\ i=1 CLP (n, i + k \u22121, i) . \f8 Proof. We will prove that A(n; F) \u2286CDB(n, b, k), CDB(n, b, k) \u2286Tb\u22121 i=1 CLP (n, i+k\u22121, i), and Tb\u22121 i=1 CLP (n, i+ k \u22121, i) \u2286A(n; F). These three containment proofs will imply the claim of the theorem. First, we prove that if c is a sequence in A(n; F) then c is a sequence in CDB(n, b, k). Let c = (c1, c2, . . . , cn) be any sequence in A(n, F), and assume for the contrary that c \u0338\u2208CDB(n, b, k). This implies that there exist integers i, j \u2208[n \u2212k + 1] such that 1 \u2264p = j \u2212i \u2264b \u22121 and c[i, i + k \u22121] = c[j, j + k \u22121]. Hence, c[i, j+k\u22121] = (ci, ci+1, . . . , cj+k\u22121) is a substring with period p and length p+k. Therefore c[i, j+k\u22121] \u2208F, a contradiction since c \u2208A(n; F). Thus, c \u2208CDB(n, b, k) which implies that A(n; F) \u2286CDB(n, b, k) Next, we prove that if c is a sequence in CDB(n, b, k) then c is a sequence of Tb\u22121 i=1 CLP(n, i+k \u22121, i). Let c = (c1, c2, . . . , cn) be any sequence in CDB(n, b, k), and assume for the contrary that c \u0338\u2208Tb\u22121 i=1 CLP(n, i + k \u22121, i). Hence, there exists p \u2208[b \u22121] such that c \u0338\u2208CLP (n, p + k \u22121, p). Therefore, c contains a period p substring of length p + k. Let c[i, i + p + k \u22121] be such a period p substring. Hence, c[i, i + k \u22121] = c[i + p, i + p + k \u22121], a contradiction since c \u2208CDB(n, b, k). Thus, C \u2286Tb\u22121 i=1 CLP (n, i + k \u22121, i) which implies that CDB(n, b, k) \u2286Tb\u22121 i=1 CLP(n, i + k \u22121, i). Finally, if c is a sequence in Tb\u22121 i=1 CLP (n, i + k \u22121, i), then by Lemma 1 we have that c is a sequence in A(n; F). Thus, Tb\u22121 i=1 CLP(n, i + k \u22121, i) \u2286A(n; F). Corollary 1. For all given admissible n, b, k, and any code C \u2282\u03a3n, the following three statements are equivalent 1) C is a subset of A(n; F). 2) C is a subset of CDB(n, b, k). 3) C is a subset of Tb\u22121 i=1 CLP(n, i + k \u22121, i). The set F of all forbidden patterns of an F-avoiding code C is never a minimal minimal, i.e., there exists another set F\u2032 such that F\u2032 \u2282F and C is also an F\u2032-avoiding code. It implies that there always exist two different sets, F1 and F2, where F1 \u2282F2, such that A(n; F1) = A(n; F2). But, there always exists one such set F whose size is smaller than the sizes of all the other sets To see that, let F be a set of forbidden sequences over \u03a3. As long as F contains \u03c3 sequences of length \u2113+ 1 which form the set S = {u\u03b1 : \u03b1 \u2208\u03a3}, where u is a string of length \u2113, these \u03c3 sequences of F which are contained in S can be replaced by the sequence u. When this process comes to its end, instead of the original set F of forbidden patterns we have a set F\u2032 of forbidden patterns for the same code. The set of sequences F\u2032 will be called the forbidden reduced set of F. Lemma 2. If F is a set of forbidden sequences and F\u2032 is its forbidden reduced set, then A(n; F) = A(n; F\u2032). Proof. The proof is an immediate observation from the fact that all the sequences of length n which do not contain the patterns in S = {u\u03b1 : \u03b1 \u2208\u03a3}, where n is greater than the length of u, as substrings do not contain the pattern u as a substring. Hence, A(n; F) \u2286A(n; F\u2032). Clearly, all the sequences of length n which do not contain u as a substring do not contain any pattern from S as a substring. Hence, A(n; F\u2032) \u2286A(n; F). Thus, A(n; F) = A(n; F\u2032). IV. ENUMERATION OF CONSTRAINED DE BRUIJN SEQUENCES In this section we consider enumeration of the number of sequences in CDB(n, b, k). Note, that we are considering only acyclic sequences. A \u03c3-ary code C of length n is a set of \u03c3-ary sequences of length n, that is C \u2286\u03a3n. For each code C of length n, we de\ufb01ne the rate of the code C to be R(C) = log\u03c3(|C|)/n, and \f9 the redundancy of the code C to be r(C) = n \u2212log\u03c3(|C|), where |C| is the size of the code C. We de\ufb01ne the maximum asymptotic rate of (b, k)-constrained de Bruijn codes to be RDB(b, k) = lim supn\u2192\u221e log\u03c3 |CDB(n,b,k)| n .1 A. Trivial Bounds In this section trivial bounds on the number of constrained de Bruijn sequences are considered. The number of (cyclic) de Bruijn sequences of length \u03c3n over an alphabet of size \u03c3 is (\u03c3!)\u03c3n\u22121/\u03c3n [1] which implies the following simple result (for acyclic sequences). Theorem 5. For any given a positive integer \u03c3 \u22652, a positive integer k, and n \u2265\u03c3k + k \u22121, |CDB(n, \u03c3k, k)| = (\u03c3!)\u03c3n\u22121 . Corollary 2. For any given a positive integer \u03c3 \u22652 and a positive integer k, RDB(\u03c3k, k) = 0 . Corollary 2 can be generalized for RDB(b, k), where \u03c3k \u2265b \u2265b0, where b0 is an integer whose value depends on \u03c3 and k.The reason that the rates are zeroes for so many values is that once we have a long simple path of length b in G\u03c3,k (and the number of such paths is very large [48]), to continue the path for a long sequence which is a (b, k)-constrained de Bruijn sequence, the sequence will be almost periodic (with a possibility of some small local changes. In the case of Theorem 5, where the path is a de Bruijn sequence, there is only one way to continue the path without violating the constraint. There are some intriguing questions in this context. The \ufb01rst one, question to be is to be speci\ufb01c in the value of b0. Another question is to \ufb01nd a good bound on RDB(b, k), where b is large compared to k and RDB(b, k) > 0. Such a bound will be given in Section V, but we have no indication how good it is. The other extreme case is when b = 1 and hence the sequence is not constrained and we have the following trivial result. Theorem 6. For any given positive integer \u03c3 \u22652, a positive integer k, and n \u2265k, |CDB(n, 1, k)| = \u03c3n and RDB(1, k) = 1. Except for these cases, to \ufb01nd the rates of constrained de Bruijn sequences, where b > 1, is not an easy task. When the rate is approaching one, we are interested in the redundancy of CDB(n, b, k). Unfortunately, \ufb01nding the redundancy of CDB(n, b, k) is even more dif\ufb01cult than to \ufb01nd the rate. Fortunately, for small values of b we can use the theory of constrained coding, and for k large enough we can even show that the redundancy is one symbol. The last trivial case is the (b, 1)-constrained de Bruijn codes. The sequences of such a code will be used for error correction of synchronization errors in racetrack memories in Section VII-A. Since this constraint implies that any b consecutive elements in the sequence will be distinct, we must have that \u03c3 \u2265b to obtain any valid sequence. If \u03c3 \u2265b, then we have to choose b elements from the \u03c3 alphabet letters to start the sequence and in any other position we can choose any of the alphabet letters, except for the previous b \u22121 positions in the sequence. Hence, we have 1The lim sup can indeed be replaced by a proper lim [47]. \f10 Theorem 7. For any b \u22651 and any alphabet of size \u03c3 we have |CDB(n, b, 1)| = \uf8f1 \uf8f2 \uf8f3 0 if b > \u03c3 \u0000\u03c3 b \u0001 b!(\u03c3 \u2212b + 1)n\u2212b if \u03c3 \u2265b . Corollary 3. For any b \u22651 and any alphabet \u03c3 we have that RDB(b, 1) = 0 if \u03c3 \u2264b and RDB(b, 1) = log\u03c3(\u03c3 \u2212b + 1) if \u03c3 > b. B. (b, k)-Constrained de Bruijn Codes with Redundancy 1 We observe that the asymptotic rate RDB(b, k) is getting close to 1 when b is \ufb01xed and k tends to in\ufb01nity. In this case, we are interested in the redundancy of the code. For a subset A of a set U, the complement of A, Ac, is the subset which consists of all the elements of U which are not contained in A. For a code C of length n over \u03a3, Cc = \u03a3n \\ C. The following simple lemma will be used in the next theorem. Lemma 3. If Ai, 1 \u2264i \u2264m, are m sets over the same domain then |(\u2229m i=1Ai)c| \u2264 m X i=1 |Ac i|. Proof. It is well known from set theory that (\u2229m i=1Ai)c = \u222am i=1Ac i . Combining this with the trivial assertion that for any two sets A and B, |A \u222aB| \u2264|A| + |B| we have that | (\u2229m i=1Ai)c | = | \u222am i=1 Ac i| \u2264 m X i=1 |Ac i|. Theorem 8. For all \u03c3, n and b < k, |CDB(n, b, k)| \u2265\u03c3n   1 \u2212(b \u22121)n \u00b7 \u0012 1 \u03c3 \u0013k! . In particular, for k \u2265\u2308log\u03c3 n + log\u03c3(b \u22121)\u2309+ 1, the redundancy of CDB(n, b, k) is at most a single symbol. Proof. By Theorem 4, CDB(n, b, k) = Tb\u22121 i=1 CLP(n, i + k \u22121, i). Combining this fact with Lemma 3 we have that |CDB(n, b, k)| = \u03c3n\u2212|CDB(n, b, k)c| = \u03c3n\u2212| \u0010 \u2229b\u22121 i=1CLP (n, i + k \u22121, i) \u0011c | \u2265\u03c3n\u2212 b\u22121 X i=1 |CLP (n, i+k\u22121, i)c|. (1) For each i, 1 \u2264i \u2264b \u22121, a word c is contained in CLP(n, i + k \u22121, i)c if and only if it has length n and a substring of period i whose length is at least i + k. There are at most n \u2212(i + k \u22121) possible starting positions for such a substring. Once the \ufb01rst i symbols in this substring, of length i + k, are known, its other k symbols are uniquely determined. The remaining n \u2212(i + k) symbols in the word of length n can be chosen arbitrarily (note, that some choices cause other substrings of the same period and larger length in this word. They might cause other substrings with larger period than i, so the computation which follows have many repetitions and \f11 some sequences which do not have to be computed.). Hence, the number of words in CLP (n, i + k \u22121, i)c is upper bounded by | (CLP(n, i + k \u22121, i))c | \u2264(n \u2212i \u2212k + 1) \u00b7 \u03c3i \u00b7 \u03c3n\u2212(i+k) < \u03c3n \u00b7 n \u00b7 \u0012 1 \u03c3 \u0013k . Therefore, b\u22121 X i=1 | (CLP(n, i + k \u22121, i))c | \u2264(b \u22121) \u00b7 \u03c3n \u00b7 n \u00b7 \u0012 1 \u03c3 \u0013k , and hence using (1) we have, |CDB(n, b, k)| \u2265\u03c3n \u2212(b \u22121) \u00b7 \u03c3n \u00b7 n \u00b7 \u0012 1 \u03c3 \u0013k = \u03c3n   1 \u2212(b \u22121) \u00b7 n \u00b7 \u0012 1 \u03c3 \u0013k! . In particular, for k \u2265\u2308log\u03c3(n) + log\u03c3(b \u22121)\u2309+ 1, we obtain |CDB(n, b, k)| \u2265\u03c3n   1 \u2212(b \u22121) \u00b7 n \u00b7 \u0012 1 \u03c3 \u0013log\u03c3 n+log\u03c3(b\u22121)+1! \u2265\u03c3n \u00b7 \u03c3 \u22121 \u03c3 \u2265(\u03c3 \u22121) \u00b7 \u03c3n\u22121. Thus, the redundancy of CDB(n, b, k) is at most a single symbol. A weaker bound than the one in Theorem 8 for \u03c3 = 2 was given in [13] (Theorem 13). Finally, to encode the (b, k)-constrained de Bruijn code ef\ufb01ciently with only a single symbol of redundancy, we may use sequence replacement techniques [62]. C. Representation as a Graph Problem The key for enumeration of the number of (b, k)-constrained de Bruijn sequences of length n is a representation of the enumeration as a graph problem. The de Bruijn graph G\u03c3,k was the key for the enumeration of the cyclic (as implies also the acyclic) de Bruijn sequences of length \u03c3k, i.e., the number of (\u03c3k, k)-constrained de Bruijn sequences of length n \u2265\u03c3k + k \u22121. This number is equal to the number of Hamiltonian cycles in the graph and also the number of Eulerian cycles in G\u03c3,k\u22121. This number was found for example with a reduction to the number of spanning trees in the graph [25]. The (not necessary simple) paths in G\u03c3,k (where vertices are considered as the k-tuples) represent the (1, k)-constrained de Bruijn sequences, i.e. sequences with no constraints. Can the de Bruijn graph or a slight modi\ufb01cation of it represents other constraints. The answer is de\ufb01nitely yes. If we remove from G\u03c3,k the \u03c3 self-loops, then the paths (represented by the vertices) in the new graph represents all the (2, k)-consrained de Bruijn sequences. This can be generalized to obtain a graph for the (3, k)-constrained de Bruijn sequences and in general the (b, k)-constrained de Bruijn sequences, when b is a \ufb01xed integer. For the (3, k) constraint, we form from G\u03c3,k a new graph G\u2032 \u03c3,k as follows. First, we remove all the self-loops from G\u03c3,k to accommodate the (2, k) constraint. Next, we consider all the cycles of length two in G\u03c3,k. These cycles have the form x \u2192y \u2192x, where x = (x1, x2, . . . , xk) and y = (y1, y2, . . . , yk), xi = yj = a, for odd i and even j, and xi = yj = b, for even i and odd j, where a and b are two distinct letters in \u03a3. In G\u2032 \u03c3,k, x and y are replaced by four vertices x1, x2 instead of x, and y1, y2 instead of y. All the in-edges and the out-edges to and from x (except for the ones from and to y) are duplicated and are also in-edges and out-edges to and from x1 and x2. Similarly, all the in-edges and the out-edges in and from y (except for the ones from and to x) are duplicated and are also in-edges and out-edges to and from y1 and y2. There is also one edge from x1 to y1 and from y2 to x2. All the paths (related to the vertices) in the constructed graph G\u2032 \u03c3,k form all the (3, k)-constrained de Bruijn sequences. The total number of cycles of length two in \f12 000 001 100 010A 010B 101B 101A 011 110 111 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 0 Fig. 1. Graph representation for the (3, 3)-constrained de Bruijn sequences, via the de Bruijn graph G\u03c3,k is \u0000\u03c3 2 \u0001 and hence the number of vertices in G\u2032 \u03c3,k is \u03c3k +\u03c3(\u03c3 \u22121). An example for the representation of the graph (i.e., state diagram) for the (3, 3)-constrained de Bruijn sequences via the de Bruijn graph is presented in Figure 1. A second representation is by using the theory of constrained coding [47]. One have to construct the state diagram of the constrained system. From the state diagram one has to generate the related adjacency matrix to compute the rate of the constrained de Bruijn code by computing the largest eigenvalue of the adjacency matrix. If \u03bb is the largest eigenvalue of the adjacency matrix then asymptotically there are \u03bbn sequences and the rate of the code is log\u03c3 \u03bb. An example will be given in the next subsection. While the \ufb01rst representation is related to the code CDB(n, b, k), the second representation is related to the code A(n; F), where F are all the substrings which are forbidden in a (b, k)-constrained sequence. By Theorem 4 these two codes are equal. Each vertex in the state diagram (of the constrained code representation) represents a substring, which is not in F. When the last vertex in the path is generated by the current pre\ufb01x s\u2032 of the sequence s is vertex v, it implies that the substring represented by vertex v is the longest suf\ufb01x of s\u2032 which is a pre\ufb01x of a forbidden pattern in F. Each one of the \u03c3 out-edges exist in v if and only each one does not lead to a forbidden substring in F. Eventually, both representations are equivalent and one can be derived from the other by using a reduction of the state diagram [47]. But, each representation has some advantages on the other one. The advantage of using the state diagram of the constraint, based on the forbidden substrings, is a simple de\ufb01nition for the state diagram from which the theory of constrained codes can be used. The advantage of using a graph like a de Bruijn graph is that each point on the sequence is related to a speci\ufb01c k-tuple and it is identi\ufb01ed by a vertex (note that some k-tuples are represented by a few vertices) which represents this k-tuple. An example of the state diagram for the (3, 3)-constrained de Bruijn sequences via the forbidden patterns is depicted in Figure 2. \f13 x 0 1 00 01 11 10 000 010 101 111 0101 1010 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 Fig. 2. State diagram for the (3, 3)-constrained de Bruijn sequences, via the forbidden patterns D. State Diagrams and Rates based on the Forbidden Subsequences As a sequence of a constrained code, the (b, k)-constrained de Bruijn sequence has several forbidden patterns, s1, s2, . . . , s\u2113. W.l.o.g. we assume that si is not a pre\ufb01x of sj for i \u0338= j. The state diagram has an initial vertex x, from which there are \u2113paths. The ith path has length smaller by one than the length of the ith forbidden pattern si. Paths share a pre\ufb01x if their related sequences share a corresponding pre\ufb01x. Each vertex in the state diagram, except for the ones at the end of the \u2113paths, has out-degree \u03c3, one edge for each possible symbol of \u03a3 that can be seen in the corresponding read point of the constrained sequence. The edges are labelled by the related symbols. The last symbol of si does not appear of an edge from the last vertex of the related path. A vertex in the state diagram is labelled by the pre\ufb01x of the path which it represents. The initial vertex x is labelled with the sequence of length zero. Thus, to construct the state diagram we have to determine all the forbidden substrings in the (b, k)-constrained de Bruijn sequence. This representation coincides with the representation of the constrained system and the related example for the (3, 3)-constrained de Bruijn sequences is depicted in Figure 2. To determine exactly the maximum asymptotic rates of (b, k)-constrained de Bruijn codes, we use the wellknown Perron-Frobenius theory [47]. When b and k are given, by using the the de Bruijn graph for the given constraint or the related state diagram, we can build a \ufb01nite directed graph with labelled edges such that paths in the graph generate exactly all (b, k)-constrained de Bruijn sequences. For example, when (b, k) = (3, 3) the adjacency matrix is AG = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \f14 Its largest eigenvalue is \u03bb \u22481.73459. Hence, the capacity of this constrained system, which is the maximum asymptotic rate of (3, 3)-constrained de Bruijn code, is log \u03bb = 0.7946. Similarly, we can compute the maximum asymptotic rates of (b, k)-constrained de Bruijn codes for other values of b, k. Table I presents some values for the asymptotic rates of the constrained systems for small parameters. TABLE I THE MAXIMAL ASYMPTOTIC RATES OF (b, k)-CONSTRAINED DE BRUIJN CODES. k = 2 k = 3 h = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10 b = 2 0.6942 0.8791 0.9468 0.9752 0.9881 0.9942 0.9971 0.9986 0.9993 b = 3 0.4056 0.7946 0.9146 0.9614 0.9817 0.9912 0.9957 0.9978 0.9989 b = 4 0 0.6341 0.8600 0.9392 0.9719 0.9865 0.9934 0.9966 0.9978 b = 5 0 0.4709 0.7973 0.9150 0.9615 0.9818 0.9912 0.9957 0.9978 b = 6 0 0.4517 0.7289 0.88412 0.94815 0.97574 ? ? ? The asymptotic rate can be evaluated for in\ufb01nite pairs of (b, k) as proved in the next theorem. Theorem 9. For any positive integer k > 1, the maximum asymptotic rate of a binary (3, k)-constrained de Bruijn code is log2 \u03bb, where \u03bb is the largest root of the polynomial x2k\u22121 = x2k\u22123 + 2x2k\u22124 + \u00b7 \u00b7 \u00b7 + (k \u2212 2)xk + (k \u22121)xk\u22121 + (k \u22121)xk\u22122 + (k \u22122)xk\u22123 + \u00b7 \u00b7 \u00b7 + 2x + 1. Proof. Recall that Fp,p+k is the set of all period p sequences of length p + k. Let F1 = F1,k+1 \u222aF2,k+2, i.e., F1 contains the all-zero word of length k + 1, the all-one word of length k + 1, and the two words of length k + 2 in which any two consecutive positions has distinct symbols. By Corollary 1, CDB(n, 3, k) = A(n; F1), i.e. binary (3, k)-constrained de Bruijn code of length n is an F1-avoiding code of length n. Let F2 by the set which contains the all-zero word of length k and the all-one word of length k+1. Consider the D-morphism de\ufb01ned \ufb01rst in [40], D : Bn 7\u2192Bn\u22121, B = {0, 1}, where D(x) = D(x1, x2, . . . , xn) = y = (y2, . . . , yn), with yi = xi + xi\u22121, 2 \u2264i \u2264n. It was proved in [40] that the mapping D is a 2 to 1 mapping. Furthermore, x \u2208A(n; F1) if and only if y \u2208A(n\u22121; F2). Hence, |A(n; F1)| = 2|A(n\u22121; F2)|. This implies that lim n\u2192\u221e log2 |A(n; F2)| n = lim n\u2192\u221e log2 |A(n; F1)| n . Hence, A(n; F1) and A(n; F2) have the same maximum asymptotic rate, and hence A(n; F2) can be computed instead of A(n; F1). Let A0(n; F2) be the set of all F2-avoiding words of length n which start with a zero and let A1(n; F2) be the set of all F2-avoiding words of length n which start with a one. Clearly, the asymptotic rates of A(n; F2), A0(n; F2), A1(n; F2) are equal. Let \u03a61 : A0(n; F2) 7\u2192\u222ah\u22121 i=1 A1(n \u2212i; F2) be the mapping for which \u03a61(x = (0, x2, . . . , xn)) = (xi+1, . . . , xn) \u2208A1(n \u2212i; F2), where i is the smallest index that xi+1 = 1. Since A0(n; F2) avoids the all-zero sequence of length k, it follows that the mapping \u03a61 is a well-de\ufb01ned bijection. Therefore, |A0(n; F2)| = k\u22121 X i=1 |A1(n \u2212i; F2)|. (2) \f15 Similarly, we can de\ufb01ne the bijection \u03a62 : A1(n; F2) 7\u2192\u222ak i=1A0(n \u2212i; F2) and obtain the equality |A1(n; F2)| = k X i=1 |A0(n \u2212i; F2)|. (3) Equations (2) and (3) imply that |A0(n; F2)| = k\u22121 X i=1 |A1(n \u2212i; F2)| = k\u22121 X i=1 k X j=1 |A0(n \u2212i \u2212j; F2)| = k X \u2113=2 (\u2113\u22121)|A0(n \u2212\u2113; F2)| + k\u22121 X \u2113=1 \u2113|A0(n \u22122h + \u2113; F2)| = |A0(n \u22122; F2)| + 2|A0(n \u22123; F2)| + \u00b7 \u00b7 \u00b7 + (k \u22121)|A0(n \u2212k; F2)| + (k \u22121)|A0(n \u2212k \u22121; F2)| + (k \u22122)|A0(n \u2212k \u22122; F2)| + \u00b7 \u00b7 \u00b7 + |A0(n \u22122k + 1; F2)|. It is again easy to verify that the maximum asymptotic rates of A0(n \u2212\u2113; F2) for all 2 \u2264\u2113\u22642k \u22121 are equal. Let \u03bbn be this maximum asymptotic rate. The recursive formula can be solved now for \u03bb and the maximum asymptotic rate of A0(n; F2) will be log2 \u03bb, where \u03bb is computed as the largest root of the polynomial x2k\u22121 = x2k\u22123 + 2x2k\u22124 + \u00b7 \u00b7 \u00b7 + (k \u22122)xk + (k \u22121)xk\u22121 + (k \u22121)xk\u22122 + (k \u22122)xk\u22123 + \u00b7 \u00b7 \u00b7 + 2x + 1. Using recursive formulas in Equations (2) and (3), we can compute the exact size of A(n; F2) ef\ufb01ciently. Hence, we can rank/unrank all words in A(n; F2) ef\ufb01ciently using enumerative technique [17]. Computing the largest eigenvalue of the adjacency matrix is the key for computing the asymptotic rate. The size of the matrix and the number of its nonzero entries is important for reducing the complexity of the computation. Fortunately we can evaluate some of these parameters to get some idea for which parameters it is feasible to compute the largest eigenvalue. For lack of space and since this is mainly a combinatorial problem we omit this computation and leave it for a future work. Finally, we note that we can use the idea in the proof of Theorem 9 to reduce the number of forbidden patterns by half, while keeping the same asymptotic rate. This is done in the following example. Example 1. Consider the (3, 3)-constrained de Bruijn sequences. The set of patterns which should be avoided is {0000, 1111, 01010, 10101}. As was illustrated in Figure 1 and Figure 2, 10 nodes are required to represent a graph of constrained sequences avoiding these four patterns. Using Theorem 9, it can be observed that the asymptotic size of this code is the same as the code avoiding all patterns in set {000, 1111}. Hence, only 5 nodes are required to represent the state diagram of the constrained sequences avoiding these two patterns as depicted in Figure 3. V. CODES WITH A LARGE CONSTRAINED SEGMENT After considering in Section IV enumerations and constructions for (b, k)-constrained de Bruijn sequences which are mainly ef\ufb01cient for small b compared to k, we will use the theory of shift registers for a construction which is applied for considerably larger b compared to k. Such constructions for (\u03c3k, k)-constrained de Bruijn codes are relatively simple as each sequence in the code is formed by concatenations of the same de Bruijn \f16 x 0 1 00 11 111 0 1 1 0 0 1 1 1 0 0 Fig. 3. Reduced state diagram to count the number of (3, 3)-constrained de Bruijn sequences sequence. The disadvantage is that the rate of the related code is zero. In this section we present a construction of codes in which b is relatively large compare to k and the rate of the code is greater than zero. Let Pk be the set of all primitive polynomial of degree k over Fq. By Theorem 1, there are \u03c6(qk\u22121) k polynomials in Pk. Each polynomial is associated with a linear feedback shift register and a related m-sequence of length qk \u22121. Let SPk be this set of m-sequences. In each such sequence, each nonzero k-tuple over Fq appears exactly once as a window in one period of the sequence. We can single out that in each such sequence there is a unique run of k \u22121 zeroes and a unique run of length k for each other symbol. This run of the same symbol is the longest such run in each sequence. But, there is another window property for all the sequences in SPk. Lemma 4. Each (2k)-tuple over Fq appears at most once as a window in one of the sequences of SPk. Proof. Let f1(x) and f2(x) be two distinct primitive polynomials in Pk, whose state diagrams contain the cycles of length qk \u22121, C1 and C2, respectively. By Theorem 2, the polynomial f1(x)f2(x) has degree 2k and its state diagram contains the two cycles C1 and C2. Each (2k)-tuple appears exactly once in a window of length 2k in one of the cycles of the state diagram related to f1(x)f2(x). Hence, each such (2k)-tuple appears at most once in either C1 or C2. The windows of length k are distinct within each sequence and the windows of length 2k are distinct between any two sequences which implies the claim of the lemma. We are now in a position to describe our construction of constrained de Bruijn codes with a large constrained segment. Construction 1. Let SPk be the set of all m-sequences of order k over Fq. Each sequence will be considered as an acyclic sequence of length qk \u22121, where the unique run of k \u22121 zeroes is at the end of the sequence. We construct the following code C \u225c{(0\u01eb0=0s10\u01eb1, s20\u01eb2, . . . , s\u2113\u221210\u01eb\u2113\u22121, s\u21130\u01eb\u2113) : si \u2208SPk, 0 \u2264\u01ebi \u2264k + 1, 1 \u2264i \u2264\u2113} . Theorem 10. The code C contain (qk \u22121, 2k)-constrained de Bruijn sequences, each one of length at least \u2113(qk \u22121) and at most \u2113(qk + k). The size of C is M\u2113, where M = \u03c6(qk\u22121)(k+2) k and k \u22653. \f17 Proof. We \ufb01rst prove that each codeword of C is a (qk \u22121, 2k)-constrained de Bruijn sequence. Let s = 0\u01ebi\u22121si0\u01ebi, 1 \u2264i \u2264\u2113, be a substring of a codeword in C. We \ufb01rst note that except for the windows of length 2k having more than k \u22121 zeroes at the start or at the end of s, all the other windows of length 2k contained in s appear in the cyclic sequence si. These new windows (having more than k \u22121 zeroes) as well as the windows which contain the run of zeroes 0\u01ebi+k\u22121 appear only between two sequences sj and sj+1 and as such they are separated by at least qk \u2212k symbols. Moreover, each window of length 2k containing zeroes from one such run is clearly unique. This implies that each sequence of C is a (qk \u22121, 2k)-constrained de Bruijn sequence (as repeated windows of length 2k can occur only between different si\u2019s). Since each si has length qk \u22121 and 0 \u2264\u01ebi \u2264k +1, it follows that the length of a codeword is at least \u2113(qk \u22121) and at most \u2113(qk +k). The number of sequences that can be used for the substring si0\u01ebi is \u03c6(qk\u22121)(k+2) k since si can be chosen in \u03c6(qk\u22121) k ways (see Theorem 1) and 0\u01ebi can be chosen in k + 2 distinct ways. It implies that |C| = M\u2113. The codewords in the code C obtained via Construction 1 can be of different lengths. We will now construct a similar code in which all codewords have the same length. Let C\u2032 be a code which contains all the pre\ufb01xes of codewords from C. Let C1 def ={(s1s2) : s1 \u2208C, s2 \u2208C\u2032, length(s1s2) = \u2113(qk + k)} Theorem 11. The code C1 contain (qk \u22121, 2k)-constrained de Bruijn sequences, of length n = \u2113(qk + k) and its size is at least M\u2113, where M = \u03c6(qk\u22121)(k+2) k . Proof. The codewords in C1 are formed from the codewords in C by lengthening them to the required length with pre\ufb01xes of codewords in C. The lengthening does not change the structure of the codewords, it just change their length. Hence, the proof that the codewords are (qk \u22121, 2k)-constrained de Bruijn sequence is the same as in the proof of Theorem 10. The number of codewords is the same as in C if there is exactly one way to lengthen each codeword of C. Since there is usually more than one such way, it follows that the code will be of a larger size. Corollary 4. The rate of the code C1 is k qk+k. Construction 1 yields acyclic sequences. Can we \ufb01nd a related construction for cyclic constrained de Bruijn sequences with similar parameters? The answer is de\ufb01nitely positive. Construction 1 can be viewed as a construction for cyclic sequences of different length. To have a code with cyclic (qk \u22121, 2k)-constrained de Bruijn sequences of the same length (as the acyclic sequences in C1, we can restrict the values in the \u01ebi\u2019s. For example, we can require that P\u2113 i=1 \u01ebi = \u230a\u2113(k+1) 2 \u230b. This will imply similar results for the cyclic code as for the acyclic code. The code C1 can be slightly improve by size, but as it does not make a signi\ufb01cant increase in the rate we ignore the possible improvements. The same construction can be applied with a larger number of cycles whose length is \u03c3k \u22121 (\u03c3 not necessarily a prime power) and with a the same window length or even a smaller one. We do not have a general construction for such cycles, but we outline the simple method to search for them. Let D(\u03c3, k) be the set of de Bruijn sequences in G\u03c3,k. Let \u03b4 be a positive integer. Construct a graph whose vertices are all the de Bruijn sequences in D(\u03c3, k). Two vertices (de Bruijn sequences) are connected by an edge if they have the same window of length greater than k +\u03b4. Now let T be an independent set in the graph. Two sequences related to this independent set do not share a window of length k + \u03b4 or less, so we can apply \f18 Construction 1 and its variants, where 0 \u2264\u01ebi \u2264\u03b4. Note that in Construction 1, the set of sequences in SPk form an independent set with \u03b4 = k. A computer search for the 2048 binary de Bruijn sequences of length 32 was performed. An independent set of size 8 was found for \u03b4 = 5 compared to the 6 m-sequences of length 31. Furthermore, for \u03b4 = 4, an independent set of size 4 was found, and for \u03b4 = 2 the size of the independent set that was found was only 2. This implies that this direction of research can be quite promising. VI. APPLICATION TO THE \u2113-SYMBOL READ CHANNEL In this section, we show that cyclic (b, k)-constrained de Bruijn codes can be used to correct synchronization errors in the \u2113-symbol read channel [8], [14], [63]. Previously, only substitution errors were considered in such a channel. Each cyclic (b, k)-constrained de Bruijn sequence which forms a codeword in the channel can be used to correct a number of limited synchronization errors which might occurred. The correction does not depend on the other codewords of the code. The mechanism used in this section will be of help in correcting such errors in racetrack memory as discussed in the next section. We will consider now Fq as our alphabet although the method can be applied on alphabets of any size. The reason is that other error-correcting codes, for this purpose, are de\ufb01ned over Fq. De\ufb01nition 4. Let x = (x1, x2, . . . , xn) \u2208Fn q be a q-ary sequence of length n. In the \u2113-symbol read channel, if x is a codeword then the corresponding \u2113-symbol read vector of x is \u03c0\u2113(x) = ((x1, . . . , x\u2113), (x2, . . . , x\u2113+1), . . . , (xn, x1 . . . , x\u2113\u22121)). Note \ufb01rst, that x might not be a cyclic sequence, but the channel read the symbols cyclically, i.e., after the last symbol, the \ufb01rst symbol, second symbols and so on are read to complete an \u2113-symbol read in each position of \u03c0\u2113(x). In this channel, the \u2113-symbol read sequence \u03c0\u2113(x) is received with no error, it is simple to recover the codeword x. However, several types of errors, such as substitution errors and synchronization errors, can occur. Substitution errors were considered in [8], [14], [63], but a synchronization error might be a more common one. We focus now on the \u2113-symbol read channel with only synchronization errors which are deletions and sticky insertions. A sticky insertion is the error event when an \u2113-tuple (xi, . . . , xi+\u2113\u22121) is repeated, i.e., read more than once. A deletion is the error event when an \u2113-tuple is deleted. A burst of deletions of length at most b is an error in which at most b consecutive \u2113-tuples are deleted. Example 2. Let x = (0, 1, 0, 0, 1, 0, 0, 0) be a codeword. When \u2113= 2, the 2-symbol read sequence of x is \u03c02(x) = ((0, 1), (1, 0), (0, 0), (0, 1), (1, 0), (0, 0), (0, 0), (0, 0)). If there is a sticky insertion at the second location and a burst of deletions of length two at the 6-th and 7-th positions, we obtain the 2-symbol read sequence \u03c02(y) = ((0, 1), (1, 0), (1, 0), (0, 0), (0, 1), (1, 0), (0, 0)). The goal of this section is to show that a cyclic constrianed de Bruijn code is a code correcting sticky insertions and bursts of deletions in the \u2113-symbol read channel. Theorem 12. Let b and k be two positive integers such that qk \u2265b \u22652 and \u2113= k+b\u22122. The code C\u2032 DB(n, b, k) can correct any number of sticky-insertions and any number of bursts of deletions (depending only on n and b) whose length is at most b \u22122 (including one possible cyclic burst of this size) in the \u2113-symbol read channel. \f19 Proof. Let c = (c0, c1, . . . , cn\u22121) be a codeword and \u03c0\u2113(c) = (((c0, . . . , c\u2113\u22121), (c1, . . . , c\u2113), . . . , (cn\u22121, . . . , cn+\u2113\u22122)), where indices are taken modulo n, be a corresponding \u2113-symbol read sequence of c. Let \u03c0\u2113(y) = ((y1,1, . . . , y1,\u2113), (y2,1, . . . , y2,\u2113), . . . , (yt,1, . . . , yt,\u2113)) be a received \u2113-symbol read sequence. Clearly, since no more than b\u22122 consecutive deletions occurred, it follows that there exists an i, 1 \u2264i \u2264b\u22121, such that (ci, ci+1, . . . , ci+\u2113\u22121) = (y1,1, . . . , y1,\u2113). Similarly, there also exists a j, 0 \u2264j \u2264b \u22121, such that (ci+j, ci+j+1, . . . , ci+j+\u2113\u22121) = (y2,1, . . . , y2,\u2113). Since \u2113= k +b\u22122 and at most b\u22122 deletions occurred between the \ufb01rst two consecutive \u2113-reads, it follows that either the substring (y2,1, . . . , y2,k) is either a substring of (y1,1, . . . , y1,\u2113) or (y1,b, y1,b+1, . . . , y1,\u2113) = (y2,1, . . . , y2,k\u22121). If (y2,1, . . . , y2,k) is a substring of (y1,1, . . . , y1,\u2113), then since any b consecutive k-tuples of c are distinct, it follows that there exists a unique j, 0 \u2264j \u2264b \u22122, such that (ci+j, ci+j+1, . . . , ci+j+k\u22121) = (y2,1, . . . , y2,k). If j = 0 then a sticky insertion has occurred and if j = 1 no error occurred in the second read, and if 2 \u2264j \u2264b \u22122, then a burst of j \u22121 deletions has occurred. If a burst of b \u22122 deletion has occurred, then (y1,b, y1,b+1, . . . , y1,\u2113) = (y2,1, . . . , y2,k\u22121). The same process continue between the second read and the third read and so on until the last read and the \ufb01rst read which behave in the same way since also cyclically the longest run of deletions has at most length b \u22122. Thus, using the overlaps between consecutive reads we can recover the codeword c. VII. APPLICATIONS TO RACETRACK MEMORIES Racetrack memory is an emerging non-volatile memory technology which has attracted signi\ufb01cant attention in recent years due to its promising ultra-high storage density and low power consumption[51], [59]. The basic information storage element of a racetrack memory is called a domain, also known as a cell. The magnetization direction of each cell is programmed to store information. The reading mechanism is operated by many read ports, called heads. In order to read the information, each cell is shifted to its closest head by a shift operation. Once a cell is shifted, all other cells are also shifted in the same direction and in the same speed. Normally, along the racetrack strip, all heads are \ufb01xed and distributed uniformly[64]. Each head thus reads only a block of consecutive cells which is called a data segment. A shift operation might not work perfectly. When the cells are not shifted (or under shifted), the same cell is read again in the same head. This event causes a repetition (or sticky insertion) error. When the cells are shifted by more than a single cell location (or over shifted), one cell or a block of cells is not read in each head. This event causes a single deletion or a burst of consecutive deletions. We note that the maximum number of consecutive deletions is limited or in other words, the burst of consecutive deletions has limited length. An experimental result shows that the cells are over shifted by at most two locations with extremely high probability[64]. In this paper, we study both kinds of errors and refer to these errors as limited-shift errors. Since limited-shift errors can be modeled as sticky insertions and bursts of consecutive deletions with limited length, sticky-insertion/deletion-correcting codes can be applied to combat these limited-shift errors. Although there are several known sticky-insertion-correcting codes [18], [37], [46], deletion-correcting codes [5], [34], [43], single-burst-deletion-correcting codes [15], [56], and multiple-burst-deletion-correcting codes [33], there \f20 is a lack of knowledge on codes correcting a combination of multiple bursts of deletions and sticky insertions. Correcting these type of errors are especially important in the racetrack memories. In this section, motivated by the special structure of having multiple heads in racetrack memories, we study codes correcting multiple bursts of deletions and sticky insertions. To correct shift errors in racetrack memories with only a single head, Vahid et al. [60] recently studied codes correcting two shift errors of deletions and/or insertions. Another approach to combat limited-shift errors is to leverage the special feature of racetrack memories where it is possible to add some extra heads to read cells. If there is no error, the information read in these extra heads is redundant. However, if there are limited-shift errors, this information is useful to correct these errors. Recently, several schemes have been proposed to leverage this feature [12], [13], [64] in order to tackle this problem. However, in [12], [13], each head needs to read all the cells while in this model, each head only needs to read a single data segment. Our goal in this section is to present several schemes to correct synchronization errors in racetrack memories, all of them are based on constrained de Bruijn sequences and codes. In some of these schemes we add extra head and some are without adding extra heads. A. Correcting Errors in Racetrack Memories without Extra Heads Our \ufb01rst goal in this subsection is to construct q-ary b-limited t1-burst-deletion-correcting codes to combat synchronization errors in racetrack memories. Such a code can correct t1 bursts of deletions if the length of each such burst is at most b, i.e., at most b deletions occurred in each such burst and each two of these bursts are separated by symbols which are not in any error. Lemma 5. Let s be a (b, 1)-constrained de Bruijn sequence over an alphabet of size q. Let s(\u2206\u2212) be the sequence obtained from s after deleting all symbols speci\ufb01ed by the locations in the set \u2206\u2212such that the number of consecutive deletions is at most b \u22121. Then the set \u2206\u2212is uniquely determined from s and s(\u2206\u2212). Proof. Assume that \u2206\u2212= {i1, i2, . . . , it} where i1 < i2 < \u00b7 \u00b7 \u00b7 < it is the set of t locations of all t deleted symbols. Since i1 is the leftmost index in which a deletion has occurred, it follows that s[1, i1 \u22121] = s(\u2206\u2212)[1, i1 \u22121]. Furthermore, since s is a (b, 1)-constrained de Bruijn sequence, it follows that the symbols s[i] for i1 \u2264i \u2264i1 + b \u22121 are distinct. Since s[i1] was deleted and the maximum number of consecutive deletions is at most b \u22121, it follows that s(\u2206\u2212)[i1] \u0338= c[i1]. So, i1 is the leftmost index in which s and s(\u2206\u2212) differ. Therefore, we can determine i1 from the two vectors c and s(\u2206\u2212). Let \u2206\u2212 1 \u225c\u2206\u2212\\ {i1} = {i2, . . . , it}. To correct the \ufb01rst error, we insert the symbol s[i1] into the i1-th position of s(\u2206\u2212) and obtain the vector s(\u2206\u2212 1 ). Similarly, we can continue now to determine, one by one, the other positions where deletions have occurred using words s and s(\u2206\u2212 1 ). Thus, the set \u2206\u2212can be determined from c and c(\u2206\u2212) and the lemma is proved. We are now ready to present a construction of q-ary b-limited t1-burst-deletion-correcting codes. We will show that the maximum rate of these codes is close to the maximum rate of codes correcting multiple erasures, especially when q is large. A q-ary t-erasure-correcting code of length \u2113is a set of q-ary words of length \u2113in which one can correct any set of t-erasures, i.e. t known positions whose values are not known. Construction 2. Let s = (s1, s2, . . . , s\u2113) be a (b, 1)-constrained de Bruijn sequence over an alphabet of size q1. Let Cq2(\u2113, t) be a q2-ary t-erasure-correcting code of length \u2113and let q = q1 \u00b7 q2. For each word c = \f21 (c1, c2, . . . , c\u2113) \u2208Cq2(\u2113, t), we de\ufb01ne f(c, s) = (f1, f2, . . . , f\u2113), where fi = (ci, si) for all 1 \u2264i \u2264\u2113. Construct the q-ary code of length \u2113which is denoted by Cq(b, \u2113, t) Cq(b, \u2113, t) \u225c{f(c, s) : c \u2208Cq2(\u2113, t)} . Theorem 13. The code Cq(b, \u2113, t) obtained in Construction 2 is a q-ary (b \u22121)-limited t1-burst-deletioncorrecting code where t = t1 \u00b7 (b \u22121). Proof. Let f = (f1, . . . , f\u2113) \u2208Cq(b, \u2113, t) be a codeword of length \u2113. Let \u2206\u2212= {i1, . . . , it} be the set of all deleted positions such that i1 < \u00b7 \u00b7 \u00b7 < it and let f(\u2206\u2212) denote the received word. Since there are at most t1 bursts of deletions whose length is at most b \u22121, it follows that |\u2206\u2212| = t \u2264t1 \u00b7 (b \u22121) and there are at most b \u22121 consecutive integers in \u2206\u2212. From the received word f(\u2206\u2212), we can extract the unique pair of words (c(\u2206\u2212), s(\u2206\u2212)), where c(\u2206\u2212) and s(\u2206\u2212) are two words obtained from c and s, respectively, after deleting all the symbols speci\ufb01ed by the locations in the set \u2206\u2212. By Lemma 5, the set \u2206\u2212can be determined from the words s(\u2206\u2212) and s since s is a (b, 1)-constrained de Bruijn sequence. This imply that now we have a word of length \u2113in which t positions have unknown values, i.e. t erasures. Moreover, Cq2(\u2113, t) can correct up to t erasures. Hence, using the decoder of Cq2(\u2113, t), the codeword c can be recovered from c(\u2206\u2212) and \u2206\u2212. Thus, the codeword f = (f1, . . . , f\u2113) such that fi = (ci, si) for 1 \u2264i \u2264\u2113, can be recovered from c and s, and the theorem is proved. The proof of Theorem 13 implies a simple decoding algorithm to recover f. Let R(Cq2(\u2113, t)) = logq2 |Cq2(\u2113, t)| \u2113 denote the rate of the code Cq2(\u2113, t). By Theorem 7 there exists a (b, 1)-constrained de Bruijn sequences \u03c0 over an alphabet of size q1 if q1 \u2265b. By Construction 2, if s is a (b, 1)-constrained de Bruijn sequence of length \u2113 and there exists a q2-ary t-erasure-correcting code of length \u2113, then |Cq(b, \u2113, t)| = |Cq2(\u2113, t)|. If q1 = q/q2 = b, then the rate of the q-ary code Cq(b, \u2113, t) is R = logq |Cq(b, \u2113, t)| \u2113 = logq q2 \u00b7 logq2 |Cq2(\u2113, t)| \u2113 = logq(q/q1) \u00b7 R(Cq2(\u2113, t)) = (1 \u2212logq b) \u00b7 R(Cq2(\u2113, t)). It is well-known that a code with minimum Hamming distance t + 1 can correct t erasure errors. Moreover, for any 0 < \u01eb, \u03b4 < 1, there exists a \u201cnear MDS\u201d code [2] of length \u2113with minimum Hamming distance t = \u03b4 \u00b7 \u2113 and rate R(Cq2(\u2113, t)) \u22651\u2212\u03b4 \u2212\u01eb [32]. Hence, there exists a code of length \u2113correcting t = \u03b4 \u00b7\u2113erasures whose rate is R(Cq2(\u2113, t)) \u22651 \u2212\u03b4 \u2212\u01eb. Therefore, we have the following theorem. Theorem 14. Given 0 < \u03b4, \u01eb < 1, there exists a q-ary b-limited t1-burst-deletion-correcting code of length \u2113 such that its rate R satis\ufb01es R \u2265(1 \u2212logq(b + 1)) \u00b7 (1 \u2212\u03b4 \u2212\u01eb) , where t1 \u00b7 b = \u03b4 \u00b7 \u2113. The next goal is to study error-correcting codes to combat both under shift and limited-over-shift errors. For this purpose we start by generalizing Lemma 5. \f22 Lemma 6. Let s be a (b, 1)-constrained de Bruijn sequence over an alphabet of size q. Let s(\u2206\u2212, \u2206+) be the sequence obtained from s after deleting symbols in locations speci\ufb01ed by \u2206\u2212= {i1, . . . , it} and insertion of symbols in locations speci\ufb01ed by \u2206+. Assume further that i1 < \u00b7 \u00b7 \u00b7 < it and there are at most b\u22122 consecutive numbers in \u2206\u2212. Then the sets \u2206\u2212and \u2206+ are uniquely determined from s and s(\u2206\u2212, \u2206+). Proof. Since s is a (b, 1)-constrained de Bruijn sequence, it follows that between any two equal symbols there are at least b\u22121 different symbols. Hence, all the sticky insertions can be located and corrected. Now, Lemma 5 can be applied to \ufb01nd the positions of the deletions, i.e. to determine \u2206\u2212. Since, the locations of the sticky insertions are already known, it follows that \u2206\u2212can be now determined. It is now straightforward to generalize Theorem 13. An q-ary b-limited t1-burst-deletion sticky-insertioncorrecting code is a code over Fq which corrects any number of sticky insertions and t1 bursts of deletions, where each burst has length at most b. Theorem 15. The code Cq(b, \u2113, t) obtained in Construction 2 is a q-ary (b\u22122)-limited t1-burst-deletion stickyinsertion-correcting code where t = t1 \u00b7 (b \u22122). Corollary 5. Given 0 < \u03b4, \u01eb < 1, there exists a q-ary b-limited t1-burst-deletion sticky-insertion-correcting code of length \u2113, where t1 \u00b7 b = \u03b4 \u00b7 \u2113whose rate R satis\ufb01es R \u2265(1 \u2212logq(b + 2)) \u00b7 (1 \u2212\u03b4 \u2212\u01eb). It is clear that an upper bound on the maximum rate of our codes is at most 1 \u2212\u03b4, Since \u01eb is arbitrarily small, when b is small and q = 2m is large, it follows that the rates of our codes are close to the upper bounds and hence they are asymptotically optimal. B. An Acyclic \u2113-symbol Read Channel In this subsection, we consider a slight modi\ufb01cation of the \u2113-symbol read channel, where the symbols are not read cyclically. This acyclic \u2113-symbol read channel will be used in subsection VII-C to correct synchronization errors in the racetrack memories with extra heads. De\ufb01nition 5. Let x = (x1, x2, . . . , xn) \u2208Fn q be a q-ary sequence of length n. In the acyclic \u2113-symbol read channel, if x is the codeword then the corresponding \u2113-symbol read sequence of x is \u03c0\u2113(x) = ((x1, . . . , x\u2113), (x2, . . . , x\u2113+1), . . . , (xn, . . . , xn+\u2113\u22121)), where xi is chosen arbitrarily for i > n. In this channel, as in the cyclic \u2113-symbol read channel two type of synchronization errors can occur, bursts of deletions where the length of each one is restricted to at most b \u22122 deletions and sticky insertions. Example 3. Let x = (0, 1, 0, 0, 1, 0, 0, 0) be a stored information word. When \u2113= 2, the 2-symbol read sequence of x is \u03c02(x) = ((0, 1), (1, 0), (0, 0), (0, 1), (1, 0), (0, 0), (0, 0), (0, \u2217)), where \u2217can be any value. If there is a sticky insertion at the second location and a burst of deletions of length two at the 6-th and 7th locations, we obtain the 2-symbol read sequence \u03c02(y) = ((0, 1), (1, 0), (1, 0), (0, 0), (0, 1), (1, 0), (0, \u2217)) . \f23 \u21d3 \u21d3 c1,1 c1,2 c1,3 c1,4 c1,5 c2,1 c2,2 c2,3 c2,4 c2,5 c3,1 c3,2 c3,3 c3,4 c3,5 | {z } | {z } | {z } 1st data segment 2nd data segment 3rd data segment Fig. 4. Racetrack memory with two extra heads Theorem 16. Let b and k be two positive integers such that qk \u2265b \u22652, where \u2113= k + b \u22122. If the \ufb01rst \u2113-symbol read has no errors, then the code CDB(n, b, k) can correct any number of sticky-insertions and any number of bursts of deletions of length at most b \u22122 in the acyclic \u2113-symbol read channel. The proof of Theorem 16 is identical to the proof of Theorem 12 with only one distinction. In the cyclic channel we used the relation between the last \u2113-read and the \ufb01rst \u2113-read to determine where they overlap. This cannot be done in the acyclic case. Instead, we assummed that the \ufb01rst \u2113-read has no error. Hence, we can start generating the codeword from the \ufb01rst symbol and thus we can complete its decoding. C. Correcting errors in Racetrack Memories with Extra Heads In this subsection, we present our last application for (b, k)-constrained de Bruijn codes in the construction of codes correcting shift-errors in racetrack memories. Let N, n, m be three positive integers such that N = n \u00b7 m. The racetrack memory comprises of N cells and m heads which are uniformly distributed. Each head will read a segment of n cells. For example, in Fig. 4, the racetrack memory contains 15 data cells and three heads are placed initially at the positions of cells c1,1, c2,1, and c3,1, respectively. Each head reads a data segment of length 5. In general, if c = (c1, c2, . . . , cN) is the stored data then the output of the i-th head is ci = (ci,1, . . . , ci,\u2113) where ci,j = c(i\u22121)\u00b7n+j for 1 \u2264i \u2264m and 1 \u2264j \u2264n. Hence, an output matrix describing the output from all m heads (without error) is: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c1 c2 . . . cm \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c1,1 c1,2 . . . c1,n c2,1 c2,2 . . . c2,n . . . . . . ... . . . cm,1 cm,2 . . . cm,n \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . When an under-shift error (a sticky-insertion error) occurs, one column is added to the above matrix by repeating a related column of the matrix. When over-shift errors (a deletion error or a burst of b consecutive deletions) occur, one or a few consecutive columns in the matrix are deleted. Our goal is to combat these shift errors in racetrack memories. We note that each column in the above matrix can be viewed as a symbol in an alphabet of size q = 2m. In particular, let \u03a6m : Fm 2 7\u2192Fq be any bijection. For each column b cj = (c2,j, . . . , cm,j)T , \u03a6m(b cj) = vj \u2208Fq. Hence, by Corollary 5 , it is straightforward to obtain the following result. Proposition 1. Given 0 < \u03b4, \u01eb < 1, there exists a code correcting any number of under-shift errors and t1 over-shift errors, each of length at most b, with rate R \u2265m\u2212log2(b+2) m (1 \u2212\u03b4 \u2212\u01eb). Another way to combat these type of errors is to add some consecutive extra heads next to the \ufb01rst head. For example, in Fig. 4, there are two extra heads next to the \ufb01rst head. We assume in this section that there are \u2113\u22121 extra heads. Since there are two types of heads, we call the \u2113\u22121 extra heads secondary heads, \f24 while the \ufb01rst m uniformly distributed heads are the primary heads. Hence, there are \u2113heads which read the \ufb01rst data segment together, the \ufb01rst primary head and all the \u2113\u22121 secondary heads. For 2 \u2264i \u2264m, each other primary head will read one data segment individually. The output from the last (m \u22121) primary heads is c[n + 1, N] = (c2, . . . , cm), where \uf8eb \uf8ec \uf8ec \uf8ed c2 . . . cm \uf8f6 \uf8f7 \uf8f7 \uf8f8= \uf8eb \uf8ec \uf8ec \uf8ed c2,1 c2,2 . . . c2,n\u22121 c2,n . . . . . . ... . . . . . . cm,1 cm,2 . . . cm,n\u22121 cm,n \uf8f6 \uf8f7 \uf8f7 \uf8f8. This matrix can be viewed as a q2-ary word of length n where each column is a symbol in the alphabet of size q2 = 2m\u22121. In particular, let \u03a6m\u22121 : Fm\u22121 2 7\u2192Fq2 be any bijection. For each column b cj = (c2,j, . . . , cm,j)T , \u03a6m\u22121(b cj) = vj \u2208Fq2, and de\ufb01ne \u03a6(c[n + 1, N]) \u225c(\u03a6m\u22121(b c1), \u03a6m\u22121(b c2), . . . , \u03a6m\u22121(b cn)) = (v1, . . . , vn) = v \u2208Fn q2 . The output from all the \u2113heads in the \ufb01rst segment is \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c1,1 c1,2 . . . c1,\u2113 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed c1,1 c1,2 . . . c1,n\u22121 c1,n c1,2 c1,3 . . . c1,n \u2217 . . . . . . ... . . . . . . c1,\u2113 c1,\u2113+1 . . . \u2217 \u2217 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . It is readily veri\ufb01ed that this is the acyclic \u2113-symbol read sequence \u03c0\u2113(c[1, n]) for the \ufb01rst data segment c[1, n] = (c1,1, . . . , c1,n). This motivates the following construction. Construction 3. Let m, n, b, k, \u2113be positive integers such that \u2113= k + b \u22122, let c1 = c[1, n] = (c1, . . . , cn) \u2208 CDB(n, b, k), let Cq2(n, t) be a q2-ary t-erasure-correcting code of length n, where t = (b \u22122) \u00b7 t1, for some integer t1 and q2 = qm\u22121, and let c[n + 1, N] = (c2, . . . , cm), where \u03a6(c[n + 1, N]) \u225c(\u03a6m\u22121(b c1), \u03a6m\u22121(b c2), . . . , \u03a6m\u22121(b cn)) = (v1, . . . , vn) = v \u2208Cq2(n, t) . De\ufb01ne, C3(N, t1, b \u22122) \u225c{(\u03b11, \u03b12) : \u03b11 \u2208c[1, n], \u03b12 \u2208c[n + 1, N]}. Theorem 17. The code C3(N, t1, b \u22122) has size |CDB(n, b, k)| \u00b7 |Cq2(n, t)| and using \u2113\u22121 extra heads it can correct any number of sticky insertions and t1 bursts of deletions whose length is at most b \u22122. Proof. The size of the code is an immediate observation from the de\ufb01nition of the code C3(N, t1, b \u22122). The \ufb01rst data segment of length n consists of any sequence s from CDB(n, b, k). By Theorem 16, we can recover the sequence in the \ufb01rst data segment when there are any number sticky-insertions and bursts of deletions of length at most b \u22122. Moreover, we can also determine the locations of these errors. In the output from the last m \u22121 heads, all sticky insertions can be corrected easily and all deletions become erasures since we know the locations of these errors. There are at most t = t1 \u00b7 (b \u22122) erasures and the decoding procedure of the code Cq2(n, t) can be applied to correct these erasures. Thus, the sequence s can be recovered. Thus, the code C3(N, t1, b \u22122) can correct all sticky insertions and at most t1 bursts of deletions whose length at most b \u22122. \f25 Corollary 6. Consider a racetrack memory comprising of N = m \u00b7 n cells and m primary heads which are uniformly distributed. Using \u2113\u22121 extra secondary heads, it is possible to construct a code correcting a combination of any number of sticky insertions and t1 bursts of deletions whose length is at most b \u22122 such that its asymptotic rate satis\ufb01es lim N\u2192\u221e log2 |C3(N, t1, b \u22122)| N \u2265m \u22121 m \u00b7 (1 \u2212\u03b4 \u2212\u01eb) + RDB(b, k) m where \u2113\u2212b + 2 = h, t1 \u00b7 (b \u22122) = \u03b4 \u00b7 n and 0 < \u03b4, \u01eb < 1. Proof. We note that there exists a code Cq2(n, t) of length n correcting t = t1 \u00b7 (b \u22122) erasures with the asymptotic rate at least lim n\u2192\u221e logq2 |Cq2(n, t)| n \u22651 \u2212\u03b4 \u2212\u01eb. Since q2 = 2m\u22121 and N = mn, we have lim n\u2192\u221e log2 |Cq2(n, t)| N \u2265m \u22121 m (1 \u2212\u03b4 \u2212\u01eb). (4) Moreover, the asymptotic rate of the constrained de Bruijn code is RDB(b, k) = lim n\u2192\u221e log2 |CDB(n, b, k)| n . (5) Therefore, by (4), (5), and Theorem 17, the rate of the code C3(N, t1, b\u22122) in construction 3 can be computed as follows: lim N\u2192\u221e log2 |C3(N, t1, b \u22122)| N = lim N\u2192\u221e log2(|Cq2(n, t)||CDB(n, b, k)|) N \u2265m \u22121 m (1 \u2212\u03b4 \u2212\u01eb) + RDB(b, k) m . Corollary 6 can be compared with the result in Proposition 1. When b = h = 3, by Table I, we have that RDB(3, 3) \u22480.7946. Hence, using two more extra heads, the asymptotic rate of the relate code is m\u22121 m (1 \u2212 \u03b4 \u2212\u01eb) + 0.7946 m = 1 \u2212\u03b4 \u2212\u01eb + 0.7946\u22121+\u03b4+\u01eb m . We note that, without using extra head, the maximal asymptotic rate is 1 \u2212\u03b4. Hence, when 1 \u2212\u03b4 < 0.7946, using two extra heads, the asymptotic rate of our constructed code is higher than the maximal asymptotic rate of codes without extra heads. VIII."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1912.11617v1",
+                    "title": "Efficient Algorithm for the Linear Complexity of Sequences and Some Related Consequences",
+                    "abstract": "The linear complexity of a sequence $s$ is one of the measures of its\npredictability. It represents the smallest degree of a linear recursion which\nthe sequence satisfies. There are several algorithms to find the linear\ncomplexity of a periodic sequence $s$ of length $N$ (where $N$ is of some given\nform) over a finite field $F_q$ in $O(N)$ symbol field operations. The first\nsuch algorithm is The Games-Chan Algorithm which considers binary sequences of\nperiod $2^n$, and is known for its extreme simplicity. We generalize this\nalgorithm and apply it efficiently for several families of binary sequences.\nOur algorithm is very simple, it requires $\\beta N$ bit operations for a small\nconstant $\\beta$, where $N$ is the period of the sequence. We make an analysis\non the number of bit operations required by the algorithm and compare it with\nprevious algorithms. In the process, the algorithm also finds the recursion for\nthe shortest linear feedback shift-register which generates the sequence. Some\nother interesting properties related to shift-register sequences, which might\nnot be too surprising but generally unnoted, are also consequences of our\nexposition.",
+                    "authors": "Yeow Meng Chee, Johan Chrisnata, Tuvi Etzion, Han Mao Kiah",
+                    "published": "2019-12-25",
+                    "updated": "2019-12-25",
+                    "primary_cat": "cs.CR",
+                    "cats": [
+                        "cs.CR",
+                        "cs.SC"
+                    ],
+                    "main_content": "INTRODUCTION Binary sequences with good pseudorandomness and complexity properties are widely used as keystreams in cryptographic applications [17], [18]. Among the measures commonly used to measure the complexity of a sequence s is its linear complexity c(s), de\ufb01ned to be the length of the shortest linear feedback shift-register that generates s. Sequences of low linear complexity are fully determined via a solution of c(s) linear equations if 2c(s) consecutive terms of the sequence are known. Hence, high linear complexity is a prerequisite for cryptographic applications, and during the last \ufb01fty years there has been an extensive research to \ufb01nd the linear complexity of sequences. The linear complexity of a sequence s of length N over a \ufb01nite \ufb01eld Fq can be determined with the well-known Berlekamp-Massey algorithm [2], [15] in O(N2) symbol \ufb01eld operations. This algorithm was implemented during the years in various ways, e.g. [10], [19]. The complexity of this algorithm was improved to O(N(log N)2 log log N) in [3], [4], [5]. All these algorithms considered sequences of any length over any \ufb01nite \ufb01eld Fq. But, in many applications, only periodic sequences are considered and hence the algorithm to \ufb01nd the linear complexity of such sequences, can be considerably improved. Games and Chan [11] proposed a simple algorithm which \ufb01nds the linear complexity c of any sequence s with period N = 2n. Implementation of their algorithm requires N bit operations on the sequence s and another n bit operations reduced from integer operations to compute c. In the following two decades a few algorithms were suggested to generalize this algorithm to binary sequences with other periods and periodic sequences over Fq. The complexity of these algorithms for sequences with period N were kept as low as \u03b2N for some constant \u03b2, but relatively much higher than the N + log N bit operations required for The Games-Chan Algorithm. The generalization for sequence with period pn over Fpt was given in [8], [13]. Xiao et al. [20], [21] gave an algorithm to compute the linear complexity of sequence with period N \u2208{pt, 2pt} over Fq, when q is a primitive root modulo p2. Chen [6] gave an algorithm for sequences over Fpt with period \u2113\u00b7 2n, where 2n|pt \u22121 and g.c.d.(\u2113, pt \u22121) = 1. Chen [7] generalized this algorithm to determine the linear complexity of sequences with period \u2113\u00b7 n over Fpt, where \u2113|pt \u22121 and g.c.d.(n, pt \u22121) = 1. The main idea in [7] is to reduce the calculation of the sequence with period \u2113\u00b7 n over Fpt to the calculation of the linear complexity of \u2113sequences with period n over Fpt. The \f2 algorithms in [6], [7], [20], [21] are designed for sequences over a \ufb01eld of odd order. The ideas in [6], [7] are generalized in [16] for binary sequences. The [16] Meidl presents the most ef\ufb01cient algorithm for computing the linear complexities of binary sequences of period N = \u2113\u00b72n. To apply Meidl\u2019s algorithm on a sequence s, one forms a family of sequences of length 2n from s and apply The Games-Chan Algorithm to each of these sequences. Then for speci\ufb01c values of \u2113, Meidl showed that the algorithm requires \u03b2N, where \u03b2 is a small constant. The algorithm in [16] is of interest for large N, a small odd integer \u2113such that the smallest k for which \u2113divides 2k \u22121 is not large. In this paper we consider as in [16] only the most important class of sequences, namely the binary sequences. We present an algorithm which is very similar in nature to The Games-Chan Algorithm and can be viewed as its direct generalization compared to previous algorithms, but it does not reduce most of the computations to The Games-Chan Algorithm. Moreover, the algorithm can handle ef\ufb01ciently sequences of even period as [16] and also binary sequences of some odd periods, which are not considered by previous algorithms. The algorithm is always ef\ufb01cient for large N, but also small N, depending on tradeoff between \u2113and n. Moreover, the polynomial which generates the sequence is also computed in the algorithm, a feature that does not exist in the algorithm of [16]. The algorithm can also compute the linear complexity of binary sequences of length pn1 1 pn2 2 \u00b7 \u00b7 \u00b7 pnt t , where the pi\u2019s are primes, the ni\u2019s are positive integers, and the polynomial Ppj\u22121 i=0 xipj m is primitive, for 0 \u2264m \u2264nj \u22121. The algorithm requires \u03b2N bit operations to compute the linear complexity of a binary sequence s of length N, where the constant \u03b2 is relatively small. Furthermore, our exposition presents also interesting, even if not surprising, results on properties of sequences and their linear complexities. The rest of this paper is organized as follows. In Section II, we present the necessary de\ufb01nitions for sequences, linear complexity, the related polynomials, and \ufb01nally The Games-Chan Algorithm is discussed. In Section III, our general method and its algorithm is presented. In Section IV, we consider a generalization of The Games-Chan Algorithm, for sequences generated by a power of a primitive polynomial. In Section V, we present the main idea of our algorithm on sequences with period p \u00b7 2n, where p is a prime. The cases which will be discussed are p = 3 and p \u22611(mod 4), where 2 is a generator modulo p. We compare the complexity of our algorithm to the algorithm presented in [16]. A generalization for odd period is presented in Section VI. This paper includes appendices which explain some claims in the version for the International Symposium of Information Theory 2019. All the computations are left for the full version of this paper. II. PRELIMINARIES Let s = {si}i\u22650 be an in\ufb01nite binary sequence. The sequence has a period N, if N is the least positive integer such that si = si+N for each i \u22650. Such a sequence is considered as a cyclic sequence and is be denoted by [s0, s1, . . . , sN\u22121]. Any periodic sequence satis\ufb01es a linear recursion si+m = a1si+m\u22121 + \u00b7 \u00b7 \u00b7 + am\u22121si+1 + amsi , i \u22650, (1) of order m \u2264N, where aj \u2208{0, 1} and all computations are done modulo 2. The linear complexity c(s) of s is de\ufb01ned as the least m for which (1) holds. Clearly c(s) \u2264N, for any sequence with period N, since si+N = si by de\ufb01nition. In terms of the shift operator E, de\ufb01ned as Esi = si+1 or E[s0, s1, . . . , sN\u22121] = [s1, . . . , sN\u22121, s0], the linear recursion (1) takes the form   Em + m X j=1 ajEm\u2212j ! si = 0 , i \u22650. (2) Since m is the smallest such integer, it implies that am \u0338= 0. If we de\ufb01ne f(E) \u225cEm + m X j=1 ajEm\u2212j , \f3 then we have f(E)si = 0 for each i \u22650. Replacing the shift operator E by the variable x, we have that f(x) = xm + Pm j=1 ajxm\u2212j, where am \u0338= 0. Let cm\u2212j = aj for 1 \u2264j \u2264m, and hence f(x) = xm + Pm j=1 cm\u2212jxm\u2212j = xm + Pm\u22121 j=0 cjxj, where c0 \u0338= 0. This implies that (2) takes the form f(E)si =   Em + m\u22121 X j=0 cjEj ! si = 0 , i \u22650. If f(x) = xm + Pm\u22121 j=0 cjxj and the sequence s = [s0, s1, . . . , sN\u22121] satis\ufb01es si+m = cm\u22121si+m\u22121 + cm\u22122si+m\u22122 + \u00b7 \u00b7 \u00b7 + c1si+1 + c0si , for each i \u22650, then we say that the polynomial f(x) generates the sequence s (or s is generated by f(x)). It implies the following observation. Lemma 1. If for s = [s0, s1, . . . , sN\u22121] the linear recurrence of the least degree m satis\ufb01es si+m = cm\u22121si+m\u22121 + cm\u22122si+m\u22122 + \u00b7 \u00b7 \u00b7 + c1si+1 + c0si , for each i \u22650, then   Em + m\u22121 X j=0 cjEj ! si = 0 , for each i \u22650. Henceforth, we denote by 0 (1, respectively) any sequence of any length which contains only zeroes (ones, respectively). Corollary 1. For any nonzero polynomial f(x) and any cyclic nonzero sequence s, f(E)s = 0, if and only if the sequence s is generated by f(x). Corollary 2. If s is a nonzero cyclic sequence generated by the nonzero polynomial f(x)m and f(E)m\u22121s = r, then s is not generated by f(x)m\u22121 if and only if the sequence r is a nonzero sequence generated by f(x). The sequence s generated by the polynomial f(x), can be described in terms of a linear feedback shiftregister sequence with a feedback function xm+1 = f(x1, x2, . . . , xm), where (x1, x2, . . . , xm) can be any binary m-tuple. As a special case, these m-tuples can be also those of the sequence s. The fundamental theory of shift-register sequences is given in [12]. A polynomial f(x) is irreducible if it has no nontrivial divisor polynomial of a smaller degree. An irreducible polynomial of degree k is called primitive if its nonzero roots are generators (primitive elements) of the \ufb01eld F2k. The nonzero cyclic sequence s generated by a primitive polynomial of degree k has length 2k \u22121, and each nonzero k-tuple appears exactly once in a window of length k in s. Such a sequence is called an m-sequence for maximal length linear shiftregister sequence [12]. The period of sequences generated by other irreducible polynomials (which are not primitive) can be also calculated by using the theory in [12]. For this purpose the following de\ufb01nition is given. The exponent of a polynomial f(x) is the smallest integer e such that f(x) divides xe \u22121. There is a connection between the exponent of any polynomial f(x) and the periods of the related sequences generated by its shift-register. It does not give immediately the period of the sequences for every polynomial f(x), but it does when f(x) is an irreducible polynomial. Theorem 3. If f(x) is an irreducible polynomial, then the nonzero sequences which it generates have period which is equal to the exponent of f(x). Each sequence s can be generated by several distinct polynomials, but it appears that the structure of the polynomials which generate s is determined by the polynomial of least degree which generates s. For a sequence s, the polynomial f(E) is a minimal zero polynomial for s if f(E) is a polynomial of the least degree such that f(E)s = 0. In other words, in view of Corollary 1, f(x) is a polynomial of \f4 least degree which generates s. The polynomial f(x) will be called a minimal (connection) polynomial that generates s. Lemma 2. If the two polynomials f(x) and g(x) generate the same sequence s, then h(x) = g.c.d.(f(x), g(x)) also generates s, where g.c.d.(\u03b1(x), \u03b2(x)) is the the greatest common divisor of the polynomials \u03b1(x) and \u03b2(x). Proof. By Corollary 1, f(x) generates s if and only if f(E)s = 0. Similarly, g(x) generates s if and only if g(E)s = 0. By the Euclidian algorithm, there exists two polynomials a(x) and b(x) such that h(x) = a(x)f(x) + b(x)g(x). Hence, h(E)s = a(E)f(E)s + b(E)g(E)s, and since f(E)s = 0 and g(E)s = 0, it follows that h(E)s = 0. Therefore, by Corollary 1, the polynomial h(x) generates s. Corollary 4. If s is a nonzero sequence, then it has a unique minimal zero polynomial. The linear complexity of the sequence s can be computed using the greatest common divisors of two related polynomials as follows. Let s(x) be the generating function of s considered as an in\ufb01nite sequence, de\ufb01ned by s(x) = s0 + s1x + s2x2 + s3x3 + \u00b7 \u00b7 \u00b7 , or by it periodic sequence sN(x) = s0 + s1x + s2x2 + s3x3 + \u00b7 \u00b7 \u00b7 + sN\u22121xN\u22121 . The generating function s(x) can be written as s(x) = sN(x) 1 \u2212xN = sN(x)/g.c.d.(sN(x), 1 \u2212xN) (1 \u2212xN)/g.c.d.(sN(x), 1 \u2212xN) = g(x) fs(x), where g(x) = sN(x)/g.c.d.(sN(x), 1 \u2212xN) , fs(x) = (1 \u2212xN)/g.c.d.(sN(x), 1 \u2212xN) . Obviously, g.c.d.(g(x), fs(x)) = 1, deg g(x) < deg fs(x), the polynomial fs(E) is the minimal zero polynomial of s, and deg fs(x) = c(s) [8]. Since the sequence s can be any binary sequence of period N, it follows that computing the linear complexity of s is equivalent to the computation of the degree of the greatest common divisor of sN(x) and XN \u22121. Finding the minimal zero polynomial of s is equivalent to the computation of greatest common divisor of sN(x) and XN \u22121. Hence, our algorithm can be regarded as an algorithm for the greatest common divisor of some cases. The computation of the linear complexity in [6], [7], [16], [20], [21] is based on these computations of the greatest common divisor. Our method is different as it is based on the direct de\ufb01nition of the linear complexity and on simple computations as done in the well-known Games-Chan Algorithm [11]. Binary sequences of period 2n are very important among all binary sequences. The linear complexity c(s) of a binary sequence s = [L R] of period 2n, where L and R are sequences of length 2n\u22121, can be recursively computed by The Games-Chan Algorithm as follows [11]: when L+R = 0, then c(s) = c(L); otherwise we set c(s) = 2n\u22121 + c(L + R). This algorithm is described in more details as follows. The Games-Chan Algorithm: The input to the algorithm is a sequence s of period 2n. If s \u0338= 0, the complexity c of s is computed recursively as follows. Initially, set cn = 0 and An = s. At a typical step of the algorithm the left half of Am, L(Am) = [b0, \u00b7 \u00b7 \u00b7 , b2m\u22121\u22121], is added to the right half, R(Am) = [b2m\u22121, \u00b7 \u00b7 \u00b7 , b2m\u22121], the result being a sequence Bm, of length 2m\u22121. If Bm = 0, Am is replaced by Am\u22121 = L(Am), and the complexity is left unchanged, i.e., cm\u22121 = cm. If Bm \u0338= 0, Am is replaced by Am\u22121 = Bm, and cm is replaced by cm\u22121 = cm + 2m\u22121. The complexity of s is given by c(s) = c0 + 1. \f5 The number of recursive steps in the algorithm is n for a sequence of period N = 2n, and this is the number of integer computations (on integers of length at most n bits) to obtain c(s). Since each such integer addition is for a distinct power of two, no more than a total of n bit operations are required. The algorithm also has to make at most N bit computations (comparisons or additions) for a sequence of period N = 2n. The reason is that in the \ufb01rst step 2n\u22121 such operations are required; and the number of operations is reduced by half from one step to the following step. Thus, Theorem 5. The complexity in The Games-Chan Algorithm is at most N +n bit operations, for a sequence of period N = 2n. III. A GENERAL METHOD FOR ANY BINARY SEQUENCE Let s be a binary sequence of period at most N, which implies that (EN \u22121)s = 0. Assuming that we can factorize xN \u22121 (or equivalently EN \u22121) ef\ufb01ciently into irreducible factors, we want to \ufb01nd the minimal zero polynomial g(E) of s, i.e. the smallest factor of EN \u22121 for which g(E)s = 0 (or equivalently the factor of xN \u22121, with the smallest degree, which generates s). Let xN \u22121 = q1(x)\u03b11q2(x)\u03b12 . . . qt(x)\u03b1t, where the qi(x)\u2019s are distinct irreducible polynomials and 1 \u2264\u03b1i \u22642\u03b3i for some nonnegative integer \u03b3i, 1 \u2264i \u2264t. We want to \ufb01nd the polynomial of the smallest degree g(x) = q1(x)\u03b41q2(x)\u03b42 . . . qt(x)\u03b4t, such that g(E)s = 0, where 0 \u2264\u03b4i \u22642\u03b3i, 1 \u2264i \u2264t. Since at least one of the \u03b4i\u2019s is nonzero, we assume w.l.o.g. that \u03b4t \u0338= 0. For any irreducible polynomial q(x), let S(q(x)) be the set of all nonzero cyclic sequences generated by q(x). For example, S(x4 + x3 + x2 + x + 1) = {00011, 01010, 11011}. Lemma 3. Let q(x) be an irreducible polynomial and r \u2208S(q(x)). Then for any polynomial f(x), such that g.c.d.(q(x), f(x)) = 1, we have that f(E)r \u2208S(q(x)). Proof. Suppose that s = f(E)r, for some sequence r \u2208S(q(x)). Assume \ufb01rst that s = 0. It implies by Corollary 1 that f(x) generates r. Since q(x) also generates r, it follows by Lemma 2 that g.c.d.(q(x), f(x)) generates r. But g.c.d.(q(x), f(x)) = 1 and hence g.c.d.(q(x), f(x)) does not generates r. Thus, s is a nonzero sequence. Since r \u2208S(q(x)) it follows that q(E)s = q(E)(f(E)r) = f(E)(q(E)r) = f(E)0 = 0. Since s is a nonzero sequence, it follows that s = f(E)r \u2208S(q(x)). Lemma 3 leads to two interesting consequences. Corollary 6. If q(x) is an irreducible polynomial and r is a nonzero sequence generated by q(x), then for any polynomial f(x), f(E)r = 0 if and only if f(x) is divisible by q(x). Proof. If f(x) is divisible by q(x), i.e. f(x) = h(x)q(x), then f(E)r = h(E)q(E)r = h(E)0 = 0. Now, suppose that f(E)r = 0 and assume that f(x) is not divisible by q(x). Since q(x) is an irreducible polynomial, it implies that gcd(q(x), f(x)) = 1. Therefore, by Lemma 3, f(E)r \u2208S(q(x)) and hence f(E)r \u0338= 0, a contradiction. Thus, f(x) is divisible by q(x). The second consequence from Lemma 3 is the well known Shift and Add property for m-sequences [12]. The version given here generalizes the one which is usually used. Corollary 7. Let s be an m-sequence generated by a polynomial q(x). If f(x) is any other polynomial for which g.c.d.(q(x), f(x)) = 1, then f(E)s = s. Proof. Since s is an m-sequence, it follows that q(x) is a primitive polynomial and s is the only sequence in S(q(x)). The claim is now an immediate consequence from Lemma 3. \f6 Theorem 8. Let s be a binary sequence, whose minimal polynomial is g(x) = q1(x)\u03b41 . . . qt(x)\u03b4t, where the qi\u2019s are distinct irreducible polynomials and \u03b4i \u22651, 1 \u2264i \u2264t. Then, q1(E)\u03b41q2(E)\u03b42 . . . qt\u22121(E)\u03b4t\u22121qt(E)\u03b4t\u22121s \u2208S(qt(x)). Furthermore, for every 1 \u2264i \u2264t \u22121, let di be an integer such that di \u2265\u03b4i. Then, q1(E)d1q2(E)d2 . . . qt\u22121(E)dt\u22121qt(E)\u03b4t\u22121s \u2208S(qt(x)). Proof. If s\u2032 = q1(E)\u03b41q2(E)\u03b42 . . . qt\u22121(E)\u03b4t\u22121qt(E)\u03b4t\u22121s, then since g(x) is the minimal polynomial of s, it follows that s\u2032 \u0338= 0 and qt(E)s\u2032 = 0. This implies that s\u2032 \u2208S(qt(x)). Since s\u2032 \u0338= 0, qt(E)s\u2032 = 0, and g.c.d.(q1(x)d1\u2212\u03b41q2(x)d2\u2212\u03b42 . . . qt\u22121(x)dt\u22121\u2212\u03b4t\u22121, qt(x)) = 1, it follows by Lemma 3 that q1(E)d1\u2212\u03b41q2(E)d2\u2212\u03b42 . . . qt\u22121(E)dt\u22121\u2212\u03b4t\u22121s\u2032 \u2208S(qt(x)), and hence q1(E)d1q2(E)d2 . . . qt\u22121(E)dt\u22121qt(E)\u03b4t\u22121s \u2208S(qt(x)). Now, for our given sequence s of period N, we want to \ufb01nd the smallest \u03b4t \u22651 such that q1(E)\u03b41 . . . qt(E)\u03b4ts = 0. By Theorem 8 it is suf\ufb01cient to \ufb01nd the smallest \u03b4t \u22651, such that q1(E)2\u03b31q2(E)2\u03b32 . . . qt\u22121(E)2\u03b3t\u22121qt(E)\u03b4t\u22121s \u2208S(qt(x)). The algorithm to \ufb01nd \u03b4t for s: Let r = q1(E)2\u03b31q2(E)2\u03b32 . . . qt\u22121(E)2\u03b3t\u22121s. If r = 0, then \u03b4t = 0, and proceed to \ufb01nd \u03b4t\u22121. Otherwise, set r\u2032 = r and \u03b4t = 0 (0 < \u03b4t \u22642\u03b3t in this case). For each j from 1 to \u03b3t, let Aj = q2\u03b3t\u2212j t (E)r\u2032 Case 1 : If Aj = 0, then proceed to the next iteration. Case 2 : If Aj \u0338= 0, then set r\u2032 = Aj, \u03b4t = \u03b4t + 2\u03b3t\u2212j. Set \u03b4t = \u03b4t + 1 (as r\u2032 \u2208S(qt(x))) and either proceed to \ufb01nd \u03b4t\u22121 for s or set s = qt(E)r\u2032 and proceed to \ufb01nd \u03b4t\u22121 for s. IV. POWERS OF A PRIMITIVE POLYNOMIAL The Games-Chan Algorithm can be generalized in a trivial way to sequences generated by a power of a primitive polynomial. If the primitive polynomial f(x) has degree k, then the period of such sequences is given by the following lemma. Lemma 4. Let f(x) be a primitive polynomial of degree k. The polynomial f(x)m generates the all-zero sequence and 2(\u2113\u22121)k\u2212i sequences whose period are (2k \u22121) \u00b7 2i, for each 1 \u2264\u2113\u2264m, where i = \u2308log \u2113\u2309. Lemma 4 is a generalization of a similar result for f(x) = x + 1 [9]. Sequences of period (2k \u22121) \u00b7 2i can have also other minimal zero polynomials and they will be discussed later, as the linear complexity of sequences with these periods should be computed by the algorithm presented in this paper. At this point we will consider the linear complexity of such sequences which are generated by a polynomial f(x)m, i.e., we would like to \ufb01nd the minimal m for such sequences of period (2k \u22121) \u00b7 2n. This will be done with an algorithm which is a straightforward generalization of The Games-Chan Algorithm. Power of a Primitive Polynomial (PPP) Algorithm: The input for the algorithm is a primitive polynomial f(x) of degree k (whose m-sequence is r), and a nonzero sequence s of length (2k \u22121)2n, whose minimal zero polynomial is f(E)m. The output of the algorithm is m, such that f(E)m is the minimal zero polynomial of s. We are looking for the minimal m such that f(E)ms = 0. By Corollary 2, it implies that f(E)m\u22121s = r. Hence, if n = 0 then m = 1 (this will be recognized by the formal steps of the algorithm) and the algorithm comes to its end. \f7 Set sn = s and mn = 0 and apply the n iterations of the algorithm; at iteration j, 1 \u2264j \u2264n, we have a sequence sn\u2212j+1 = [Ln\u2212j+1 Rn\u2212j+1], whose length is (2k \u22121) \u00b7 2n\u2212j+1. We also have a variable mn\u2212j+1, where initially mn = 0. In iteration j, we perform the following computations. Let f(E)2n\u2212jsn\u2212j+1 = s\u2032, where s\u2032 has length (2k\u22121)\u00b72n\u2212j (possibly of a smaller period) and distinguish between two cases: Case 1: If s\u2032 \u0338= 0, then set mn\u2212j = mn\u2212j+1 + 2n\u2212j, sn\u2212j = s\u2032, and proceed to the next iteration. Case 2: If s\u2032 = 0, then m \u22121 < mn\u2212j+1 + 2n\u2212j. Hence, set mn\u2212j = mn\u2212j+1 and sn\u2212j = Ln\u2212j+1. After the last iteration s0 = r, we set m = m0 + 1, and the algorithm comes to its end. The linear complexity of the input sequence s is km. The PPP algorithm can be generalized for any irreducible polynomial. In fact, there is no need to make any modi\ufb01cation in the algorithm. The only difference is that the m-sequence r is replaced by a set of sequences {r1, r2, . . . , rt} (see Corollary 2), where the period of ri is equal to the exponent of the irreducible polynomial f(x). After the last iteration the sequence s0 which is obtained is ri, i.e. f(E)m\u22121s = ri, for some 1 \u2264i \u2264t. But, the algorithm does not have to know which sequence rj was obtained. Hence, r1, r2, . . . , rt or the m-sequence r are not part of the inputs to the algorithm. Finally, note that the complexity of the PPP algorithm depends on the primitive polynomial. V. IMPLEMENTATION FOR SOME PERIODS p \u00b7 2n, p PRIME We are now in a position to implement our method for speci\ufb01c lengths of sequences, \u2113\u00b7 2n, where \u2113is odd. We will consider as examples the cases in which \u2113= p is a prime such that p = 3 or p \u22611(mod 4), where 2 is a generator modulo p, and especially p = 5. We will concentrate on the number of bit operations required in these cases and compare them with the number of bit operations required by the method of Meidl [16]. The explanations for the exact steps in the implementation are long and sometimes it is required to split the sequences into a few parts to apply some tricky computations. The exact computations are given in Appendix A and Appendix B. We summarize the results in these appendices with the following concluding theorem. Theorem 9. Let s be a binary sequence of length N on which the algorithm is applied. 1) If N = 3 \u00b7 2n then 7 \u00b7 2n + 2n bit operations are required to implement the algorithm. 2) If N = 5 \u00b7 2n then 16 3 4 \u00b7 2n + 2n bit operations are required to implement the algorithm. 3) If N = p\u00b72n, where p \u22611(mod 4) and 2 is a generator modulo p, then p2+7p+7 4 \u00b72n+2n bit operations are required to implement the algorithm. As for comparison with the method of Meidl [16], one can verify that for binary sequences of length 3 \u00b7 2n, the implementation requires 8 \u00b7 2n + 4n bit operations, while for sequences of length 5 \u00b7 2n, the implementation requires 20 \u00b7 2n + 10n bit operations. For general length \u2113\u00b7 2n it is too dif\ufb01cult to compare since it is dif\ufb01cult to compute the exact number of bit operations required in the algorithm of [16]. We just note that in some case the algorithm in [16] is more ef\ufb01cient, e.g. for N = 7 \u00b7 2n. VI. ALGORITHM FOR SEQUENCES WITH ODD PERIOD When we are given a binary sequence of odd period, the situation is more complicated and our algorithm is not ef\ufb01cient in many cases. We consider two cases in which our algorithm is ef\ufb01cient for a binary sequence s of odd period \u2113. Case 1: \u2113= pn, p prime, 2 is a generator modulo p. It is well known that the polynomial Pp\u22121 j=0 xj is irreducible if and only if 2 is a generator modulo p. For any given integer i \u22650 the polynomial Pp\u22121 j=0 xj\u00b7pi is also irreducible in this case [14]. Now, one can verify that xpn \u22121 is a multiplication of n + 1 irreducible polynomials, where the ith polynomial is gi(x) = Pp\u22121 j=0 xjpi, 0 \u2264i \u2264n \u22121, i.e. xpn \u22121 = (x + 1) Qn\u22121 i=0 gi(x). Our algorithm can be made now simpler and more ef\ufb01cient. The input sequence s has period at most pn. Initialize s\u2032 = s, c = 0, and f(x) = 1, where c will be the linear complexity of the sequence s and f(x) will be the minimal zero polynomial of s. Consider the mth iteration of the algorithm, 1 \u2264m \u2264n. At each iteration we have \f8 to check if s\u2032 whose length is pn\u2212m+1 has period pn\u2212m+1. For this, it is suf\ufb01cient to check if the \ufb01rst (p \u22121)pn\u2212m bits of r \u25b3 = (Epn\u2212m + 1)s\u2032 are equal to 0 using (p \u22121)pn\u2212m bit operations. 1) If r = 0, then replace s\u2032 with the \ufb01rst pn\u2212m bits of s\u2032. 2) If r \u0338= 0, then set c = c + (p \u22121)pn\u2212m, f(x) = gn\u2212m(x)f(x) and s\u2032 = gn\u2212m(E)s\u2032. After the n iterations if s\u2032 = 1, then replace c with c + 1 and f(x) with (x + 1)f(x). Each one of the two steps in the n iterations requires (p\u22121)pn\u2212m bit operations. The total number of bit operations required by the algorithm (omitting the additions to compute c) is 1 + Pn m=1 2(p \u22121)pn\u2212m = 1 + (2pn \u22122) \u22642pn = 2N. Case 2: \u2113= pn1 1 pn2 2 \u00b7 \u00b7 \u00b7 pnt t , pi prime, 2 is a generator modulo pi, ni is a positive integer. This case is solved similarly to Case 1, by considering t steps, one for each prime. ACKNOWLEDGMENT Y. M. Chee and H. M. Kiah were supported by the Singapore Ministry of Education under Grant MOE2015-T2-2-086. J. Chrisnata and T. Etzion were supported by the ISF grant no. 222/19."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1906.12073v1",
+                    "title": "Access Balancing in Storage Systems by Labeling Partial Steiner Systems",
+                    "abstract": "Storage architectures ranging from minimum bandwidth regenerating encoded\ndistributed storage systems to declustered-parity RAIDs can be designed using\ndense partial Steiner systems in order to support fast reads, writes, and\nrecovery of failed storage units. In order to ensure good performance,\npopularities of the data items should be taken into account and the frequencies\nof accesses to the storage units made as uniform as possible. A proposed\ncombinatorial model ranks items by popularity and assigns data items to\nelements in a dense partial Steiner system so that the sums of ranks of the\nelements in each block are as equal as possible. By developing necessary\nconditions in terms of independent sets, we demonstrate that certain Steiner\nsystems must have a much larger difference between the largest and smallest\nblock sums than is dictated by an elementary lower bound. In contrast, we also\nshow that certain dense partial $S(t, t+1, v)$ designs can be labeled to\nrealize the elementary lower bound. Furthermore, we prove that for every\nadmissible order $v$, there is a Steiner triple system $(S(2, 3, v))$ whose\nlargest difference in block sums is within an additive constant of the lower\nbound.",
+                    "authors": "Yeow Meng Chee, Charles J. Colbourn, Hoang Dau, Ryan Gabrys, Alan C. H. Ling, Dylan Lusi, Olgica Milenkovic",
+                    "published": "2019-06-28",
+                    "updated": "2019-06-28",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.CO",
+                        "math.IT"
+                    ],
+                    "main_content": "Introduction Distributed storage systems [14, 30], systems for batch coding [31], and multiserver private information retrieval systems [17] have each employed combinatorial designs for data placement, so that elements of the design are associated with data items and blocks with storage units. In these contexts, the most common types of designs employed are t-designs and t-packings. A t-(v, k, \u03bb) packing is a pair (X, B), where X, the point set, is a v-set and and B is a collection of k-subsets (blocks) of X such that every t-subset of X is contained in at most \u03bb blocks. The packing is a t-(v, k, \u03bb) design when every t-subset of X is a subset of exactly \u03bb blocks. A t-(v, k, 1) design is a Steiner system, denoted by S(t, k, v). A 2-(v, 3, 1) design is a Steiner triple system of order v, denoted by STS(v). When \u03bb = 1, a t-(v, k, 1) packing is also referred to as a partial S(t, k, v) or partial Steiner system. When data items are of the same size, and data is placed on storage units using a t-design, placement of data is uniform across the storage units. Indeed in t-(v, k, \u03bb) design, every point appears in exactly r = \u03bb(v t) (k t) blocks; this is the replication number of the design. In order to understand why Steiner systems can be employed in data placement, we outline some examples. Large-scale distributed storage systems (DSS) must address potential loss of storage units, while not losing data. One solution is to replicate each data item and distribute these replicas among multiple storage nodes; systems such as the Hadoop Distributed File System and the Google File System employ this strategy [8]. One can further mitigate information loss by sensibly organizing the data. For example, exact Minimum Bandwidth Regenerating (MBR) codes [14] consist of two subcodes, an outer MDS code along with an inner fractional repetition code (FRC) that support redundancy and repairability, respectively. To make this precise, an (n, k, d)-DSS with k \u2264d \u2264n consists of n storage nodes in which a read can be accomplished by access to k nodes and a failed node recovered by access to d nodes. A fractional repetition code C [14] with repetition degree \u03c1 for an (n, k, d)-DSS is a collection C of n subsets V1, V2, . . . , Vn of a set V , |V | = v, and of cardinality d each, satisfying the condition that each element of V belongs to exactly \u03c1 di\ufb00erent sets in the collection. The rate of the FRC is minI\u2282[n],|I|=k | \u222ai\u2208I Vi|. To optimize the rate and ensure correct repetition and repair, we require that |Vi \u2229Vj| \u22641 whenever i \u0338= j. When \u03c1 = v\u22121 d\u22121, such an FRC is a Steiner 2-(v, d, 1) design with replication number \u03c1, where the set of (coded) \ufb01le chunks V is the set of points and the set of storage nodes {V1, . . . , Vn} is the set of blocks of the design. Steiner systems also prove useful for applications needing both high data availability and throughput, such as transaction processing. The storage systems underlying these applications require uninterrupted operation, satisfying user requests for data even in the event of disk failure and repairing these failed disks, on-line, in parallel. Continuous operation alone is not su\ufb03cient, because such systems cannot a\ufb00ord to su\ufb00er signi\ufb01cant loss of performance during disk failures. Declustered-parity RAIDS (DPRAIDs) are designed to satisfy these requirements [7,21]. Like standard RAIDs (short for \u201cRedundant Arrays of Inexpensive Disks\u201d), DPRAIDs handle disk failure by using parity-encoded redundancy, in which subsets of the stored data (called parity stripes) are XORed together to store a single-errorcorrection code. Unlike standard RAIDs, however, all disks in the DPRAID cooperate in the 2 \freconstruction of all the data units on a single failed disk. One can represent a DPRAID as a t-(v, k, \u03bb) design (X, B), with X (|X| = v) being the set of disks in the array, and B being the set of all parity stripes, each of size k. Then each disk occurs in the same number c of parity stripes, guaranteeing that the reconstruction e\ufb00ort is distributed evenly. Although designs arise naturally in balancing data placement, little attention has been paid to the relative popularity of the data items. However, one can exploit popularity information in order to improve the relative equality of access among the storage units. Dau and Milenkovic [11] formulate a number of problems to address access balancing, by labeling the points of the underlying design. In order to introduce their problems and results, we \ufb01rst present more de\ufb01nitions and known results concerning designs. Although storage systems handle \u201chot\u201d (frequently accessed) and \u201ccold\u201d (infrequently accessed) data categories di\ufb00erently, they do not take the long-term popularity of the data items within each category into account, which may result in unbalanced access frequencies to the storage units. Access balancing can be achieved in part by selecting an appropriate packing or design, and by appropriate association of data items with elements of the packing or design. Dau and Milenkovic [11] propose a combinatorial model that ranks data items by popularity, and then strives to ensure that the sums of the ranks of the data elements in each block are not too small, not too large, or not too di\ufb00erent from block to block. In \u00a72 we summarize their model, state elementary bounds on various block sums, and provide a small but important improvement in the lower bound on the smallest possible di\ufb00erence among the block sums in a Steiner triple system. In \u00a73 we establish a close connection between such block sums and the size of a maximum independent set of elements in the packing or design. For certain designs, this connection can be used to show that, no matter how data items are associated with the elements of the design, the block sums must be far from the values dictated by the elementary bounds from \u00a72. Indeed, in order to approach the elementary bounds, one must select designs or packings with very speci\ufb01c properties; we pursue this in \u00a74. The described \ufb01ndings indicate the need to \ufb01nd speci\ufb01c S(t, k, v) designs, or at least \u2018dense\u2019 t-(v, k, 1) packings, to match the elementary bounds more closely. In \u00a75, we explore a construction of t-(v, t + 1, 1) packings that asymptotically match the bounds and contain almost the same number of blocks as the full Steiner system S(t, t + 1, v). Completion of the dense t-(v, t + 1, 1) packings to a Steiner system S(t, t + 1, v) appears problematic for general t; doing so without dramatically changing the block sums appears to be even more challenging. Nevertheless, in \u00a76, we pursue this to establish, for every admissible order v, the existence of a Steiner triple system of order v whose di\ufb00erence in block sums is at most an additive constant more than the elementary lower bound. 2 Point Labelings and Block Sums Let D = (V, B) be a t-(v, k, \u03bb) packing. A point labeling of D is a bijection rk : V 7\u2192 {0, . . . , v \u22121}; our interpretation is that rk maps an element to its rank by popularity. The reverse rk of a point labeling rk has rk(i) = v \u22121 \u2212rk(i) for each i \u2208{0, . . . , v \u22121}. With respect to a speci\ufb01c point labeling rk, de\ufb01ne sum(B, rk) = P x\u2208B rk(x) when B \u2208B. Then 3 \fde\ufb01ne MinSum(D, rk) = min(sum(B, rk) : B \u2208B); MaxSum(D, rk) = max(sum(B, rk) : B \u2208B); Di\ufb00Sum(D, rk) = MaxSum(D, rk) \u2212MinSum(D, rk); RatioSum(D, rk) = MaxSum(D, rk)/MinSum(D, rk). Following [11], one primary objective is to choose point labelings to maximize the MinSum and/or to minimize one of the other three. Access balancing is concerned primarily with minimizing the Di\ufb00Sum or RatioSum; because of the similarity between these two entities we often focus on the Di\ufb00Sum. Let RD denote the set of all point labelings of D. Noting that MaxSum(D, rk) = k(v \u22121) \u2212MinSum(D, rk), we de\ufb01ne MinSum(D) = max(MinSum(D, rk) : rk \u2208RD); MaxSum(D) = k(v \u22121) \u2212MinSum(D); Di\ufb00Sum(D) = min(Di\ufb00Sum(D, rk) : rk \u2208RD); RatioSum(D) = min(RatioSum(D, rk) : rk \u2208RD). If the storage system dictates the data layout and data items have the same size, we are free to permute the data items; this is captured by the selection of the point labeling rk. If we are also free to choose the t-(v, k, 1) packing that determines the data layout, we may select a packing to improve the sum metrics de\ufb01ned. In order to capture this, let Dt,k,v,b denote the set of all t-(v, k, 1) packings having exactly b blocks. Then de\ufb01ne MinSum(t, k, v, b) = max(MinSum(D) : D \u2208Dt,k,v,b); MaxSum(t, k, v, b) = k(v \u22121) \u2212MinSum(t, k, v, b); Di\ufb00Sum(t, k, v, b) = min(Di\ufb00Sum(D) : D \u2208Dt,k,v,b); RatioSum(t, k, v, b) = min(RatioSum(D) : D \u2208Dt,k,v,b). When b = (v t) (k t), the packing is a Steiner system S(t, k, v); in these cases we omit b from the notation to get MinSum(t, k, v) and similarly for all other entities. Theorem 2.1. [11] When D is a Steiner system S(t, k, v), MinSum(D) \u2264MinSum(t, k, v) \u22641 2(v(k \u2212t + 1) + k(t \u22122)); MaxSum(D) \u2265MaxSum(t, k, v) \u22651 2(v(k + t \u22121) \u2212kt); Di\ufb00Sum(D) \u2265Di\ufb00Sum(t, k, v) \u2265(v \u2212k)(t \u22121); RatioSum(D) \u2265RatioSum(t, k, v) \u2265 v(k+t\u22121)\u2212kt v(k\u2212t+1)+k(t\u22122). When k = t + 1, MinSum(D) \u2264(v \u22121) + \u0000t 2 \u0001 , MaxSum(D) \u2265t(v \u22121) \u2212 \u0000t 2 \u0001 , Di\ufb00Sum(D) \u2265 (t \u22121)(v \u2212t \u22121), and RatioSum(D) \u2265RatioSum(t, t + 1, v) \u2265 t(v\u22121)\u2212(t 2) (v\u22121)+(t 2) . When in addition t = 2 (D is a Steiner triple system), the stronger bounds Di\ufb00Sum(D) \u2265 v and RatioSum(D) \u22652 hold. 4 \fTheorem 2.1 provides bounds on the metrics across all Steiner systems S(t, k, v) and all point labelings of them. In previous work, the focus has been on the MinSum (or equivalently, by reversal, the MaxSum). Dau and Milenkovic [11] use the Bose [3] and Skolem [20,32] constructions of Steiner triple systems to establish the existence of an STS(v) D with MinSum(D) = v, the largest possible by Theorem 2.1 (These results are extended by Brummond [4] for Kirkman systems). They accomplish this by specifying a particular point labeling that meets the MinSum bound, but unfortunately the labeling chosen yields a MaxSum near 8 3v, a Di\ufb00Sum near 5 3v, and a RatioSum near 8 3, far from the bounds of 2v, v, and 2, respectively. The reversal of this labeling yields a MinSum far from optimal, the same Di\ufb00Sum, and a larger RatioSum. One might hope to improve the Di\ufb00Sum and RatioSum by choosing a di\ufb00erent labeling or by choosing a di\ufb00erent Steiner system S(t, k, v). In Section 3, we show that certain S(t, k, v)s cannot meet any of the bounds in Theorem 2.1. 2.1 Improved bounds for STSs There is an STS(7) with MinSum = 6 and MaxSum = 13 with blocks 016, 024, 035, 123, 145, 256, and 346 (here we write abc for {a, b, c}). There is an STS(9) with MinSum = 9 and MaxSum = 18 with blocks 018, 027, 036, 045, 126, 135, 147, 234, 258, 378, 468, and 567. However, we establish that these are the only two Steiner triple systems with Di\ufb00Sum = v, and indeed the only STS(v) with RatioSum = 2 is the STS(9). We \ufb01rst prove a useful lemma. Lemma 2.2. A 2-(x, 3, 1) packing on {0, . . . , x\u22121} with MaxSum x\u22121 has at most \u230a\u03c6(x)/3\u230b triples, where \u03c6(x) = x(x \u22121) 4 \u2212 jx 6 k \u2212 \uf8f1 \uf8f2 \uf8f3 0 if x \u22610, 1, 4 (mod 6) 1 2 if x \u22612, 3 (mod 6) 1 if x \u22615 (mod 6) Proof. We determine an upper bound on the number of pairs that could appear in triples of the packing. In total there are \u0000x 2 \u0001 pairs; of these, exactly \u230ax 2\u230bhave sum equal to x \u22121. For each pair {a, b} with a+b < x\u22121, the pair {x\u22121\u2212a, x\u22121\u2212b} has sum 2x\u22122\u2212(a+b) > x\u22121. It follows that the number of pairs with sum at most x\u22121 is x2 4 when x is even, and x2\u22121 4 when x is odd. Not all of these can appear together in a packing, as follows. Let a \u2208{0, . . . , \u230ax\u22122 3 \u230b}. Consider the pairs Pa = {{a, x \u22121 \u2212b} : a \u2264b \u22642a}. To place a pair of Pa in a triple of sum at most x \u22121, the third element must be from {0, . . . , a}, but it cannot be a. By the pigeonhole principle, at least one pair of Pa cannot be in a triple of the packing, reducing the number of pairs available by \u230ax+1 3 \u230b. Simple calculations now show that the number \u03c6(x) of pairs available is as given in the statement. Theorem 2.3. Let D be a Steiner triple system of order v \u226513. Then Di\ufb00Sum(D) \u2265v + 1 and RatioSum(D) > 2. Proof. Let v = 2x + 1, and consider an STS(2x + 1) D on elements {0, . . . , 2x}, noting that x \u22656. Partition {0, . . . , 2x} into three classes V0 = {0}, Vs = {1, . . . , x}, and V\u2113= 5 \f{x + 1, . . . , 2x}. Suppose to the contrary that Di\ufb00Sum(D) = v. Without loss of generality, MinSum(D) \u2208{v \u22121, v}, for otherwise we can apply the argument to the reversal of D. Let m = MinSum(D) and M = MaxSum(D). Because m \u2265v \u22121, all triples containing 0 contain one element of Vs and one of V\u2113, as follows. Consider the pair {0, w} with w \u2208Vs. The third element y completing its triple satis\ufb01es y \u22652x \u2212w \u22652x \u2212x = x. Now y \u0338= x because {0, x, x} cannot be a triple, so y > x, and hence y \u2208V\u2113. This accounts for all triples involving 0. Call a pair mixed if it contains an element of Vs and one from V\u2113, pure otherwise. Similarly a triple is pure if it lies entirely on Vs or V\u2113, mixed when it has two from one and one from the other. The number of mixed triples can be calculated as follows. There are x(x \u22121) mixed pairs not contained in triples containing 0, and each must be contained in a mixed triple. Because each mixed triple contains two mixed pairs, there are exactly 1 2x(x \u22121) mixed triples. Each mixed triple covers one pure pair. Hence the number of pure pairs to be covered by pure triples is x(x \u22121) \u22121 2x(x \u22121) = 1 2x(x \u22121), and there are 1 6x(x \u22121) pure triples. Form a collection Ds of triples on {0, . . . , x \u22121} by including {a, b, c} whenever {a + 1, b + 1, c + 1} is a pure triple on Vs, so that Ds contains triples each having sum at least m \u22123. The reversal Es then has each sum at most 3x \u2212m. Form a collection D\u2113of triples on {0, . . . , x \u22121} by including {a, b, c} whenever {2x \u2212a, 2x \u2212b, 2x \u2212c} is a pure triple on V\u2113, so that D\u2113contains triples each having sum at least 6x \u2212M. The reversal E\u2113then has each sum at most M \u22123x \u22123. Case 1. MinSum(D) = v and hence Maxsum(D) = 2v. Then Es and E\u2113both have maximum sum at most x \u22121. Applying Lemma 2.2 to Es and to E\u2113, D can contain at most 2\u03c6(x) pairs in pure triples, but 2\u03c6(x) < 1 2x(x \u22121), which yields the contradiction. (Only when v = 9 would there be no contradiction.) Case 2. MinSum(D) = v \u22121 and hence Maxsum(D) = 2v \u22121. Then Es has maximum sum x and E\u2113has maximum sum x \u22122. Because no edge involving element x \u22121 can appear in a triple of E\u2113, the number of pairs covered by triples is at most \u03c6(x \u22121) by Lemma 2.2. By a similar argument, the number of pairs covered by triples of Es is at most \u03c6(x+ 1) by Lemma 2.2. Because (x\u22121)(x\u22122) 4 + x(x+1) 4 = x(x\u22121) 2 + 1 2, \u03c6(x \u22121) + \u03c6(x + 1) < 1 2x(x \u22121), which yields the contradiction. (Only when v = 7 would there be no contradiction.) 3 Independent Sets Let D = (V, B) be a t-(v, k, \u03bb) packing. An independent set in D is a subset X \u2286V such that there is no B \u2208B with B \u2286X. An independent set I is maximal if there is no independent set Y with X \u2282Y , and maximum if there is no independent set Y such that |Y | > |X|. The independence number of D, denoted \u03b1(D), is the size of a maximum independent set. There is a close connection between the independence number of a packing and the quality of any of its labelings. Prior to establishing this fact, we \ufb01rst improve the lower bounds on the Di\ufb00Sum and RatioSum for Steiner triple systems. 6 \fLemma 3.1. A t-(v, k, \u03bb) packing D has MinSum at most k\u03b1(D) \u2212 \u0000k 2 \u0001 , MaxSum at least k(v \u22121 \u2212\u03b1(D)) + \u0000k 2 \u0001 , and Di\ufb00Sum at least k(v + k \u22122 \u22122\u03b1(D)). Proof. It su\ufb03ces to prove the statement for MinSum. No matter how D is given a point labeling, on elements with ranks in {0, . . . , \u03b1(D)}, there is a block. The sum of this block is at most Pk i=1(\u03b1(D) \u2212(i \u22121)). Corollary 3.1.1. Meeting the bound on MinSum in Theorem 2.1 for a t-(v, k, 1) packing D requires that \u03b1(D) \u2265v(k \u2212t + 1) 2k + k + t \u22123 2 . For example, Corollary 3.1.1 states that a necessary condition for a partial Steiner triple system D to have MinSum equal to v is that \u03b1(D) \u2265v 3 + 1. A single maximum independent set yields an upper bound on the MinSum or a lower bound on the MaxSum, but cannot be used simultaneously for both. We prove this next. Suppose that a t-(v, k, \u03bb) packing D contains two disjoint independent sets of sizes \u03b3D and \u03b4D, respectively, with \u03b3D \u2265\u03b4D. Set \u03b3\u2032 D = min \u0010 \u03b3D, v(k\u2212t+1) 2k + k+t 2 \u22121 \u0011 , \u03b4\u2032 D = min \u0010 \u03b4D, v(k\u2212t+1) 2k + k+t 2 \u22121 \u0011 . Two independent sets form a maximum independent pair when \u03b3\u2032 D+\u03b4\u2032 D is as large as possible. Lemma 3.2. A t-(v, k, \u03bb) packing D with a maximum independent pair of sizes (\u03b3D, \u03b4D) has Di\ufb00Sum at least k(v + k \u22122 \u2212\u03b4\u2032 D \u2212\u03b3\u2032 D). Proof. Form a point labeling of D in which the smallest x for which a block appears on {0, . . . , x \u22121} also has a block on {v \u2212x, . . . , v \u22121}; reverse the labeling if necessary to do this. Suppose to the contrary that this labeling has Di\ufb00Sum less than k(v +k \u22122\u2212\u03b4\u2032 D \u2212\u03b3\u2032 D). If no block appears on {0, . . . , v(k\u2212t+1) 2k + k+t 2 \u22122} set c = v(k\u2212t+1) 2k + k+t 2 \u22121. Otherwise let c be the smallest value for which a block appears on {0, . . . , c}, so that {0, . . . , c\u22121} forms an independent set. Proceed similarly to select d so that {v \u2212d, . . . , v \u22121} is an independent set. This labeling has MaxSum at least k(v \u22121 \u2212d) + \u0000k 2 \u0001 and MinSum at most kc \u2212 \u0000k 2 \u0001 , based on Theorem 2.1 and Lemma 3.1. It follows that the Di\ufb00Sum for this labeling is at least k(v + k \u22122 \u2212d \u2212c), so we have c + d > \u03b3\u2032 D + \u03b4\u2032 D. This contradicts the requirement that the maximum independent pair have sizes (\u03b3D, \u03b4D). Corollary 3.2.1. Meeting the bound on Di\ufb00Sum in Theorem 2.1 for a t-(v, k, 1) packing D requires that D have a maximum independent pair of sizes ( v(k\u2212t+1) 2k + k+t 2 \u22121, v(k\u2212t+1) 2k + k+t 2 \u22121). For a pair of independent sets to be maximum in this context, there is no requirement that either be maximum, nor is it required that their combined size be as large as possible. For a Steiner triple system, for example, Corollary 3.2.1 asks only for two disjoint independent sets, each of size at least v 3 + 1, for a combined size of 2v 3 + 2. Applying the 2v + 1 construction [9] 7 \ftwice to an STS(v), we form an STS(4v + 3) having a maximum independent pair of sizes (2v + 2, v + 1); despite the fact that the combined size is over 3 4 of the size of the STS, such a pair could not lead to a Di\ufb00Sum that meets the bound of Theorem 2.1, because the second largest of the pair is too small. Corollary 3.2.1 gives a necessary condition, not a su\ufb03cient one. Nevertheless, some bounds on the metrics can be stated. Lemma 3.3. When a t-(v, k, 1) packing D has two disjoint independent sets of sizes \u03b1 and \u03b2, there is a point labeling with MinSum(D) \u2265\u03b1 + \u0000k 2 \u0001 and (for the same labeling) MaxSum(D) \u2264k(v \u22121) \u2212\u03b2 \u22121 \u2212 \u0000k 2 \u0001 , so Di\ufb00Sum(D) \u2264k(v \u2212k) \u2212\u03b2 \u2212\u03b1 \u22121. Proof. Any point labeling assigning labels {0, . . . , \u03b1 \u22121} to the points of the independent set of size \u03b1, labels {v \u2212\u03b2, . . . , v \u22121} to the points of the independent set of size \u03b2, and labels {\u03b1, . . . , v \u2212\u03b2 \u22121} to the remaining points, meets the stated bounds. A Steiner system S(t, k, v) is 2-chromatic if its elements can be partitioned into two classes, both being independent sets. When a 2-chromatic S(3, 4, v) D exists (see, for example, [12,22,27]), Lemma 3.3 establishes that Di\ufb00Sum(D) \u22643v \u221217. Recall that Dau and Milenkovic [11] use the Bose and Skolem constructions of Steiner triple systems. In retrospect, this choice is well-justi\ufb01ed because the Bose construction leads to maximum independent pairs of sizes ( v 3 + 1, v 3 + 1) when v \u22613 (mod 6) and the Skolem construction leads to maximum independent pairs of sizes ( v+2 3 + 1, v+2 3 + 1) when v \u22611 (mod 6). One must hence focus on Steiner triple systems, and on t-(v, k, 1) packings in general, having large sizes in maximum independent pairs. This choice is important, because not all such systems have even a single large independent set, as we explain next. 4 Small Maximum Independent Sets Can one choose an arbitrary t-(v, k, 1) packing, and by cleverly choosing a point labeling optimize one or more of the sum metrics? If not, how far from the bound of Theorem 2.1 can the best point labeling be? In order to discuss these questions, de\ufb01ne \u03b1min(t, k, v) = min{\u03b1(D) : D is a t-(v, k, 1) packing}, and \u03b1\u22c6 min(t, k, v) = min{\u03b1(D) : D is an S(t, k, v)}. When an S(t, k, v) exists, \u03b1min(t, k, v) \u2264\u03b1\u22c6 min(t, k, v). Erd\u02dd os and Hajnal [15] establish that \u03b1min(2, 3, v) \u2265\u230a \u221a 2v\u230b; indeed a simple greedy algorithm produces an independent set of this size. A t-(v, k, 1) packing has each element in at most \u0000v\u22121 t\u22121 \u0001 / \u0000k\u22121 t\u22121 \u0001 = Qt\u22121 i=1 v\u2212i k\u2212i blocks. Applying a result of Spencer [33] generalizing Tur\u00b4 an\u2019s theorem for graphs, we obtain \u03b1min(t, k, v) \u2265ck v \u0000Qt\u22121 i=1 v\u2212i k\u2212i \u0001 1 k\u22121 8 \ffor ck a constant independent of v. For partial Steiner triple systems, this asserts that \u03b1min(2, 3, v) \u2265c \u00b7 v \u221a 2/\u221av \u22121, a small improvement on the Erd\u02dd os-Hajnal result. State-of-the-art lower bounds rely heavily on the following theorem, and all di\ufb00er only by constant factors. Theorem 4.1. [1] Let \u03ba \u22652 be a \ufb01xed integer. Let G be a (\u03ba + 1)-uniform hypergraph on n vertices. Then there are constants t0(\u03ba) and n0(\u03ba, \u03c4) so that whenever 1. G is uncrowded (i.e., has no 2-, 3-, or 4cycles); 2. the maximum degree \u2206(G) satis\ufb01es \u2206(G) \u2264\u03c4 \u03ba where \u03c4 \u2265t0(\u03ba); and 3. n \u2265n0(\u03ba, \u03c4), one has that \u03b1(G) \u2265.98 e \u00b7 10\u22125/\u03ba \u00b7 n \u03c4 \u00b7 (ln \u03c4)1/\u03ba. In order to apply this result to all t-(v, k, 1) packings, typically one selects a large subset of the blocks that are uncrowded. The approach is used to establish the lower bound in the following, while the upper bound is shown using the Lov\u00b4 asz Local Lemma: Theorem 4.2. [13,28] For \ufb01xed k and t, there are absolute constants c and d for which cv k\u2212t k\u22121(log v) 1 k\u22121 \u2264\u03b1min(t, k, v) \u2264dv k\u2212t k\u22121(log v) 1 k\u22121, Variations in Theorem 4.2 have resulted in improvements in the constants; see [16,23,24, 34]. It is possible in principle that restricting to Steiner systems, rather than packings, one might observe di\ufb00erent behaviour in the minima. However, Phelps and R\u00a8 odl [26] establish that the bounds of Theorem 4.2 apply to Steiner triple systems, not just to partial ones; that is, c \u221a v ln v \u2264\u03b1\u22c6 min(2, 3, v) \u2264d \u221a v ln v for absolute constants c and d. Grable, Phelps and R\u00a8 odl [19] establish similar statements when t \u2208{2, 3} for all k > t. For the applications intended, it is of interest to \ufb01nd independent sets of (at least) the size guaranteed e\ufb03ciently. For research in this vein, see [2,18]. Of course, one wants to \ufb01nd a pair of disjoint maximum independent sets whose total size is as large as possible, but this is NP-complete even for 3-uniform hypergraphs [25]. Remarkably, there is a polynomial time algorithm to determine whether an S(3, 4, v) contains two independent sets, each of size v/2 [10], but the ideas used do not appear to generalize. Nevertheless, the bounds on sizes of smallest maximum independent sets provides bounds on the best sum metrics one can hope to achieve. We state one such bound explicitly, showing that some Steiner systems have only point labelings far from the bounds of Theorem 2.1. Theorem 4.3. For in\ufb01nitely many orders v, there exists an STS(v) D with MinSum(D) \u2264 3c \u221a v ln v \u22123 and MaxSum(D) \u22653v \u22123c \u221a v ln v, and hence Di\ufb00Sum(D) \u22653v \u22126c \u221a v ln v+3. Hence we must focus on speci\ufb01c Steiner systems or packings, if we are to obtain sum metrics at or near the basic bounds. 9 \f5 Dense t-(v, t + 1, 1) Packings We establish next that one can obtain metrics close to the optimal when k = t+1 for packings that contain all but a vanishingly small fraction of the blocks of an S(t, t + 1, v) as v \u2192\u221e. The independent set requirements indicate that we must have a maximum independent pair having large sizes. To accomplish this, we partition all (t + 1)-subsets of Zv according to their sum modulo v, and choose one class of the partition to form the blocks of the packing. The basic strategy dates back at least a century to Bussey [6], and perhaps earlier. Prior to establishing our result, we note that this is not a mere theoretical curiosity; as Chen et al. observe in [7], declustered-parity RAIDs do not in practice need to have their loads perfectly balanced. Hence, for practical reasons, one may choose to omit some blocks from the design. Theorem 5.1. Let t and v be integers with v > \u0000t+2 2 \u0001 + \u0000t+1 2 \u0001 and (v, t + 1) = 1. For each of the following statements, there exists a t-(v, t + 1, 1) packing D on elements Zv having \u0000 v t+1 \u0001 v = v \u2212t v \u0000v t \u0001 \u0000t+1 t \u0001 blocks. (1) MinSum(D) = v + \u03c3 and MaxSum(D) = tv + \u03c3 whenever \u2212 \u0000t+2 2 \u0001 + 1 \u2264\u03c3 < \u0000t+1 2 \u0001 . (2) MinSum(D) = v + \u0000t+1 2 \u0001 \u22121. (3) MaxSum(D) = tv \u2212 \u0000t+2 2 \u0001 + 1. (4) Di\ufb00Sum(D) = (t \u22121)v. (5) RatioSum(D) = tv+(t+1 2 )\u22121 v+(t+1 2 )\u22121 . Proof. It su\ufb03ces to prove statement (1); the other results follow directly from it. Partition all (t + 1)-subsets of Zv into v classes {B\u03c3 : 0 \u2264\u03c3 < v} by placing set S = {x1, . . . , xt+1} in class B\u03c3 if and only if \u03c3 \u2261\u03a3t+1 i=1xi (mod v). Because for any t-subset T of Zv and each \u03c3 with 0 \u2264\u03c3 < v there is a unique element s for which \u03c3 \u2261s+\u03a3x\u2208Tx ( mod v), each B\u03c3 is a t \u2212(v, t + 1, 1) packing. Without restrictions on v, these v packings need not have the same number of blocks. We now use the restriction that (v, t + 1) = 1. Consider the orbits of (t + 1)-subsets of Zv under the cyclic action of Zv. When S is a (t + 1)-subset of Zv with sum \u03c3, let S + \u03b1 be the subset of Zv obtained by adding \u03b1 (modulo v) to each element in S. Then the orbit containing S is {S + \u03b1 : 0 \u2264\u03b1 < v}. For 0 \u2264\u03b1 < v, the sum of S + \u03b1 is \u03c3 + (t + 1)\u03b1 (mod v). Now if S + \u03b1 and S + \u03b2 have the same sum modulo v, we have (t + 1)\u03b1 \u2261(t + 1)\u03b2 (mod v), which can happen only when \u03b1 \u2261\u03b2 (mod v). Hence every orbit contains exactly v blocks, one in each of the v classes. It follows that each B\u03c3 contains ( v t+1) v blocks. 10 \fNow we prove statement (1). First we treat the cases when \u03c3 \u22650. Choose \u03c3 so that 0 \u2264\u03c3 < \u0000t+1 2 \u0001 and consider the packing D = (Zv, B\u03c3). Suppose to the contrary that S is a (t + 1)-subset of Zv with smallest sum \u03c4 < v + \u03c3. When \u03c4 \u2261\u03c3 (mod v) and \u03c4 < v + \u03c3, it must happen that \u03c3 = \u03c4 = \u03a3x\u2208Sx. But \u03a3x\u2208Sx \u2265\u03a3t i=0i = \u0000t+1 2 \u0001 > \u03c3, which is a contradiction. Hence MinSum(D) \u2265v + \u03c3. Because \u03c3 \u22650, tv + \u03c3 is the largest integer less than (t + 1)v that is congruent to \u03c3 modulo v, and hence MaxSum(D) \u2264tv + \u03c3. Next we address the cases when \u2212 \u0000t+2 2 \u0001 + 1 \u2264\u03c3 < 0. Let \u03c9 = v + \u03c3, and consider the packing D = (Zv, B\u03c9). Then MinSum(D) \u2265\u03c9 = v + \u03c3 < v, because of the congruence requirement. For MaxSum(D), suppose to the contrary that S is a (t + 1)-subset of Zv with largest sum \u03c4 > tv +\u03c3 = (t\u22121)v +\u03c9. Then \u03c4 = tv +\u03c9. Now \u03a3t+1 i=1(v \u2212i) = (t+1)v \u2212 \u0000t+2 2 \u0001 \u2265 \u03a3x\u2208Sx. Hence \u03c9 \u2264v \u2212 \u0000t+2 2 \u0001 so \u03c3 \u2264\u2212 \u0000t+2 2 \u0001 , which is a contradiction. Statements (2), (4), and (5) follow by taking \u03c3 = \u0000t+1 2 \u0001 \u22121. Statement (3) follows by taking \u03c3 = \u2212 \u0000t+2 2 \u0001 + 1. Not surprisingly, the packings so produced contain large independent sets. For example, when \u03c3 = 0, the elements {0, . . . , \u230av t+1\u230b} form an independent set. Theorem 5.1 yields packings that are dense in the following sense. When an S(t, t + 1, v) exists, it has (v t) (t+1 t ) blocks; the packings considered have a v\u2212t v fraction of this number. Hence for \ufb01xed t the fraction of t-sets left uncovered by the packing approaches 0 as v \u2192\u221e. Moreover, the bounds established for dense t-(v, t + 1, 1) packings on MinSum and MaxSum match the values from Theorem 2.1 (which are best for Steiner systems). On the other hand, as v \u2192\u221eand t is \ufb01xed, the ratio of Di\ufb00Sum of the packing to the bound approaches 1, and the RatioSum approaches its bound of t \u22121. By generalizing to partial systems, Theorem 5.1 applies to all parameters that are large enough, whether or not an S(t, t + 1, v) exists. Although Theorem 5.1 establishes a Di\ufb00Sum of (t \u22121)v for certain dense t-(v, t + 1, 1) packings, one might hope to obtain a somewhat smaller Di\ufb00Sum when t > 2. Theorem 5.2 gives one result in this direction, producing a packing that achieves a smaller Di\ufb00Sum than that of Theorem 5.1 when t = 3, but is nearly as dense. Theorem 5.2. When v > 18 is even, there is a 3-(v, 4, 1) packing D with v\u22124 v\u22121 (v 3) (4 3) blocks, having MinSum(D) = v + 2, MaxSum(D) = 3v \u22126, and hence Di\ufb00Sum(D) = 2v \u22128. Proof. Write v = 2s. We form D on elements {0, . . . , 2s \u22121}, with blocks 1. {{a, b, c, s + d} : 0 \u2264a < b < c < s, 0 \u2264d < s, a + b + c + d \u22612 (mod s)}, and 2. {{s + a, s + b, s + c, d} : 0 \u2264a < b < c < s, 0 \u2264d < s, a + b + c + d \u2261s \u22126 (mod s)}. This forms a 3-(v, 4, 1) packing with the speci\ufb01ed number of blocks. Because a + b + c + d \u2208 {s + 2, 2s + 2, 3s + 2}, blocks of the \ufb01rst class have sum in {2s + 2, 3s + 2, 4s + 2}. Similarly, because a + b + c + d \u2208{s \u22126, 2s \u22126, 3s \u22126}, blocks of the second class have sum in {4s \u22126, 5s \u22126, 6s \u22126}. Hence MinSum(D) = 2s + 2 = v + 2 and MaxSum(D) = 6s \u22126 = 3v \u22126. 11 \f6 Sums and Steiner triple systems Despite the utility of dense packings in the intended applications, it remains desirable to employ a Steiner system when possible. In what follows, we extend Theorem 5.1 to produce Steiner triple systems in which the sum metrics are close to optimal. Building on the construction in Theorem 5.1, Schreiber [29] and Wilson [35] demonstrate that for certain values of v, the packing can be completed to an STS(v). We provide a proof of their result, in order to examine the consequences for certain sums. The construction relies on a number-theoretic property, which we state next without proof. Theorem 6.1. [29] Every cycle in an abelian group G of order n contains twice an odd number of elements if and only if, for every prime divisor p of n, the order of \u22122 (mod p) is singly even. Lemma 6.2. Let n \u22611, 5 (mod 6). Every pair in {{a, b} : a, b \u2208Zn \\ {0}, a \u2261\u22122b (mod n)} has (n + 1)/2 \u2264a + b \u2264(n \u22121)/2 + n. Proof. Consider such a pair a, b \u2208Zn with b \u2261\u22122a (mod n). We examine two cases: Case 1: 1 \u2264a \u2264(n \u22121)/2. Then b = n \u22122a and hence (n + 1)/2 \u2264a + b \u2264n \u22121. Case 2: (n+1)/2 \u2264a \u2264n\u22121: Then b = 2n\u22122a and hence n+1 \u2264a+b \u2264n+(n\u22121)/2. Theorem 6.3. Whenever v \u22611, 3 (mod 6) and for every prime p dividing v \u22122, it is the case that the order of \u22122 mod p is singly even. Hence there is an STS(v), D, with MinSum(D) \u2265v \u22122, MaxSum(D) \u22642v + 2, and hence Di\ufb00Sum(D) \u2264v + 4 and RatioSum(D) \u22642v+2 v\u22122 . Proof. Let n = v \u22122. As in the proof of Theorem 5.1, construct the 2-(v, 3, 1) packing B0 on Zn. Each triple in this packing has sum v \u22122 or 2v \u22124 at present. The pairs left uncovered by any triple are E0 = {{x, \u22122x} : x \u2208Zv\u22122 \\ {0}}, each having sum between (v \u22121)/2 and (v \u22123)/2 + v \u22122 by Lemma 6.2. Because for every prime p dividing v \u22122, it is the case that the order of \u22122 mod p is singly even, Theorem 6.1 ensures that the pairs in E0 can be partitioned into two 1-factors, F1 and F2, on Zn \\ {0}. To form an STS(v) on Zv with block set C, we employ the mapping \u03c6 : Zv\u22122 7\u2192Zv \\ {(v \u22121)/2, (v + 1)/2} de\ufb01ned by \u03c6(x) = x when 0 \u2264x \u2264(v \u22123)/2 and \u03c6(x) = x + 2 when (v \u22121)/2 \u2264x < v \u22122. Then C is formed as follows. (1) When {x, y, z} \u2208B0, place {\u03c6(x), \u03c6(y), \u03c6(z)} in C; (2) For i = 1, 2, when {x, y} \u2208Fi, place {(v \u22123 + 2i)/2, \u03c6(x), \u03c6(y)} in C; (3) Place {0, (v \u22121)/2, (v + 1)/2} in C. Triples of B0 have sum v \u22122 or 2v \u22124, so triples of type (1) in C have sum between v \u22122 and v+2, or between 2v\u22122 and 2v+2. A pair {x, y} \u2208E0 has (v\u22121)/2 \u2264x+y \u2264(v\u22121)/2+(v\u22123). Applying \u03c6, we have (v \u22121)/2 \u2264\u03c6(x) + \u03c6(y) \u2264(v \u22121)/2 + (v + 1). Hence, each triple of type (2) in C has sum at least v \u22121 and at most 2v + 1. Finally, the single type (3) block has sum v. 12 \fUnlike the point labelings in [11], the labeling for the Schreiber-Wilson construction in Theorem 6.3 need not achieve the largest MinSum or smallest MaxSum. Nevertheless it yields a substantial improvement on earlier constructions with respect to the Di\ufb00Sum and RatioSum, within an additive constant of the best bound possible for the Di\ufb00Sum. Unfortunately, Theorem 6.3 requires that the order of \u22122 mod p be singly even, and so applies to an in\ufb01nite set of orders but not all admissible ones. We remedy this next, using a result from [5], but obtaining slightly weaker bounds. Theorem 6.4. Whenever v \u22611, 3 (mod 6), there is an STS(v), D, with MinSum(D) \u2265 v \u22125, MaxSum(D) \u22642v + 2, and hence Di\ufb00Sum(D) \u2264v + 7 and RatioSum(D) \u22642v+2 v\u22125 . Proof. Form B0 over Zv\u22122 as in the proof of Theorem 6.3. Remove element 0 as well as all triples {{0, x, v\u22122\u2212x} : 1 \u2264x \u2264(v\u22123)/2} to form D0. Let E0 be the set of pairs on Zv\u22122 not covered by a triple of B0. The pairs in E0, together with {{x, v \u22122 \u2212x} : 1 \u2264x \u2264(v \u22123)/2} form a 3-regular graph G on Zv\u22122 \\ {0}. By [5, Lemma 9], G can be partitioned into three 1-factors, F1, F2, and F3. To form the STS(v) on Zv with block set C, we employ the mapping \u03c8 : Zv\u22122 \\ {0} 7\u2192 Zv \\ {(v \u22123)/2, (v \u22121)/2, (v + 1)/2} de\ufb01ned by \u03c8(x) = x \u22121 when 1 \u2264x \u2264(v \u22123)/2 and \u03c8(x) = x + 2 when (v \u22121)/2 \u2264x < v \u22122. Then C is formed as follows. (1) When {x, y, z} \u2208D0, place {\u03c8(x), \u03c8(y), \u03c8(z)} in C; (2) For i = 1, 2, 3, when {x, y} \u2208Fi, place {(v \u22125 + 2i)/2, \u03c8(x), \u03c8(y)} in C; (3) Place {(v \u22123)/2, (v \u22121)/2, (v + 1)/2} in C. Triples of B0 have sum v \u22122 or 2v \u22124, so triples of type (1) in C have sum between v \u22125 and v\u22122, or between 2v\u22121 and 2v+2. By Lemma 6.2, a pair {x, y} \u2208E0 has (v\u22121)/2 \u2264x+y \u2264 (v \u22121)/2 + (v \u22123). Applying \u03c8, we have (v \u22121)/2 \u22122 \u2264\u03c8(x) + \u03c8(y) \u2264(v \u22121)/2 + (v + 1). It follows that each triple of type (2) in C has sum at least v \u22124 and at most 2v + 1. The block of type (3) in C has sum 3v\u22123 2 . Although the bounds are slightly weaker, Theorem 6.4 applies to all admissible orders for Steiner triple systems. In conjunction with Theorem 2.2, for all v \u22611, 3 (mod 6) with v \u226513 one has v + 1 \u2264Di\ufb00Sum(2, 3, v) \u2264v + 7 and 2 + 1 v \u2264RatioSum(2, 3, v) \u22642 + 12 v\u22125. Using a simple computational search, we constructed S(2, 3, v)s with speci\ufb01ed MinSum and MaxSum, as shown next: Order v MinSum MaxSum Di\ufb00Sum RatioSum 7 v \u22121 2v \u22121 v 2 + 1 v\u22121 9 v 2v v 2 13,15,19,21,25,27 v \u22121 2v v + 1 2 + 2 v\u22121 7,15,19,21,27 v 2v + 1 v + 1 2 + 1 v 13,25 v 2v + 2 v + 2 2 + 2 v It appears plausible that Di\ufb00Sum(2, 3, v) = v + 1 when v \u226513. It also appears plausible that RatioSum(2, 3, v) \u2208{2 + 1 v, 2 + 2 v} for every v \u0338= 9, but there is insu\ufb03cient data to speculate on when it takes the larger value and when the smaller. 13 \f7 Concluding remarks Because Theorem 5.1 achieves a Di\ufb00Sum of (t \u22121)v for dense t-(v, t + 1, 1) packings, one might hope that this di\ufb00erence can be realized for S(t, t + 1, v) Steiner systems. However, Theorem 2.3 establishes that this does not happen when t = 2 unless v \u2208{7, 9}, although Theorem 6.4 is within an additive constant. The situation when t = 3 appears to be quite di\ufb00erent. There is an S(3, 4, 8) with blocks {0127, 0136, 0145, 0235, 0246, 0347, 0567, 1234, 1256, 1357, 1467, 2367, 2457, 3456}, having MinSum 10 and MaxSum 18. Adapting the construction in [12,27], one can produce an S(3, 4, v) with MinSum v + 2, MaxSum 3v \u22126, and hence Di\ufb00Sum 2v \u22128 whenever v is a power of 2. In these cases, the upper bound on the MinSum and the lower bound on the MaxSum from Theorem 2.1 are met simultaneously. We do not expect this to happen for all orders, because the smallest Di\ufb00Sum for an S(3, 4, v) when v \u2208{10, 14} appears to arise from systems with MinSum v + 1 and MaxSum 3v \u22125. It may happen that for every admissible v, an S(3, 4, v) with Di\ufb00Sum strictly smaller than 2v exists. If so, completing the packing from Theorem 5.1 could not yield the smallest Di\ufb00Sum. Nevertheless, the structure of independent sets must underlie appropriate constructions. Acknowledgements The work was supported by NSF grants CCF 1816913 (CJC) and 1814298 (OM)."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1808.09535v2",
+                    "title": "Low-Power Cooling Codes with Efficient Encoding and Decoding",
+                    "abstract": "A class of low-power cooling (LPC) codes, to control simultaneously both the\npeak temperature and the average power consumption of interconnects, was\nintroduced recently. An $(n,t,w)$-LPC code is a coding scheme over $n$ wires\nthat (A) avoids state transitions on the $t$ hottest wires (cooling), and (B)\nlimits the number of transitions to $w$ in each transmission (low-power).\n  A few constructions for large LPC codes that have efficient encoding and\ndecoding schemes, are given. In particular, when $w$ is fixed, we construct LPC\ncodes of size $(n/w)^{w-1}$ and show that these LPC codes can be modified to\ncorrect errors efficiently. We further present a construction for large LPC\ncodes based on a mapping from cooling codes to LPC codes. The efficiency of the\nencoding/decoding for the constructed LPC codes depends on the efficiency of\nthe decoding/encoding for the related cooling codes and the ones for the\nmapping.",
+                    "authors": "Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Alexander Vardy, Hengjia Wei",
+                    "published": "2018-08-16",
+                    "updated": "2019-03-20",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION Power and heat dissipation have emerged as \ufb01rst-order design constraints for chips, whether targeted for battery-powered devices or for high-end systems. High temperatures have dramatic negative effects on interconnect performance. Power-aware design alone is insuf\ufb01cient to address the thermal challenges, since it does not directly target the spatial and temporal behavior of the operating environment. For this reason, thermally-aware approaches have emerged as one of the most important domains of research in chip design today. Numerous techniques have been proposed to reduce the overall power consumption of on-chip buses (see [3] which uses coding techniques and the references therein using non-coding techniques). However, all the non-coding techniques do not directly address peak temperature minimization. Recently, Chee et al. [3] introduced several ef\ufb01cient coding schemes to directly control the peak temperature and the average power consumption. Among others, low-power cooling (LPC) codes are of particular interest as they control both the peak temperature and the average power consumption simultaneously. Speci\ufb01cally, an Earlier results of this paper were presented at the 2018 Proceedings of the IEEE ISIT [4]. \f2 (n, t, w)-LPC code is a coding scheme for communication over a bus consisting of n wires, if the scheme has the following two features: (A) every transmission does not cause state transitions on the t hottest wires; (B) the number of state transitions on all the wires is at most w in every transmission. LPC codes have both features, while cooling codes control only the peak temperature. De\ufb01nition 1. For n and t, an (n, t)-cooling code C of size M is de\ufb01ned as a collection {C1, C2, . . . , CM}, where C1, C2, . . . , CM are disjoint subsets of {0, 1}n satisfying the following property: for any set S \u2286[n] of size |S| = t and for i \u2208[M], there exists a vector u \u2208Ci such that supp(u) \u2229S = \u2205. We refer to C1, C2, . . . , CM as codesets and the vectors in them as codewords. Using partial spreads, Chee et al. [3] constructed LPC codes with ef\ufb01cient encoding and decoding schemes. When t \u22640.687n and w \u2265(n \u2212t)/2, these codes achieve optimal asymptotic rates. However, when w is small, i.e. low-power coding is used, the code rates are small and Chee et al. proposed another construction based on decomposition of the complete hypergraph into perfect matchings. While the construction results in LPC codes of large size, usually ef\ufb01cient encoding and decoding algorithms are not known. In this work, we focus on this regime (w small) and construct LPC codes with ef\ufb01cient encoding and decoding schemes. Speci\ufb01cally, our contributions are as follows. (I) We propose a method that takes a linear erasure code as input and constructs an LPC code. Using this method, we then construct a family of LPC codes of size (n/w)w\u22121 which attains the asymptotic upper bound O(nw\u22121) when w is \ufb01xed. We also use this method to construct a class of LPC codes of size (n/w)w\u2212e\u22121 which is able to correct e transmission errors. (II) We propose ef\ufb01cient encoding/decoding schemes for the LPC codes of the given construction. In particular, for the above family of LPC codes, we demonstrate encoding with O(n) multiplications over Fq and decoding with O(w3) multiplications over Fq, where q = n/w. Furthermore, the related class of LPC codes is able to correct e errors with complexity O(n3). (III) A de\ufb01nition for a new family of low-power cooling codes, called constant-power cooling (CPC in short) codes, which have the same weight for all the codewords. All our previous constructions can be applied to obtain such codes. (IV) A recursive construction for a class of (nq, tq, w)-CPC codes (and also (nq, tq, w)-LPC codes) from (n, t, w)CPC codes (and a special type of (n, t, w)-LPC codes). (V) A construction for a class of (n, t, w)-LPC codes based on a mapping from (m, t)-cooling codes. This mapping send all the binary words of length m into a Hamming ball of radius w in Fn q such that each coordinate of the obtained words in the Hamming ball is dominated by a coordinate of the binary words of length m. This property guarantees that the cooling property of the (m, t)-cooling code is preserved in the low-power cooling code. Our main target in this paper are cooling codes, but these codes, or more precisely their de\ufb01nition by codesets, might have other applications too. One such application is in the design of WOM (Once Write Memory) codes \f3 which are very important in coding for \ufb02ash memories (see [5] and references therein). This application of codesets into construction of WOM codes was written in detail in [5] and is described in short as follows. In a WOM code one is trying to write binary information words of length k into a memory of length n, where the information is written only in positions where there are zeroes. The goal is to write as many rounds as possible until there is no way to distinguish between some of the written words. Each information word will be identi\ufb01ed by a codeset, the codeword taken from the appropriate codeset should have ones on all positions where the memory has ones and hence the ones in complement of the codeword should have empty intersection with the ones of the memory. As was mentioned, this application to WOM codes was considered in [5]. We believe that other applications will arise in the future. The rest of this paper is organized as follows. In Section II we present some necessary de\ufb01nitions for our exposition, some of the known results, and new upper bounds on the sizes of low-power cooling codes and constant-power cooling codes. Finally, the known constructions are presented and a new one is suggested. Section III suggests a construction for CPC codes based on non-binary linear codes in general and on MDS codes in particular. For these codes ef\ufb01cient encoding and decoding algorithms are derived. We continue in Section IV and add error-correction capabilities for such codes and provide ef\ufb01cient algorithms also in this case. The construction that was used in Section IV is modi\ufb01ed in Section V to provide a recursive construction for (nq, tq, w)-CPC codes (and related (nq, tq, w)-LPC codes) from (n, t, w)-CPC codes (and some special (n, t, w)CPC codes). While in Section III the constructions are for t \u2264n/w \u22121, in Section V the construction is effective for larger t. In Section VI a method to transfer an (m, t)-cooling code to an (n, t, w)-LPC code is given. This method is based on a special injection from the set of all binary words of length m into binary words of length n and weight at most w. A product construction using this method implies codes with ef\ufb01cient encoding and decoding algorithms. We further analyse and compare between this construction and constructions in previous works. II. UPPER BOUNDS AND KNOWN RESULTS Given a positive integer n, the set {1, 2, . . ., n} is abbreviated as [n]. The Hamming weight of a vector x \u2208Fn q, denoted wt(x), is the number of nonzero positions in x, while the support of x is de\ufb01ned as supp(x) \u225c{i \u2208[n] : xi \u0338= 0}. A q-ary code C of length n is a subset of Fn q. If C is a subspace of Fn q, it is called a linear code. An [n, k, d]q code is a linear code with dimension k and minimum Hamming distance d. De\ufb01nition 2. For n, t and w with t + w \u2264n, an (n, t, w)-low-power cooling (LPC) code C of size M is de\ufb01ned as a collection of codesets {C1, C2, . . . , CM}, where C1, C2, . . . , CM are disjoint subsets of {u \u2208{0, 1}n : wt(u) \u2264w} satisfying the following property: for any set S \u2286[n] of size |S| = t and for i \u2208[M], there exists a vector u \u2208Ci such that supp(u) \u2229S = \u2205. In this paper, we focus on a class of (n, t, w)-LPC codes where every transmission results in exactly w state transitions. We call such codes (n, t, w)-constant-power cooling (CPC) codes. In particular, let J(n, w) \u225c{u \u2208{0, 1}n : \f4 wt(u) = w}. Then an (n, t, w)-CPC code is an (n, t, w)-LPC code such that Ci \u2286J(n, w) for each i \u2208[M]. A. Set Systems For a \ufb01nite set X of size n, 2X denotes the collection of all subsets of X, i.e., 2X \u225c{A : A \u2286X}. A set system of order n is a pair (X, B), where X is a \ufb01nite set of n points, B \u22862X, and the elements of B are called blocks. Two set systems (X, B1) and (X, B2) with the same point set are called disjoint if B1 \u2229B2 = \u2205, i.e. they don\u2019t have any block in common. A partial parallel class of a set system (X, B) is a collection of pairwise disjoint blocks. If a partial parallel class partitions the point set X, it is called parallel class. A set system (X, B) is called resolvable if the block set B can be partitioned into parallel classes. There is a canonical one-to-one correspondence between the set of all codes of length n and the set of all set systems of order n: the coordinates of vectors in {0, 1}n correspond to the points in [n], and each vector u \u2208{0, 1}n corresponds to the block de\ufb01ned by supp(u). Thus we may speak of the set system of a code or the code of a set system. By abuse of notation we sometimes do not distinguish between the two different notations and this can be readily observed in the text. B. Upper Bounds Given a t-subset S and a vector u \u2208{0, 1}n, we shall say that u avoids S if supp(u) \u2229S = \u2205. The following bounds on LPC codes and CPC codes are easily derived. Theorem 3. Let C be an (n, t, w)-LPC code of size M, then M \u2264 w \u2211 i=0 \u0012n \u2212t i \u0013 . Furthermore, if C is an (n, t, w)-CPC code, then M \u2264 \u0012n \u2212t w \u0013 . Proof. For any given t-subset S of [n], each codeset should have at least one codeword which avoids S. The number of words with weight i which avoid S is (n\u2212t i ) and hence there are no more than (n\u2212t w ) codesets in an (n, t, w)-CPC code and no more than \u2211w i=0 (n\u2212t i ) codesets in an (n, t, w)-LPC code. Theorem 3 implies that both CPC codes and LPC codes share the same asymptotic upper bound O(nw) on the number of codewords. The upper bound of Theorem 3 can be improved for some parameters. For this purpose, we need to de\ufb01ne and to introduce some results on Tur\u00b4 an systems. Let n \u2265k \u2265r, and let X be a \ufb01nite set with n distinct elements. The set (X r ) is the collection of all r-subsets of X. A Tur\u00b4 an (n, k, r)-system is a set system (X, B), where |X| = n and B \u2286(X r ) is the set of blocks such that each k-subset of X contains at least one of the blocks. The Tur\u00b4 an number T(n, k, r) is the minimum number of blocks in such a system. This number is determined only for r = 2 and some sporadic examples (see [13], [16] and references therein). De Caen [7] proved the following general lower bound on T(n, k, r). \f5 T(n, k, r) \u2265n \u2212k + 1 n \u2212r + 1 \u00b7 (n r) (k\u22121 r\u22121) . (1) The following proposition is an immediate result from the de\ufb01nition of Tur\u00b4 an systems. Proposition 4. A family of codesets {C1, C2, . . . , CM} is an (n, t, w)-CPC code if and only if the set system of each Ci is a Tur\u00b4 an (n, n \u2212t, w)-system and these M set systems are pairwise disjoint. Combining the bound in (1) and Proposition 4, we have the following upper bound on the size of CPC codes. Theorem 5. If C is an (n, t, w)-CPC code of size M, then M \u2264n \u2212w + 1 t + 1 \u0012n \u2212t \u22121 w \u22121 \u0013 . Proof. By (1) we have that T(n, n \u2212t, w) \u2265 t + 1 n \u2212w + 1 \u00b7 (n w) (n\u2212t\u22121 w\u22121 ) . (2) By Proposition 4 we have that M \u2264 (n w) T(n, n \u2212t, w). (3) Combining (2) and (3) yield that M \u2264n \u2212w + 1 t + 1 \u0012n \u2212t \u22121 w \u22121 \u0013 . Corollary 6. If C is an (n, t, w)-LPC code of size M, then M \u2264 w\u22121 \u2211 i=0 \u0012n i \u0013 + n \u2212w + 1 t + 1 \u0012n \u2212t \u22121 w \u22121 \u0013 . Proof. If we consider an (n, t, w)-CPC code C, then to form an (n, t, w)-LPC code we can add to C at most \u2211w\u22121 i=0 (n i) codesets, each one contains exactly one codeword of weight less than w. When t = \u0398(n), we have that (n \u2212w + 1)/(t + 1) = O(1), and so the upper bound for (n, t, w)-CPC codes is improved from O(nw) implied by Theorem 3 to O(nw\u22121) implied by Theorem 5. For an (n, t, w)-LPC code, we have by Corollary 6 that the size of such a code is at most w\u22121 \u2211 i=0 \u0012n i \u0013 + n \u2212w + 1 t + 1 \u0012n \u2212t \u22121 w \u22121 \u0013 , which is also O(nw\u22121) when t and n are of the same order of magnitude. C. Some Known Constructions Chee et. al [3] provided the following construction of LPC/CPC codes. \f6 Proposition 7 (Decomposition of Complete Hypergraphs). If n = (t + 1)w then there exists an (n, t, w)-CPC code of size (n\u22121 w\u22121). When w is \ufb01xed, we have that t and n are of the same order of magnitude and the above construction attains the asymptotic upper bound O(nw\u22121). Unfortunately, usually no ef\ufb01cient encoding and decoding methods are known for this construction and generally the only known encoding method involves listing all the (n\u22121 w\u22121) codesets. The exceptions are for small n or when w is very small, e.g. when w = 2 or w = 3 [1], [8]. Chee et. al [3] also proposed the following constructions of LPC codes which have ef\ufb01cient coding schemes. Proposition 8 (Concatenation). Suppose that q \u2264\u2211w\u2032 i=0 (s i) and q is a prime power and t \u2264s. (i) If t + 1 \u2264m/2, then there exists an (ms, t, mw\u2032)-LPC code of size qm\u2212t\u22121. (ii) If t + 1 \u2264m \u2264q + 1, then there exists an (ms, t, mw\u2032)-LPC code of size qm\u2212t. Proposition 9 (Sun\ufb02ower Construction). Let r + t \u2264(n + s)/2. If a linear [n, s, w + 1]2 code exists and a linear [n \u2212t, r, w + 1]2 code does not exist, then there exists an (n, t, w)-LPC code of size 2n\u2212t\u2212r. Finally, Proposition 4 suggests a new method to construct (n, t, w)-CPC codes. We just have to \ufb01nd a set with large number of pairwise disjoint Tur\u00b4 an (n, n \u2212t, w)-systems. One work in this direction was done in [11] where pairwise disjoint Tur\u00b4 an (n, w + 1, w)-systems were considered. Another possible construction based on Proposition 4 is to consider complements of pairwise disjoint Steiner systems. Such pairwise disjoint systems were considered in [2], [17] and for Steiner quadruple systems which will be used in the sequel in [9]. III. AN EFFICIENT CONSTRUCTION FOR CONSTANT-POWER COOLING CODES In this section, we present a new construction of CPC codes which has ef\ufb01cient encoding and decoding algorithms. Asymptotically, the codes obtained by the construction attain the bound of Theorem 5. As was mentioned before, there are three types of constructions for LPC codes in [3]. The \ufb01rst one is based on decomposition of the complete hypergraph, the second one is a concatenation method based on q-ary cooling codes, and the third one is a Sun\ufb02ower Construction. The construction in this section, is an explicit construction for CPC codes which combines the advantages of the \ufb01rst two types of constructions. We \ufb01rst rephrase the construction based on decomposition of the complete hypergraph in terms of set systems. The construction is based on the following generalization of Proposition 7. Proposition 10. Let (X, B) be a set system of order n, where B is partitioned into M partial parallel classes P1, P2, . . . , PM. If B \u2286(X w) and each Pi has at least t + 1 blocks, then the codesets P1, P2, . . . , PM form an (n, t, w)-CPC code. Proof. By de\ufb01nition, each codeword of a codeset has weight w. Hence, to show that P1, P2, . . . , PM form an (n, t, w)-CPC code, we only have to prove that given a t-subset S of X with the list of hottest wires and a codeset Pi, 1 \u2264i \u2264M, there exists a block B \u2208Pi such that B \u2229S = \u2205. Since Pi is a partial parallel class with at least t + 1 codewords, it follows that S intersects at most t blocks of Pi. Hence, there exists a block B \u2208Pi such that B \u2229S = \u2205. \f7 The complete k-uniform hypergraph G = (V, E) has a vertex set V with n \u2265k vertices, and each subset of (V k) is connected by an hyperedge. The decomposition of G is a partition of the set of edges E in G into disjoint perfect matching. In other words, a partition into vertex-disjoint sets of edges, where each vertex of V appears exactly once in each set of the partition. The celebrated Baranyai\u2019s theorem [18, p. 536] asserts that such a decomposition always exists if k divides the number of vertices in V. Therefore, since a decomposition of the complete k-uniform hypergraph with vertex set X is a resolvable set system (X, (X k)), if k divides |X|, we recover Proposition 7. A. CPC Codes Based on Linear Codes Let C be an [N, K, D]q code. Using the codewords of C, we will show how to construct a set system with qK\u22121 partial parallel classes, each one has blocks of the same size, and as a consequence Proposition 10 yields a CPC code D. To equip D with ef\ufb01cient encoding and decoding schemes, we utilize the erasure-correcting algorithms of the linear code C. These schemes are discussed in detail in Section III-B. For a set of coordinates T and a vector \u03c3 \u03c3 \u03c3 \u2208F|T| q , we say that \u03c3 \u03c3 \u03c3 appears \u03bb times in C at T if there are \u03bb codewords in C whose restriction on T is \u03c3 \u03c3 \u03c3. Since any two codewords of C differ in at least D coordinates, it follows that they agree in at most N \u2212D positions. Hence, we have the following observations. Lemma 11. Let C be an [N, K, D]q code. (i) For any (N \u2212D + 1)-subset of coordinates T and any \u03c3 \u03c3 \u03c3 \u2208FN\u2212D+1 q , \u03c3 \u03c3 \u03c3 appears in at most one codeword of C at T. (ii) For any (N \u2212D)-subset of coordinates T and any \u03c4 \u03c4 \u03c4 \u2208FN\u2212D q , \u03c4 \u03c4 \u03c4 appears in at most q codewords of C at T. Proof. (i) If \u03c3 \u03c3 \u03c3 \u2208FN\u2212D+1 q appears twice in codewords of C at an (N \u2212D + 1)-subset of coordinates T, then the two related codewords have distance at most D \u22121, a contradiction. (ii) If \u03c4 \u03c4 \u03c4 \u2208FN\u2212D q appears in q + 1 codewords of C at an (N \u2212D)-subset of coordinates T, then let t be a coordinate not in T. In at least two of the related codewords coordinate t has the same symbol. We add this symbol to \u03c4 \u03c4 \u03c4 to obtain \u03c3 \u03c3 \u03c3 \u2208FN\u2212D+1 q which appears in two codewords of C at the (N \u2212D + 1)-subset T \u222a{t}, contradicting claim (i) of this lemma. For a code C and a subset of coordinates T, let C|T denotes the set of codewords restricted to the coordinates of T, i.e., the projection of C into the set of coordinates indexed by T. For a word u, let u|T denotes the restriction of u to the coordinates of T. Finally, for a matrix G, let G|T denotes the submatrix of G obtained from the columns indexed by T. Lemma 12. Let C be an [N, K, D]q code. If G is a generator matrix of C, then every K \u00d7 (N \u2212D) submatrix of G has rank either K or K \u22121. Furthermore, there exists a K \u00d7 (N \u2212D) submatrix of G whose rank is K \u22121. Proof. Let T be a subset of N \u2212D coordinate positions and assume the corresponding K \u00d7 (N \u2212D) submatrix G|T has rank r, where r \u2264K. Let \u03c6T be the linear map from FK q to FN\u2212D q de\ufb01ned by \u03c6T(x) = xG|T. Clearly, the \f8 dimension of the kernel of \u03c6T is K \u2212r. Hence, the all-zeroes vector of length N \u2212D appears in qK\u2212r codewords of C|T. By Lemma 11 the all-zeroes vector appears in at most q codewords of C|T at T which implies that K \u2212r \u22641 and therefore r \u2265K \u22121. Let u be a codeword of C with minimum weight D, T be the subset of [N] not in the support of u, i.e. T = [N] \\ supp(u). Let x be the information vector of length K such that u = xG. Since u has weight D, it follows that |supp(u)| = D and hence the size of T is N \u2212D. Since u has zeroes in the coordinates of T, it follows that for the K \u00d7 (N \u2212D) submatrix G|T of G we have that xG|T = 0. Therefore, the rank of GT is at most K \u22121. Thus the rank of G|T is at least K \u22121 by the \ufb01rst part of the proof and at most K \u22121 by the second part of the proof, which implies that the rank of G|T is K \u22121. We are now ready to present the new ef\ufb01cient construction for CPC codes. Construction 1. Let C be an [N, K, D]q code and G be a generator matrix of C, where the last N \u2212D columns of G form a K \u00d7 (N \u2212D) submatrix of G whose rank is K \u22121. \u2022 Partition the codewords of C into disjoint codesets C1, C2, . . . , CM such that two codewords u and v are in the same codeset if and only if they agree on their last N \u2212D symbols. \u2022 For each i \u2208[M], truncate the codewords in Ci to length w by removing their last N \u2212w symbols. In other words, set C\u2032 i \u225c{u|[w] : u \u2208Ci} for each i \u2208[M]. \u2022 For each i \u2208[M] construct the set system (X, Di), where X = Fq \u00d7 [w] and Di = {(xj, j) : x = x1x2 \u00b7 \u00b7 \u00b7 xw \u2208C\u2032 i, j \u2208[w]}. Theorem 13. If N \u2212D + 1 \u2264w \u2264D, then the collection of codesets D = {D1, D2, . . . , DM} is an (n, t, w)-CPC code of size M = qK\u22121, where n = qw and t = q \u22121. Proof. Clearly, by the de\ufb01nition of the construction we have that n = qw and each codeword has weight w. Hence, to complete the proof it is suf\ufb01cient to show that M = qK\u22121, the M codesets are pairwise disjoint, and for any t-subset S of coordinates from X and each i \u2208[M], there exists a codeword u in the codeset Di such that supp(u) \u2229S = \u2205. 1) Let G\u2032 be the K \u00d7 (N \u2212D) submatrix of G formed from the last N \u2212D columns of G. Consider the linear map \u03c6 from FK q to FN\u2212D q de\ufb01ned by \u03c6(x) = xG\u2032. Since the rank of G is K \u22121, it follows that the image of \u03c6 has dimension K \u22121 and the kernel of \u03c6 has dimension one. It follows that M = qK\u22121 and |Ci| = q for each i \u2208[M]. 2) The minimum distance of C is D and hence each two codewords of C can agree in at most N \u2212D coordinates, i.e. they differ in any subset of N \u2212D + 1 coordinates. Since w \u2265N \u2212D + 1 and the codewords of C were shortened in their last N \u2212w coordinates, it follows that all the shortened codewords of C are distinct. Thus C\u2032 1, C\u2032 2, . . . , C\u2032 M are pairwise disjoint and |C\u2032 i| = |Ci| = q. Now, it can be easily veri\ufb01ed by the de\ufb01nition of Di that D1, D2, . . . , DM are pairwise disjoint and |Di| = q for each i \u2208[M]. \f9 3) Each two codewords of Ci agree on their last N \u2212D coordinates and since their distance is at least D, it follows that they differ in the \ufb01rst D coordinates. Since w \u2264D, this implies that any two codewords of C\u2032 i differ in all their w coordinates. Hence, by the de\ufb01nition of Di it implies that each two codewords of Di differ in their nonzero coordinates. Therefore, the codewords in Di are pairwise disjoint, i.e., Di is a partial parallel class. We also have that |Di| = q for each i \u2208[M]. Hence, Di is a parallel class and as in the proof of Proposition 10 we have that for any t-subset S, Di has a codeword u such that supp(u) \u2229S = \u2205. Thus, the the required claims were proved and hence the collection of codesets D = {D1, D2, . . . , DM} is a (qw, q \u22121, w)-CPC code of size M = qK\u22121. For a given [N, K, D]q code C and its generator matrix G in Construction 1, we need to \ufb01nd a minimum weight codeword in C in order to determine a K \u00d7 (N \u2212D)-submatrix of G with rank K \u22121, i.e., to \ufb01nd a permutation of the columns of G such that the last N \u2212D coordinates of G will have rank K \u22121. Finding the minimum distance of a code is an NP-hard problem and the decision problem is NP-complete [19]. Therefore, we focus on certain families of codes where it is computationally easy to \ufb01nd minimum weight codewords. One such family is the maximum distance separable (MDS) codes. Recall that a linear [N, K, D]q code is an MDS code if D = N \u2212K + 1 [14, Ch.11]. If the code C in Construction 1 is an MDS code, then every K columns of G are linearly independent and hence each K \u00d7 (N \u2212D) submatrix of G has rank K \u22121 since N \u2212D = K \u22121. Therefore, we may use any N \u2212D coordinate as the last N \u2212D coordinates of C. It is well known that MDS codes exist for the following parameters. Theorem 14 (see [14, Ch.11]). Let q be a prime power. If D \u22653, then there exists an [N, K, D]q MDS code if N \u2264q + 1 for all q and 2 \u2264K \u2264q \u22121, except when q is even and K \u2208{3, q \u22121}, in which case N \u2264q + 2. Setting N = q + 1, K = w, D = q \u2212w + 2, and using an [N, K, D]q MDS code as the code C in Construction 1, we have that w \u2264D = q \u2212w + 2, i.e., q \u22652w \u22122. Hence, Theorem 13 yields the following corollary. Corollary 15. Let n, t and w be positive integers. If q = n/w is a prime power and q \u22652w \u22122, then there exists an (n, q \u22121, w)-CPC code of size (n/w)w\u22121. In Corollary 15, when w is \ufb01xed, t = q \u22121 has the same order of magnitude as n. Hence, the codes constructed in this case asymptotically attain the upper bound O(nw\u22121). We also note that for some parameters, these CPC codes are much larger than the LPC codes provided by Propositions 8 and 9. Example 16. By choosing n = 96, w = 6 and t = 15, Corollary 15 yields a (96, 15, 6)-CPC code of size 165 = 220. In contrast, suppose we use Proposition 8 to construct a (96, t, 6)-LPC code with t \u226415. The largest size 165 = 220 is obtained by choosing m = 6, t = 1, s = 16, w\u2032 = 1, and q = 16. The resulting (96, 1, 6)-LPC code has the same size as the CPC obtained by Corollary 15, but the cooling capability of the former is clearly much weaker. Proposition 9, on the other hand, yields a (96, 15, 6)-LPC code of size 216 by choosing s = 81 and r = 65. This code has similar parameters, but its size is much smaller. \f10 Example 17. If we choose a [17, 8, 9]9 code (see [12]). If w = 9 and t = 8 in Construction 1, then we obtain an (81, 8, 9)-CPC code of size 97 \u2248222.189. In contrast, the largest (81, 8, 9)-LPC code obtained from Proposition 8 has size 9 \u224823.17 by choosing m = s = q = 9, w\u2032 = 1. Proposition 9, on the other hand, yields an (81, 8, 9)-LPC of size 221 by choosing s = 54 and r = 52. B. Encoding and Decoding Schemes We continue in this subsection and discuss the encoding and decoding schemes for the code D obtained in Construction 1. Let G be a generator matrix of the [N, K, D]q code C, where the last N \u2212D columns of G form a K \u00d7 (N \u2212D) submatrix G\u2032 whose rank is K \u22121. Furthermore, w.l.o.g. we assume that G has the form G = \uf8eb \uf8edA IK\u22121 \u03b2 \u03b2 \u03b2K 0 \u00b7 \u00b7 \u00b7 0 \uf8f6 \uf8f8, where IK\u22121 is the identity matrix of order K \u22121. Each codeset in D will be identi\ufb01ed by the unique vector from FK\u22121 q . This is possible since the number of codesets is qK\u22121. For \u03c3 \u03c3 \u03c3 \u2208FK\u22121 q , let C\u03c3 \u03c3 \u03c3 be the set of q codewords from C whose suf\ufb01x of length K \u22121 is \u03c3 \u03c3 \u03c3. Furthermore, let C\u2032 \u03c3 \u03c3 \u03c3 and D\u03c3 \u03c3 \u03c3 be the derived codesets as de\ufb01ned in Construction 1. Given a t-subset S of Fq \u00d7 [w] and a word \u03c3 \u03c3 \u03c3 = (\u03c31, \u03c32, . . . , \u03c3K\u22121) \u2208FK\u22121 q , our objective for encoding of Construction 1 is to \ufb01nd a codeword u \u2208D\u03c3 \u03c3 \u03c3 such that supp(u) \u2229S = \u2205. Let \u03b2 \u03b2 \u03b2i be the i-th row of G. Let r = \u03c3 \u03c3 \u03c3A|[w] = K\u22121 \u2211 i=1 \u03c3i\u03b2 \u03b2 \u03b2i|[w], and hence the codeset C\u2032 \u03c3 \u03c3 \u03c3 is C\u2032 \u03c3 \u03c3 \u03c3 = {r + \u03bb\u03b2 \u03b2 \u03b2K|[w] : \u03bb \u2208Fq}. The codeset D\u03c3 \u03c3 \u03c3 is derived from C\u2032 \u03c3 \u03c3 \u03c3 as indicated in Construction 1, and hence we can consider the intersection of each one of the q blocks in D\u03c3 \u03c3 \u03c3 with S to \ufb01nd the block B such that B \u2229S = \u2205. Hence, for the encoding, O(n) multiplications over Fq are required to \ufb01nd D\u03c3 \u03c3 \u03c3. During this computation we can also check whether each codeword of D\u03c3 \u03c3 \u03c3 has nontrivial intersection with B or not. Therefore, there is no need for further computations to \ufb01nd B. For the decoding, suppose that we have a codeword {(x1, 1), (x2, 2), . . . , (xw, w)}. By our choice we have that w \u2265N \u2212D + 1 which implies that D \u22121 \u2265N \u2212w and hence we can correct any N \u2212w erasures in any codeword of C. Hence, the N \u2212w erasures in (x1, x2, . . . , xw, ?, ?, . . ., ?) can be recovered and the last K \u22121 symbols, xN\u2212K+2, xN\u2212K+3, . . . , xN are the information symbols. In particular, if the code C is a Reed-Solomon code, then by using Lagrange interpolation, O(w3) multiplications are enough to perform the decoding, e.g. [15]. IV. ERROR-CORRECTING CPC CODES In this section we consider CPC codes that can correct transmission errors (\u20180\u2019 received as \u20181\u2019, or \u20181\u2019 received as \u20180\u2019). An (n, w, t)-CPC which can correct up to e errors will be called an (n, t, w, e)-CPECC (constant weight \f11 power error-correcting cooling) code. First, Construction 1 will be used to produce CPECC codes by examining the minimum distance of the constructed codes. Theorem 18. If the code C used for Construction 1 is an [N, K, D]q code, then the code D obtained by Construction 1 is an (n, t, w, e)-CPECC code of size M = qK\u22121, where n = qw, t = q \u22121, and e \u2265w + D \u2212N \u22121. Proof. All the parameters of the code except for e = w + D \u2212N \u22121 were proved in Theorem 13. Since the minimum distance of C is D and the code C was punctured in the last N \u2212w coordinates to obtain the code C\u2032 (the union of the codesets C\u2032 i, 1 \u2264i \u2264M), it follows that the minimum distance of C\u2032 is at least D \u2212(N \u2212w). By the de\ufb01nition of D\u2032 (the union of the codesets D\u2032 i, 1 \u2264i \u2264M) we have that if u, u\u2032 \u2208C\u2032 differ in \u2113coordinates, then the related codewords in D differ in 2\u2113positions. Hence, the minimum distance of D is at least 2(D + w \u2212N) and thus the number of errors that it can correct is e \u2265w + D \u2212N \u22121. Next, an algorithm which demonstrates the error-correction for an (n, t, w, e)-CPECC code will be given. For simplicity, we will focus on a special example, where our starting point is a Reed-Solomon code C (which is of course an MDS code), where K = N \u2212D + 1 = w \u2212e. Construction 2. Let w and e be positive integers and q be a prime power such that q \u22652w \u2212e \u22121. Let a1, a2, . . . , aw, b1, b2, . . . , bw\u2212e\u22121 be 2w \u2212e \u22121 distinct elements of Fq. \u2022 For each polynomial f(X) \u2208Fq[X], de\ufb01ne the following block on the point set Fq \u00d7 [w], Cf = {( f(aj), j) : j \u2208[w], deg( f) \u2264w \u2212e \u22121}. \u2022 For each \u03c3 \u03c3 \u03c3 = (\u03c31, \u03c32, . . . , \u03c3w\u2212e\u22121) \u2208Fw\u2212e\u22121 q , let E\u03c3 \u03c3 \u03c3 = {Cf : f \u2208Fq[X], deg( f) \u2264w \u2212e \u22121, f(bi) = \u03c3i for each i \u2208[w \u2212e \u22121]}. Theorem 19. The code E = {E\u03c3 \u03c3 \u03c3 : \u03c3 \u03c3 \u03c3 \u2208Fw\u2212e\u22121 q } is an (n, t, w, e)-CPECC code of size qw\u2212e\u22121, where n = qw and t = q \u22121. Proof. It is an immediate observation from the de\ufb01nition of the point set Fq \u00d7 [w] and the codeword Cf that each codeword has length qw and weight w. The rest of the proof has four steps. In the \ufb01rst one we will prove that for each \u03c3 \u03c3 \u03c3,\u03c3 \u03c3 \u03c3\u2032 \u2208Fw\u2212e\u22121 q , E\u03c3 \u03c3 \u03c3 and E\u03c3 \u03c3 \u03c3\u2032 are disjoint whenever \u03c3 \u03c3 \u03c3 \u0338= \u03c3 \u03c3 \u03c3\u2032. In the second step we will prove that for each \u03c3 \u03c3 \u03c3 \u2208Fw\u2212e\u22121 q the blocks in E\u03c3 \u03c3 \u03c3 are pairwise disjoint. As a result, by a simple counting argument in the third step it will be proved that E has qw\u2212e\u22121 codesets, each one has parallel class of size q, and as a consequence E is a (qw, q \u22121, w)-CPC code. In the last step we will \ufb01nd the minimum Hamming distance of E and as a result the number of errors e that it can correct. 1) Assume that there exist two codewords Cf \u2208E\u03c3 \u03c3 \u03c3 and Cg \u2208E\u03c3 \u03c3 \u03c3\u2032 such that \u03c3 \u03c3 \u03c3 \u0338= \u03c3 \u03c3 \u03c3\u2032 and Cf = Cg. Then f and g agree on at least w points and since the degrees of the polynomials are less than w, it follows that f = g. It implies that \u03c3i = f(bi) = g(bi) = \u03c3\u2032 i for all i \u2208[w \u2212e \u22121] and hence \u03c3 \u03c3 \u03c3 = \u03c3 \u03c3 \u03c3\u2032, a contradiction. Thus, E\u03c3 \u03c3 \u03c3 and E\u03c3 \u03c3 \u03c3\u2032 are disjoint whenever \u03c3 \u03c3 \u03c3 \u0338= \u03c3 \u03c3 \u03c3\u2032. \f12 2) Assume that the blocks Cf and Cg in E\u03c3 \u03c3 \u03c3, where f \u0338= g, intersect at the point (x, i0) for some x \u2208Fq and i0 \u2208[w]. It implies that f(ai0) = g(ai0) and since Cf , Cg \u2208E\u03c3 \u03c3 \u03c3, it follows that f(bi) = g(bi) for each i \u2208[w \u2212e \u22121]. Therefore, f and g agree on at least w \u2212e points. Since the degrees of f and g are at most w \u2212e \u22121, it follows that f = g, a contradiction. Therefore, the blocks in E\u03c3 \u03c3 \u03c3 are pairwise disjoint. Recall that each block has size w and the size of the point set of these blocks Fq \u00d7 [w] is qw. Hence, each codeset E\u03c3 \u03c3 \u03c3 contains at most q blocks. 3) The number of distinct polynomials in Fq[X] whose degrees are at most w \u2212e \u22121 is qw\u2212e. Each polynomial induces exactly one codeword in E. Hence, E contains exactly qw\u2212e distinct codewords. Since there are qw\u2212e\u22121 codesets and each one contains at most q codewords, it follows that each one contains exactly q codewords. The length of a codeword is qw and the weight of a codeword is w which implies that each codeset is a parallel class. Thus, by Proposition 10, E is a (qw, q \u22121, w)-CPC code. 4) Finally, for any two distinct codewords Cf and Cg, where f and g have degree at most w \u2212e \u22121, we have that |Cf \u2229Cg| \u2264w \u2212e \u22121 since larger intersection implies that f = g. Therefore, the Hamming distance between Cf and Cg is at least 2e + 2. Thus, the code E has minimum Hamming distance at least 2e + 2 and it can correct e errors. Thus, E is an (n, t, w, e)-CPECC code of size qw\u2212e\u22121, where n = qw and t = q \u22121. The encoding scheme in Section III-B can be easily adapted for the encoding of the CPECC code E. Algorithm 1 illustrates the decoding scheme for the (n, t, w, e)-CPECC code E obtained in Construction 2. Algorithm 1 Error-Correction for the CPECC codes in Construction 2 Input: a binary word u \u2282Fq \u00d7 [w] {the word received after the transmission of a codeword} Output: a message \u03c3 \u03c3 \u03c3 \u2208Fw\u2212e\u22121 q {the information word that was sent} 1: for each i \u2208[w] do 2: if |Yi \u225c{(y, i) : (y, i) \u2208u}| = 1 then 3: yi \u2190y, where (y, i) is the unique pair in Yi; 4: else 5: yi \u2190\u2018?\u2019; 6: \u02c6 y \u2190(y1, y2, . . . , yw); 7: apply the decoding algorithm for Reed-Solomon codes on \u02c6 y 8: The output of the algorithm is a polynomial L(x) of degree w \u2212e \u22121; 9: \u03c3 \u03c3 \u03c3 \u2190(L(b1), L(b2), . . . , L(bw\u2212e\u22121)); 10: return \u03c3 \u03c3 \u03c3; Theorem 20. Suppose that the codeword c \u2208E obtained in Construction 2 was submitted and the word u was received from c with at most e errors. Then, Algorithm 1 returns the word \u03c3 \u03c3 \u03c3 \u2208Fw\u2212e\u22121 q such that c \u2208E\u03c3 \u03c3 \u03c3. Proof. Using the notation of Algorithm 1, let i \u2208[w], Yi \u225c{(y, i) : (y, i) \u2208u}, and e\u2032 = |{i : |Yi| \u0338= 1}|. If |Yi| = 0 then an erasure occurred and this is re\ufb02ected as an erasure in yi. If |Yi| > 1 then we also know that an error has occurred for at least one coordinate (y, i). This will be also re\ufb02ected as an erasure in yi. Hence, at least e\u2032 erasure errors are re\ufb02ected in \u02c6 y as a result of at least e\u2032 errors in these Yi\u2019s. For the remaining w \u2212e\u2032 Yi\u2019s, while \f13 there may be errors, we know that each of these Yi\u2019s contains either no errors or two errors. Thus, the number of other erroneous Yi\u2019s is at most \u230a(e \u2212e\u2032)/2\u230b. The vector \u02c6 y is obtained by mapping the subsets Y1, Y2, . . . , Yw to the elements of Fq \u222a{?}. The word \u02c6 y was obtained from a codeword x f of a Reed-Solomon code of length N = w, dimension K = w \u2212e, and minimum Hamming distance D = N \u2212K + 1 = e + 1. An error-correction algorithm for such a code is capable of correcting e\u2032 erasures and at most \u230a(e \u2212e\u2032)/2\u230berrors as required by Algorithm 1. Using the Berlekamp-Welch algorithm [20] we can correct the errors with O(q3) operations [20], and hence, Algorithm 1 has complexity O(n3). V. RECURSIVE CONSTRUCTION All the (n, t, w)-CPC codes obtained from Proposition 7 and Construction 1 have t = n/w \u22121. In this section, we present a recursive construction that yields (n, t, w)-CPC codes which will be designed especially for larger values of t. For this purpose, recall the conditions of Proposition 10. Let (X, B) be a set system with a point set of size n, where B \u2286(X w) and B can be partitioned into M partial parallel classes P1, P2, . . . , PM to form a code E with M codesets. Suppose further that each partial parallel class Pi has exactly q blocks. Let S be a t-subset of X and Pi be a given partial parallel class. If t \u2265q, it might not be possible to choose a block/codeword in Pi which avoids S. However, by the pigeonhole principle, we can \ufb01nd such a block/codeword which intersects S in at most \u230at/q\u230belements. Given a (w, \u230at/q\u230b, w\u2032)-LPC code C it is possible to substitute it instead of each block/codeword of B and break up each codeword into codewords of weight at most w\u2032. This will enable to \ufb01nd a block/codeword of weight w\u2032 which avoids S. The following construction is based on this idea, where the code E is constructed similarly to the code in Construction 2. Construction 3. Let q \u2265n + w \u22121 be a prime power and let a1, a2, . . . , an, b1, b2, . . . , bw\u22121 be n + w \u22121 distinct elements of Fq. \u2022 Consider the point set Fq \u00d7 [n] and let B = {Cf \u225c{( f(aj), j) : j \u2208[n]} : f \u2208Fq[x], deg( f) \u2264w \u22121}. Note that the size of each block Cf is n. \u2022 For each \u03c3 \u03c3 \u03c3 = (\u03c31, \u03c32, . . . , \u03c3w\u22121) \u2208Fw\u22121 q , let E\u03c3 \u03c3 \u03c3 = {Cf : f \u2208Fq[X], deg( f) \u2264w \u22121, f(bi) = \u03c3i for each i \u2208[w \u22121]}. Similarly to the proof of Theorem 19 one can show that B is partitioned by E\u03c3 \u03c3 \u03c3, \u03c3 \u03c3 \u03c3 \u2208Fw\u22121 q into qw\u22121 parallel classes, each one of size q. Label the parallel classes and their blocks by Pi = {Bij : j \u2208[q]} for i \u2208[qw\u22121]. \u2022 Let D be an (n, t, w)-CPC code of size m, where t \u2265n/w. \u2022 Each block Bij is replaced by the codewords of each codeset of D by using any bijection between the set of points of Bij and the point set of D. Therefore, each codeword in D corresponds to a w-subset of Bij \f14 and from each block Bij we construct codewords for m new codesets. These sets of codewords from the m codesets will be denoted by Eij\u2113for each \u2113\u2208[m]. \u2022 For (i, \u2113) \u2208[qw\u22121] \u00d7 [m], the codeset Ei\u2113is de\ufb01ned by Ei\u2113\u225cSq j=1 Eij\u2113. Along the same lines of the proof in Theorem 19 one can prove that Theorem 21. The code {Pi : 1 \u2264i \u2264qw\u22121} is an (nq, q \u22121, n)-CPC code. Theorem 22. The code E = {Ei\u2113: i \u2208[qw\u22121], \u2113\u2208[m]} is an (nq, tq, w)-CPC code of size mqw\u22121. Proof. The size of E, the length of its codewords and their weight follow immediately from the de\ufb01nition of the codewords in E. Given a (tq)-subset S \u2282Fq \u00d7 [n] and a codeset Ei\u2113, (i, \u2113) \u2208[qw\u2212l] \u00d7 [m], we should \ufb01nd a codeword u \u2208Ei\u2113 such that supp(u) \u2229S = \u2205. Since Ei\u2113was constructed from the q blocks of Pi in which the codewords of the \u2113-th codeset of D were substituted, we have to \ufb01nd \ufb01rst a block Bij \u2208Pi which contains a subset S\u2032 of S whose size is at most t. Such a block exists since the number of blocks in Pi is q and S has size tq. Since Eij\u2113is a codeset in an (n, t, w)-CPC code, we can \ufb01nd a block u in Eij\u2113which avoids S\u2032. As a consequence supp(u) \u2229S = \u2205as required. To complete the proof we have to show that all the codesets of E are pairwise disjoint, i.e. Ei\u2113and Ei\u2032\u2113\u2032 are disjoint whenever (i, \u2113) \u0338= (i\u2032, \u2113\u2032). To this end, it suf\ufb01ces to show Eij\u2113and Ei\u2032j\u2032\u2113\u2032 are disjoint for any j, j\u2032 \u2208[q]. If (i, j) \u0338= (i\u2032, j\u2032), it can be veri\ufb01ed that |Bij \u2229Bi\u2032j\u2032| \u2264w \u22121 since intersection of size w will imply that the related polynomials are equal. Hence, since each Eij\u2113is a collection of w-subsets of Bij, we have that Eij\u2113and Ei\u2032j\u2032\u2113\u2032 are disjoint. If (i, j) = (i\u2032, j\u2032) then Eij\u2113and Eij\u2113\u2032 are from the same (n, t, w)-CPC code and therefore they are disjoint. Construction 3 can be applied also on (n, t, w)-LPC code (instead of (n, t, w)-CPC code). The only condition is that there is no codeset in which there are codewords of different weight. Also, when there are codewords of weight w\u2032 < w in the codeset, the whole construction should work with w\u2032 instead of w, e.g. the degree of the polynomial must be at most w\u2032 \u22121. Corollary 23. Let q be a prime power. If t + w \u2264n and q \u2265n + w \u22121, then (i) there exists an (nq, tq, w)-CPC code of size qw\u22121; (ii) there exists an (nq, tq, w)-LPC code of size \u2211w\u22121 i=0 qi. Proof. (i) the \ufb01rst claim follows from the fact that we can use an (n, t, w)-CPC code with exactly one codeset which contains all the w-subsets of the related n-set. (ii) the second claim follows from the fact we can apply Construction 3 and claim (i) on any w\u2032 \u2264w and obtain disjoint codes that can be combined together. \f15 Example 24. We compare certain CPC codes obtained from Construction 3 and Corollary 23 with the LPC codes obtained from Proposition 9. (i) Consider the set of \ufb01ve disjoint 3-(10, 4, 1) designs constructed by Etzion and Hartman [9]. By taking the complements of the blocks we obtain a (10, 3, 6)-CPC code of size \ufb01ve. Applying Construction 3 with q = 16, we obtain a (160, 48, 6)-CPC code of size 5 \u00b7 165 \u2248222.322. In contrast, Proposition 9 yields a (160, 48, 6)-LPC code of size 217 by setting s = 137 and r = 95. (ii) Setting n = 9, t = 2, w = 7, and q = 16 in Corollary 23 yields a (144, 32, 7)-LPC code of size \u22116 i=0 16i \u2248224.093. In contrast, Proposition 9 yields a (144, 32, 7)-LPC code of size 218 by setting s = 121 and r = 94. In the regime where w is \ufb01xed and t has order of magnitude as n, we show that the codes obtained in this section are asymptotically larger than those obtained from Proposition 9. The CPC codes obtained from Construction 3 and Corollary 23 attain the asymptotic upper bound O((nq)w\u22121) when w is \ufb01xed. In contrast, if we apply Proposition 9 with s = nq \u2212\u2308log2(\u2211w\u22121 i=0 (nq\u22121 i )\u2309) (the Gilbert-Varshamov lower bound) and r = nq \u2212tq \u2212\u230alog2(\u2211 w 2 i=0 (nq\u2212tq i ))\u230b(the Hamming upper bound), we obtain an (nq, tq, w)-LPC code of smaller size O((nq)w/2), or o((nq)w\u22121). VI. LPC CODES FROM COOLING CODES In this section we use a novel method to transform cooling codes into low-power cooling codes, while preserving the ef\ufb01ciency of the cooling codes. The construction is based on an injective mapping called domination mapping which was de\ufb01ned as follows in [6] The Hamming ball of radius w in {0, 1}n is the set B(n, w) of all words of weight at most w. Explicitly, B(n, w) \u225c \b y \u2208{0, 1}n : wt(y) \u2264w \t . Given m \u2264n, we are interested in injective mappings \u03d5 from {0, 1}m into B(n, w) that establish a certain domination relationship between positions in x \u2208{0, 1}m and positions in its image y = \u03d5(x). Speci\ufb01cally, one should be able to \u201cswitch off\u201d every position j \u2208[n] in y (that is, ensure that yj = 0) by switching off a corresponding position i \u2208[m] in x (that is, setting xi = 0). More precisely, let G = \u0000[m] \u222a[n], E \u0001 be a bipartite graph with m left vertices and n right vertices. If G has no isolated right vertices, we refer to G as a domination graph. De\ufb01nition 25. Given an injective map \u03d5 : {0, 1}m \u2192B(n, w) and a graph G = \u0000[m] \u222a[n], E \u0001 , we say that \u03d5 is a G-domination mapping, or G-dominating in brief, if \u2200(x1, x2, . . . , xm) \u2208{0, 1}m, \u2200(i, j) \u2208E : if \u03d5(x1, x2, . . . , xm) = (y1, y2, . . . , yn) and xi = 0, then yj = 0 We say that \u03d5 is an (m, n, w)-domination mapping if there exists a domination graph G = \u0000[m] \u222a[n], E \u0001 , such that \u03d5 is G-dominating. Properties of domination mappings, bounds on their parameters, constructions, and existence theorems were given in [6]. For our purpose we need some results from [6] and some which will be developed in the sequel. \f16 The \ufb01rst one taken from [6] restricts the structure of the domination graph. Lemma 26. The domination graph G = \u0000[m] \u222a[n], E \u0001 of an (m, n, w)-domination mapping has a subgraph with no isolated vertices and the degrees of the right vertices is exactly one. In view of Lemma 26 we will assume in the sequel that our domination graphs have no isolated vertices and all the right vertices have degree exactly one. We will de\ufb01ne the neighbourhood of a vertex v is G as the set of vertices adjacent to v and denote it by N(v). The following lemma is an immediate consequence of these observations and de\ufb01nition. Lemma 27. If U \u2282[n] is a set of right vertices of G then N(U) \u225c{N(u) : u \u2208U} is a set of vertices in [m] and |N(U)| \u2264|U|. Next, the obvious connection between domination mappings, cooling codes, and low-power cooling codes is given in the following theorem. Theorem 28. If there exists an (m, t)-cooling code C = {C1, C2, . . . , CM} and an (m, n, w)-domination mapping \u03d5 then the code C\u2032 = {C\u2032 1, C\u2032 2, . . . , C\u2032 M}, where C\u2032 i \u225c{\u03d5(x) : x \u2208Ci}, for each 1 \u2264i \u2264M, in an (n, t, w)-LPC code. Proof. The length n and the weight which is smaller from or equal to w for the codewords of C are immediate consequences from the de\ufb01nition of the (m, n, w)-domination mapping. Now, suppose we are given a t-subset S\u2032 \u2282[n] and a codeset C\u2032 i for some 1 \u2264i \u2264M. To complete the proof we have to show that there exists a codeword u\u2032 \u2208C\u2032 i such that supp(u\u2032) \u2229S\u2032 = \u2205. The t-subset S\u2032 can be viewed as a set of right vertices in the domination graph G = \u0000[m] \u222a[n], E \u0001 . By Lemma 27, for the set of neighbours of S\u2032 \u2282[n], S \u225cN(S\u2032) \u2282[m], we have that |S| \u2264|S\u2032| and hence |S| \u2264t. Since C is an (m, t)-cooling code, it follows that there exists a codeword u in Ci such that supp(u) \u2229S = \u2205which implies by the domination property that supp(\u03d5(u)) \u2229S\u2032 = \u2205. A product construction for domination mappings was presented in [6]. Let \u03d51 : {0, 1}m1 \u2192B(n1, w1) and \u03d52 : {0, 1}m2 \u2192B(n2, w2) be arbitrary domination mappings. Then their product \u03d5 = \u03d51 \u00d7 \u03d52 is a mapping from {0, 1}m1+m2 into B(n1+ n2, w1+ w2) de\ufb01ned as follows: \u03d5(x1, x2) = \u0000\u03d51(x1), \u03d52(x2) \u0001 where x1 \u2208{0, 1}m1, x2 \u2208{0, 1}m2, and (\u00b7, \u00b7) stands for string concatenation. That is, in order to \ufb01nd the image of a word x \u2208{0, 1}m1+m2 under \u03d5, we \ufb01rst parse x as (x1, x2), then apply \u03d51 and \u03d52 to the two parts. Theorem 29. If \u03d51 is an (m1, n1, w1)-domination mapping and \u03d52 is an (m2, n2, w2)-domination mapping, then their product \u03d5 = \u03d51 \u00d7 \u03d52 is an (m1+ m2, n1+ n2, w1+ w2)-domination mapping. \f17 The idea in Theorem 29 can be generalized as follows to a large number of domination mappings. Theorem 30. Let \u03d5i be an (mi, ni, wi)-domination mapping for each 1 \u2264i \u2264\u2113, and let (x1, x2, . . . , x\u2113) be a binary word, where the length of xi is mi, for each 1 \u2264i \u2264\u2113. The mapping \u03d5, de\ufb01ned by \u03d5(x1, x2, . . . , x\u2113) = (\u03d51(x1), \u03d52(x2), . . . , \u03d5\u2113(x\u2113)), is also an (m, n, w)-domination mapping for m = \u2211\u2113 i=1 mi, n = \u2211\u2113 i=1 ni, and w = \u2211\u2113 i=1 wi. Domination mappings are not dif\ufb01cult to \ufb01nd (at least for small parameters). For example, in [6] (2, 3, 1)-domination mapping, (9, 15, 3)-domination mapping, and (12, 20, 4)-domination mapping were presented. These three domination mappings have also ef\ufb01cient encoding and decoding procedures. Certainly, one can use an (m, n, w)-domination mapping to form an (n, t, w)-LPC code from an (m, t) cooling code. The only question is whether there is an ef\ufb01cient encoding and decoding schemes for the constructed LPC code. Such encoding and decoding schemes should be based on ef\ufb01cient encoding and decoding schemes for both the related cooling code and the related domination mapping. For large parameters such coding procedures might not exist. Hence, it is better to use the product constructions using many domination mappings with small parameters, but with ef\ufb01cient encoding and decoding schemes. Our next construction for (n, t, w)-LPC codes is based on this idea. For demonstration we will use a speci\ufb01c family of (n, t, w)-LPC code, but the same idea will work on any set of parameters that can be obtained from domination mappings with small parameters by using Theorem 30. We will describe the construction via its encoding scheme. Construction 4. Assume we are given w \u22656, m = 3w = 9\u03b1 + 12\u03b2, n = 5w = 15\u03b1 + 20\u03b2, t, and an (m, t)cooling code C with 2k codesets. We will construct an (n, t, w)-LPC code C\u2032. Let u be the information word of length k and let Cu be its related codeset in C. The encoder partitions the set of m coordinates into \u03b1 + \u03b2 subsets, \u03b1 subsets of size 9 and \u03b2 subsets of size 12. Similarly, it partitions the set n coordinates of the codewords from C\u2032 into \u03b1 + \u03b2 subsets, \u03b1 subsets of size 15 and \u03b2 subsets of size 20. Let \u03d51 and \u03d52 be a (9, 15, 3)-domination mapping and a (12, 20, 4)-domination mapping, respectively. Let \u03d5 be the (m, n, w)-domination mapping implied by the product construction of Theorem 30 on \u03b1 copies of \u03d51 and \u03b2 copies of \u03d52. Let T be a t-subset of [n] and let T\u2032 = N(T) be a t\u2032-subset of [m], where t\u2032 \u2264t by Lemma 27. The encoder \ufb01nds the vector v in Cu related to the set T\u2032, i.e. v has zeroes in the coordinates of T\u2032, as required. Finally the encoder parse v into v1v2 . . . v\u03b1v\u2032 1v\u2032 2 . . . v\u2032 \u03b2, where vi is of length 9 and v\u2032 i is of length 12. By using the encodings of the mappings \u03d51 and \u03d52, the encoder maps v1v2 . . . v\u03b1v\u2032 1v\u2032 2 . . . v\u2032 \u03b2 to the word \u03d5(v1v2 . . . v\u03b1v\u2032 1v\u2032 2 . . . v\u2032 \u03b2) of the (n, t, w)-LPC code, where each vi is mapped by \u03d51 to a word of length 15 and each v\u2032 i is mapped by \u03d52 to a word of length 20. The decoder is applied in reverse order to generate the information word of length k from a word of length n of the (n = 5w, t, w)-LPC code C\u2032, by \ufb01rst generating a word, of length m, from C and after that using the decoder of C to \ufb01nd the information word of length m. Note that Construction 4 can be viewed as a modi\ufb01cation of the Concatenation construction (See Proposition 8). Construction 4 has an advantage on the Concatenation of Proposition 8 and other constructions with larger size \f18 for the same weight w and the same number of hottest wires t. How good are the codes constructed by using the domination mappings. They are incomparable with the other codes which were constructed in previous sections due to their parameters. But, they can be easily compared with the codes obtained in Proposition 8. We will consider some examples by using three of the most simple (and less powerful) domination mappings, a (2, 3, 1)-domination mapping, a (3, 9, 15)-domination mapping and a (4, 12, 20)-domination mapping (note that the last two were used in Construction 4). To this end we will describe the simple and effective construction of cooling codes given in [3]. This construction is based on spreads (or partial spreads) which will be de\ufb01ned next. Loosely speaking, a partial \u03c4-spread of the vector space Fn q is a collection of disjoint \u03c4-dimensional subspaces of Fn q. Formally, a collection V1, V2, . . . , VM of \u03c4-dimensional subspaces of Fn q is said to be a partial \u03c4-spread of Fn q if Vi \u2229Vj = {0} for all i \u0338= j , Fn q \u2287V1 \u222aV2 \u222a\u00b7 \u00b7 \u00b7 \u222aVM . If the \u03c4-dimensional subspaces form a partition of Fn q then the partial \u03c4-spread is called a \u03c4-spread. It is well known that such \u03c4-spreads exist if and only if \u03c4 divides n, in which case M = (qn\u22121)/(q\u03c4\u22121) > qn\u2212\u03c4. For the case where \u03c4 does not divide n, partial \u03c4-spreads with M \u2265qn\u2212\u03c4 have been constructed in [10, Theorem 11]. Theorem 31. Let V1,V2, . . . , VM be a partial (t+1)-spread of Fn 2, and de\ufb01ne the code C = {V\u2217 1 , V\u2217 2 , . . . , V\u2217 M}, where V\u2217 i = Vi \\{0} for all i. Then C is an (n, t)-cooling code of size M \u22652n\u2212t\u22121. Assume \ufb01rst that we want to construct an (3w, t, w)-LPC code from a (2w, t)-cooling code. We consider the trivial (2, 3, 1)-domination mapping and use a (t + 1)-spread over F2w 2 , where 2(t + 1) \u22642w to obtain a (3w, t, w)LPC code C of size 22w\u22121 2t+1\u22121 > 22w\u2212t\u22121 (from a (2w, t)-cooling code) for any t + 1 \u2264w. Assume now that we want to form a comparable code using Proposition 8. There are a few options that can be taken as parameters in Proposition 8. 1) Assume \ufb01rst that we take w\u2032 = 1, s = 3, and m = w, in Proposition 8. As a consequence we can take q = 4, and hence the size of the (3w, t, w)-LPC code obtained by Proposition 8 will be 22w\u22122t\u22121, for t \u22643 and t + 1 \u2264w/2, which is clearly much smaller than C. Moreover, t is at most the minimum between 3 and w 2 \u22121 compared to t \u2264w \u22121 for the code C based on the (2, 3, 1)-domination mapping. 2) A different choice in Proposition 8 is w\u2032 = 3, s = 9, and m = w/3. As a consequence we can take q = 128, and hence the size of the (3w, t, w)-LPC code obtained by Proposition 8 will be 27w/3\u22127t\u22127 for any t + 1 \u2264w/6, but not larger than 9. Hence, t is much smaller compared to t \u2264w \u22121 for the code C based on the (2, 3, 1)-domination mapping. As for the size of the code, for the same t the code C is larger if t \u2265w 18 \u22121 We continue with our constructed (5w, t, w)-LPC code C from a (3w, t)-cooling code discussed in Construction 4. It has 23w\u2212t\u22121 codesets and it can handle any t \u22643w/2 \u22121 (if the spread construction of Theorem 31 is \f19 used). Assume now that we want to form a comparable code by using Proposition 8. There are a few options that can be taken as parameters in Proposition 8. 1) Assume \ufb01rst that we take w\u2032 = 3, s = 15, and m = w/3, in Proposition 8. As a consequence we can take q = 29, and hence the size of the (5w, t, w)-LPC code obtained by Proposition 8 will be 23w\u22129t\u22129, for any t + 1 \u2264w/6 but not larger than 15, which is clearly much smaller than C for both size and t. 2) A different choice in Proposition 8 is w\u2032 = 4, s = 20, and m = w/4. As a consequence we can take q = 212, and hence the size of the (5w, t, w)-LPC code obtained by Proposition 8 will be 23w\u221212t\u221212 for any t + 1 \u2264w/8, but not larger than 20, which is clearly much smaller than C for both size and t. It should be noted that q can be sometimes slightly larger than the one given in the example. This won\u2019t make much difference in the comparison, but the computation in a large \ufb01eld size which is not a power of 2 is more messy. We conclude that the codes obtained by this new method have in most cases larger size and better capabilities than the best codes obtained by previous known constructions. ACKNOWLEDGMENTS Y. M. Chee was supported in part by the Singapore Ministry of Education under grant MOE2017-T3-1-007. T. Etzion and A. Vardy were supported in part by the BSF-NSF grant 2016692 Binational Science Foundation (BSF), Jerusalem, Israel, under Grant 2012016. The research of T. Etzion and A. Vardy was supported in part by the National Science Foundation under grant CCF-1719139. The research of Y. M. Chee, H. M. Kiah and H. Wei was supported in part by the Singapore Ministry of Education under grant MOE2015-T2-2-086."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1807.10954v1",
+                    "title": "Domination Mappings into the Hamming Ball: Existence, Constructions, and Algorithms",
+                    "abstract": "The Hamming ball of radius $w$ in $\\{0,1\\}^n$ is the set ${\\cal B}(n,w)$ of\nall binary words of length $n$ and Hamming weight at most $w$. We consider\ninjective mappings $\\varphi: \\{0,1\\}^m \\to {\\cal B}(n,w)$ with the following\ndomination property: every position $j \\in [n]$ is dominated by some position\n$i \\in [m]$, in the sense that \"switching off\" position $i$ in $x \\in\n\\{0,1\\}^m$ necessarily switches off position $j$ in its image $\\varphi(x)$.\nThis property may be described more precisely in terms of a bipartite\n\\emph{domination graph} $G = ([m] \\cup [n], E)$ with no isolated vertices, for\nall $(i,j) \\in E$ and all $x \\in \\{0,1\\}^m$, we require that $x_i = 0$ implies\n$y_j = 0$, where $y = \\varphi(x)$. Although such domination mappings recently\nfound applications in the context of coding for high-performance interconnects,\nto the best of our knowledge, they were not previously studied.\n  In this paper, we begin with simple necessary conditions for the existence of\nan $(m,n,w)$-domination mapping $\\varphi: \\{0,1\\}^m \\to {\\cal B}(n,w)$. We then\nprovide several explicit constructions of such mappings, which show that the\nnecessary conditions are also sufficient when $w=1$, when $w=2$ and $m$ is odd,\nor when $m \\le 3w$. One of our main results herein is a proof that the trivial\nnecessary condition $|{\\cal B}(n,w)| \\ge 2^m$ for the existence of an injection\nis, in fact, sufficient for the existence of an $(m,n,w)$-domination mapping\nwhenever $m$ is sufficiently large. We also present a polynomial-time algorithm\nthat, given any $m$, $n$, and $w$, determines whether an $(m,n,w)$-domination\nmapping exists for a domination graph with an equitable degree distribution.",
+                    "authors": "Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Alexander Vardy",
+                    "published": "2018-07-28",
+                    "updated": "2018-07-28",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "Introduction Given a binary word y = (y1, y2, . . . , yn), the Hamming weight of y is the number of nonzero positions in y. Explicitly, wt(y) def = \b j \u2208[n] : yj \u0338= 0 \t . The Hamming ball of radius w in {0, 1}n is the set B(n, w) of all words of weight at most w. Explicitly, B(n, w) def = \b y \u2208{0, 1}n : wt(y) \u2a7dw \t . Given m \u2a7dn, we are interested in injective mappings \u03d5 from {0, 1}m into B(n, w) that establish a certain domination relationship between positions in x \u2208{0, 1}m and positions in its image y = \u03d5(x). Speci\ufb01cally, one should be able to \u201cswitch off\u201d every position j \u2208[n] in y (that is, ensure that yj = 0) by switching off a corresponding position i \u2208[m] in x (that is, setting xi = 0). More precisely, let G = \u0000[m] \u222a[n], E \u0001 be a bipartite graph with m left vertices and n right vertices. If G has no isolated right vertices, we refer to G as a domination graph. De\ufb01nition 1. Given an injective map \u03d5: {0, 1}m \u2192B(n, w) and a graph G = \u0000[m] \u222a[n], E \u0001 , we say that \u03d5 is a G-domination mapping, or G-dominating in brief, if \u2200(x1, x2, . . . , xm) \u2208{0, 1}m, \u2200(i, j) \u2208E : if \u03d5(x1, x2, . . . , xm) = (y1, y2, . . . , yn) and xi = 0, then yj = 0 We say that \u03d5 is an (m, n, w)-domination mapping if there exists a domination graph G = \u0000[m] \u222a[n], E \u0001 , with no isolated right vertices, such that \u03d5 is G-dominating. Example 1. Let m = 3, n = 4, w = 2, and consider the injective map \u03d5: {0, 1}3 \u2192B(4, 2) given by \u03d5(000) = 0000 \u03d5(001) = 0001 \u03d5(010) = 0010 \u03d5(011) = 0011 \u03d5(100) = 0100 \u03d5(101) = 0101 \u03d5(110) = 1000 \u03d5(111) = 1001 (1) It is easy to verify by inspection that \u03d5 is G-dominating, where G = \u0000[3] \u222a[4], E) can be chosen as any one of the following three domination graphs: 1 2 3 1 2 3 4 1 2 3 1 2 3 4 1 2 3 1 2 3 4 (2) 2 Note that the requirement of no isolated right vertices in Definition 1 means that every position j \u2208[n] in the image of \u03d5 is dominated by some position i \u2208[m] of its domain. This is motivated by the application to thermal-management coding for high-performance interconnects [2,3], brie\ufb02y described in what follows. 1.1. Applications to coding for interconnects Given an interconnect (bus) consisting of n wires, it is desirable to control its average power consumption. This can be achieved by making sure that, during each synchronized transmission, the total number of transitions on the n wires is below a speci\ufb01ed threshold w. As shown in [2], the above translates directly into the requirement that the transmitted binary word y is in the Hamming ball B(n, w). It is also desirable to control the peak temperature of an interconnect by effectively cooling its hottest wires. That is, given an arbitrary subset S of [n], which corresponds to the positions of the hottest wires, we wish to transmit a word (y1, y2, . . . , yn) such that yj = 0 for all j \u2208S. The cooling codes of [2] provide an elegant solution to this problem using spreads or partial spreads [4,12]. Domination mappings were introduced in [3] in order to simultaneously achieve both of these desirable properties. Given a subset S of [n], one can follow the edges of 1 \fthe domination graph G to \ufb01nd a subset S\u2032 of [m] so that every position in S is dominated by some position in S\u2032. Then, using spreads as described in [2], one can produce a binary word x = (x1, x2, . . . , xm) such that xi = 0 for all i \u2208S\u2032. If \u03d5 is a G-dominating mapping from {0, 1}m into B(n, w), the transmitted word y = \u03d5(x) satis\ufb01es wt(y) \u2a7dw and yj = 0 for all j \u2208S. Since \u03d5 is injective, x can be recovered from y at the receiver. For more details on this encoding/decoding procedure, we refer the reader to [2,3]. In the context of coding for high-performance interconnects, the parameters m and w would be usually given, and one would like to use an (m, n, w)-domination mapping with the lowest possible n, in order to minimize the redundancy n \u2212m of the encoding procedure. We note that although domination mappings were de\ufb01ned in [3], they were not studied therein. In particular, it is not clear when such mappings exist. 1.2. Our contributions Our primary objective in this paper is to answer the following question: for which parameters m, n, w, do (m, n, w)-domination mapping exist? A secondary objective is to study the structure of such mappings and provide explicit constructions for them. We believe that, aside from their applications to coding for interconnects, these problems are also interesting in their own right, from the combinatorial perspective. One of our main results is the following theorem, which shows that if the domain {0, 1}m is suf\ufb01ciently large with respect to w, then domination mappings always exist. Theorem 1. For every w \u2a7e1, there is a constant m0(w) with the following property: for all m \u2a7em0(w), if |B(n, w)| \u2a7e2m, then an (m, n, w)-domination mapping exists. Thus, for suf\ufb01ciently large m, whenever injections from {0, 1}m into B(n, w) exist at all, one of these injections is necessarily an (m, n, w)-domination mapping. This is somewhat surprising; it is certainly not true for small m. To prove Theorem 1, we introduce a certain bipartite graph with vertex set {0, 1}m \u222aB(n, w), which we call the compatibility graph. We then use Hall\u2019s marriage theorem [5] to show that the compatibility graph has a perfect matching, when m is suf\ufb01ciently large. The fact that the conditions of Hall\u2019s marriage theorem are satis\ufb01ed in this graph is established in a long and elaborate sequence of lemmas in Section 6. The other contributions in this paper are organized as follows. We begin in the next section by establishing some basic properties of domination graphs and domination mappings. In particular, we consider certain restrictions of a domination mapping \u03d5: {0, 1}m \u2192B(n, w) to a subset of the m positions of its domain. We will show that all such restrictions are also domination mappings. In Section 3, we use some of these results to derive necessary conditions for the existence of an (m, n, w)-domination mapping. In particular, we show that n \u2a7e2m \u2212w in any such mapping. In Section 4, we present several explicit constructions of domination mappings. In particular, we introduce the product construction; given any two (m1, n1, w1)-domination and (m2, n2, w2)-domination mappings, this construction produces an (m1 + m2, n1 + n2, w1 + w2)-domination mapping. We also construct perfect (m, n,1)-domination mappings with n = 2m\u22121, so that |B(n,1)| = 2m and \u03d5 is a bijection. We furthermore describe another construction for w = 2 and m odd; it appears that already in this case, the construction problem is far from trivial. In Section 5, we present a polynomial-time algorithm that, given any m, n, and w, determines whether an (m, n, w)-domination mapping exists for a domination graph G with an equitable degree distribution (that is, the degree of all the left vertices in G is either \u230an/m\u230bor \u2308n/m\u2309). To this end, we apply the K\u00f6nig-Egev\u00e1ry theorem [7] to the bipartite compatibility graph in order to conclude that the maximum matching and the maximum fractional matching in this graph have the same size. This leads to a linear program with an exponential number of variables, but we use symmetries of the compatibility graph to show that this problem can be solved in polynomial time. One consequence of our constructions in Section 4 is that the necessary conditions in Section 3 are also suf\ufb01cient for all m \u2a7d3w. This resolves the existence problem for (m, n, w)-domination mappings for small 2 \fvalues of m. As a consequence of Theorem 1, this problem is also resolved for large m. For intermediate values of m, we can determine whether an (m, n, w)-domination mapping exists in polynomial time using the algorithm of Section 5 \u2014 albeit only for an equitable domination graph. We believe, however, that the decision produced by the polynomial-time algorithm of Section 5 holds unconditionally (cf. Conjecture 1). 2. Basic Properties of Domination Mappings We begin with some basic properties and notation for domination graphs. Recall that, by de\ufb01nition, a domination graph is a bipartite graph G = \u0000[m] \u222a[n], E \u0001 with no isolated vertices on the right side of the bipartition. The following lemma shows that G cannot have isolated vertices on the left side either. Lemma 2. Let G = \u0000[m] \u222a[n], E \u0001 be a domination graph. If there is a vertex of degree zero in [m], then a G-domination mapping does not exist. Proof. Assume to the contrary that \u03d5 : {0, 1}m \u2192B(n, w) is a G-domination mapping. Since every position in [n] is dominated by some position in [m], we have \u03d5(0) = 0, where 0 denotes the all-zero word of the appropriate length. Suppose that the vertex i \u2208[m] has degree zero, and let ei \u2208{0, 1}m be the word of weight one with the single nonzero in position i. Then \u03d5(ei) = 0 since if deg(i) = 0, every position in [n] is dominated by a position in [m] other than i. Thus \u03d5(ei) = \u03d5(0), contradicting the fact that \u03d5 is injective. In view of Lemma 2, we henceforth assume w.l.o.g. that the degree of every vertex in a domination graph is at least one. In fact, we can furthermore assume that the degree of all the vertices on the right side of the bipartition is exactly one. Indeed, let G = \u0000[m] \u222a[n], E \u0001 be a domination graph, let \u03d5 be a G-domination mapping, and suppose that a vertex j \u2208[n] has degree \u03b4 > 1. We construct G\u2032 from G by removing, arbitrarily, some \u03b4 \u22121 of the \u03b4 edges incident upon j. Then, obviously, \u03d5 is G\u2032-dominating as well (cf. Example 1). It follows that any (m, n, w)-domination mapping is G-dominating for some graph G = \u0000[m] \u222a[n], E \u0001 with n edges and deg(j) = 1 for all j \u2208[n]. Up to a permutation of [n], any such domination graph G is uniquely described by the degrees \u03b41, \u03b42, . . . , \u03b4m of its left vertices. We say that (\u03b41, \u03b42, . . . , \u03b4m) is the degree sequence of G, and further assume (w.l.o.g., up to a permutation of [m]) that \u03b41 \u2a7d\u03b42 \u2a7d\u00b7 \u00b7 \u00b7 \u2a7d\u03b4m. With this assumption, a domination graph G can be also speci\ufb01ed in terms of its degree distribution (d1, d2, . . . , d\u2206), where di is the number of left vertices of degree i and \u2206def = max{\u03b41, \u03b42, . . . , \u03b4m}. Clearly, d1 + d2 + \u00b7 \u00b7 \u00b7 + d\u2206= m and d1 + 2d2 + \u00b7 \u00b7 \u00b7 + \u2206d\u2206= n (3) A domination graph is said to be equitable if \u230an/m\u230b= \u03b41 \u2a7d\u03b42 \u2a7d\u00b7 \u00b7 \u00b7 \u2a7d\u03b4m = \u2308n/m\u2309. Equitable domination graphs play an important role in the study of domination mappings. In fact, we conjecture as follows. Conjecture 1. An (m, n, w)-domination mapping exists if and only if there exists a mapping from {0,1}m to B(n,w) that is G-dominating for an equitable graph G. We now consider certain basic properties of (m, n, w)-domination mappings. The following easy lemma shows that the existence problem for such mappings is \u201cmonotonic\u201d in both n and w. Lemma 3. If an (m, n, w)-domination mapping exists, then there also exist an (m, n + 1, w)-domination mapping and an (m, n, w + 1)-domination mapping. Proof. Given an (m, n, w)-domination map \u03d5, we construct an injective map \u03d5\u2032 : {0, 1}m \u2192B(n + 1, w) as follows: \u03d5\u2032(x) = \u0000\u03d5(x), 0), where (\u00b7, \u00b7) stands for string concatenation. Clearly, \u03d5\u2032 is an (m, n + 1, w)domination mapping. The other claim follows trivially from the fact that B(n, w) \u2286B(n, w + 1). 3 \fWe next show that every G-domination mapping gives rise to a multitude of derived domination mappings corresponding to subgraphs of G induced by subsets of [m]. Lemma 4. Let \u03d5: {0, 1}m \u2192B(n, w) be a G-domination mapping. Fix an arbitrary left vertex i of G, and let G\u2032 be the induced subgraph of G obtained by removing i and all vertices adjacent to i. Then \u03d5 induces a G\u2032-domination mapping \u03d5\u2032 from {0, 1}m\u22121 to B(n \u2212\u03b4, w), where \u03b4 = deg(i). Proof. The mapping \u03d5\u2032 : {0, 1}m\u22121 \u2192B(n \u2212\u03b4, w) is derived from \u03d5 as follows. Given an arbitrary word x\u2032 = (x1, x2, . . . , xm\u22121) in {0, 1}m\u22121, we \ufb01rst extend it to x = (x1, x2, . . . , xi\u22121, 0, xi, . . . , xm\u22121) \u2208{0, 1}m. Let {j, j + 1, . . . , j + \u03b4 \u22121} \u2208[n] be the set of positions dominated by the vertex i (where we have assumed w.l.o.g. that these positions are consecutive). Then y = \u03d5(x) is of the form y = \u0000y1, y2, . . . , yj\u22121, 0, 0, . . . , 0 | {z } \u03b4 zeros , yj+\u03b4, . . . , yn), and we set \u03d5\u2032(x\u2032) = (y1, y2, . . . , yj\u22121, yj+\u03b4, . . . , yn). It is easy to see that the mapping \u03d5\u2032 thereby de\ufb01ned is an injection from {0, 1}m\u22121 to B(n \u2212\u03b4, w), and \u03d5\u2032 is G\u2032-dominating. The construction of the derived mapping \u03d5\u2032 in Lemma 4 is akin to the process of shortening linear codes in coding theory [8]. We restrict both the domain and the range of \u03d5 to subsets thereof consisting of those words that contain zeros in speci\ufb01ed positions, then puncture out these all-zero positions to obtain \u03d5\u2032. Iterating this construction, a single (m, n, w)-domination mapping gives rise to 2m derived domination mappings (some of which may be isomorphic or trivial) corresponding to all subsets of [m]. An immediate consequence of Lemma 4 is that the existence problem for domination mappings is also monotonic in m. That is, for all m \u2a7e2, if an (m, n, w)-domination mapping exists, we can use Lemma 4 (in conjunction with Lemma 3) to construct an (m \u22121, n, w)-domination mapping. In view of this monotonicity, it is natural to de\ufb01ne the following parameters: \u00b5(n, w) def = max \b m \u2208Z : an (m, n, w)-domination mapping exists \t \u03bd(m, w) def = min \b n \u2208Z : an (m, n, w)-domination mapping exists \t We say that an (m, n, w)-domination mapping is optimal if n = \u03bd(m, w). We next discuss necessary conditions for the existence of an (m, n, w)-domination mapping, which can be regarded as bounds on \u03bd(m, w). 3. Necessary Conditions An (m, n, w)-domination mapping is injective by de\ufb01nition, so the size of its domain {0, 1}m cannot exceed the size of its range B(n, w). While trivial, this necessary condition is fundamental; we state it as follows. Lemma 5. For any (m, n, w)-domination mapping, we have 2m \u2a7d w \u2211 j=0 \u0012n j \u0013 (4) In conjunction with the shortening procedure in Lemma 4, the trivial necessary condition of Lemma 5 leads to a considerably more elaborate bound on the parameters of an (m, n, w)-domination mapping. 4 \fLemma 6. Let G = \u0000[m] \u222a[n], E \u0001 be a domination graph with degree distribution (d1, d2, . . . , d\u2206). Then for any G-domination mapping \u03d5: {0, 1}m \u2192B(n, w), we have m \u2a7d min t1,t2,...,t\u2206 ( (t1+ t2 + \u00b7 \u00b7 \u00b7 + t\u2206) + log2 w \u2211 j=0 \u0012n \u2212t1 \u22122t2 \u2212\u00b7 \u00b7 \u00b7 \u2212\u2206t\u2206 j \u0013) (5) where the minimum is taken over all nonnegative integers t1, t2, . . . , t\u2206with ti \u2a7ddi for i = 1, 2, . . . , \u2206. Proof. Given t1, t2, . . . , t\u2206, we invoke Lemma 4 iteratively to obtain a domination graph G\u2032 with degree distribution (d1\u2212t1, d2\u2212t2, . . . , d\u2206\u2212t\u2206) and a G\u2032-domination mapping \u03d5\u2032 with parameters (m\u2032, n\u2032, w), where m\u2032 = m \u2212(t1+ t2+ \u00b7 \u00b7 \u00b7 + t\u2206) and n\u2032 = n \u2212t1 \u22122t2 \u2212\u00b7 \u00b7 \u00b7 \u2212\u2206t\u2206. Applying Lemma 5 to \u03d5\u2032 establishes (5). Observe that the minimization in (5) can be reduced to the minimum of at most m terms, parametrized by s = t1+ t2 + \u00b7 \u00b7 \u00b7 + t\u2206with s \u2a7dm \u22121. The following lemma is an important special case of Lemma 6. This lemma shows, in particular, that the trivial necessary condition of Lemma 5 is, in general, not suf\ufb01cient. Lemma 7. For any (m, n, w)-domination mapping, we have n \u2a7e2m \u2212w. Proof. Set t1 = 0 and ti = di for i = 2, 3, . . . , \u2206in Lemma 6. Then n \u2212t1 \u22122t2 \u2212\u00b7 \u00b7 \u00b7 \u2212\u2206t\u2206= d1 while t1 + t2 + \u00b7 \u00b7 \u00b7 + t\u2206= m \u2212d1 in view of (3), and the bound in (5) reduces to d1 \u2a7dlog2 w \u2211 j=0 \u0012d1 j \u0013 This is only possible if w \u2a7ed1. Now, again in view of (3), we have n \u2a7ed1 + 2(d2 + \u00b7 \u00b7 \u00b7 + d\u2206) = 2m \u2212d1, and the lemma follows. The bounds on \u03bd(m, w) resulting from Lemma 5 and Lemma 7 are illustrated in Figure 1 for w = 3, and we can see that neither of them implies the other. In fact, for all w, it can be readily shown that Lemma 7 is tighter than Lemma 5 whenever w < m \u2a7d3w, whereas Lemma 5 is tighter when m is large. We next investigate the conditions under which equality can hold in the bounds of Lemma 5 and Lemma 7. A domination mapping that establishes a bijection between {0, 1}m and B(n,w) will be called perfect; clearly an (m, n, w)-domination mapping is perfect if and only if its parameters satisfy 2m = \u2211w j=0 (n j). There are only four known cases where (an incomplete) sum of binomial coef\ufb01cients gives a power of 2. The (m, n, w) triples for these four cases are given by (m, m + 1, m/2) for m even, (m, 2m \u22121, 1), (12, 90, 2), (11, 23, 3) (6) All these cases were discovered in the context of perfect error-correcting codes [6,11] in the 1960s, and none were found since. It is well known that perfect binary codes \u2014 that is, partitions of {0, 1}n into translates of the Hamming ball B(n, w) \u2014 exist in only three of the four cases (it is not possible to partition {0, 1}90 into Hamming balls). Remarkably, we have found perfect domination mappings for each of the four triples listed in (6). The \ufb01rst set of parameters is attained for m = 2 by the (2, 3, 1)-domination mapping with degree sequence (1, 2), given by \u03d5(00) = 000, \u03d5(01) = 010, \u03d5(10) = 100, \u03d5(11) = 001 By Lemma 7, there are no (m, m + 1, m/2)-domination mappings for m \u2a7e4. In Section 4.2, we explicitly construct perfect (m, 2m \u22121, 1)-domination mappings for all m \u2a7e1. Finally, perfect domination mappings with parameters (12, 90, 2) and (11, 23, 3) were found by computer search using the methods of Section 5. 5 \fn necessary condition of Lemma 5 necessary condition of Lemma 7 perfect domination mapping mapping constructed in Sections 2 and 4 mapping exists by the results of Section 2 mapping exists by the results of Section 6 m Figure 1. Bounds, constructions, and existence of (m, n, w)-domination mappings for w = 3. In contrast to equality in (4), which is quite rare, equality in Lemma 7 can be achieved much more easily. The next lemma gives necessary and suf\ufb01cient conditions for this. Lemma 8. Let \u03d5 be an (m, n, w)-domination mapping. Then n = 2m \u2212w if and only if the degree distribution of the underlying domination graph is d1 = w, d2 = m \u2212w, and di = 0 for i \u2a7e2. Proof. It suf\ufb01ces to show that the conditions on the degree distribution are necessary, since the fact that they are suf\ufb01cient is trivial from (3). Suppose n = 2m \u2212w. Then (3) implies that 2m \u2212w \u2a7ed1 + 2d2 + 3(d3 + d4 + \u00b7 \u00b7 \u00b7 + d\u2206) = d1 + 2d2 + 3(m \u2212d1 \u2212d2) = 3m \u22122d1 \u2212d2 which simpli\ufb01es to 2d1 + d2 \u2a7em + w. Now, it follows from (the proof of) Lemma 7 that d1 \u2a7dw in any (m, n, w)-domination mapping. Hence 2d1 + d2 \u2a7em + w further implies d1 + d2 \u2a7em. But this is only possible if d1 + d2 = m and d3 + d4 + \u00b7 \u00b7 \u00b7 + d\u2206= 0. Finally, if d1 < w, then n \u2a7e2m \u2212d1 > 2m \u2212w. In the next section, for all m in the range w \u2a7dm \u2a7d3w, we present an explicit construction of domination mappings whose degree distribution satis\ufb01es the conditions of Lemma 8. By Lemma 7, such mappings are optimal, and therefore \u03bd(m, w) = 2m \u2212w when w \u2a7dm \u2a7d3w and w \u2a7e3. Finally, we note that the case m \u2a7dw is trivial. In this case, Lemma 5 reduces to n \u2a7em. Thus \u03bd(m, w) = m and the optimal (m, m, w)-domination mapping is the identity map from {0, 1}m to itself. 6 \f4. Constructions In this section, we present several explicit constructions of (optimal) domination mappings. These results are distinct from our results in Sections 5 and 6, which are concerned solely with the existence of such mappings. 4.1. Product construction We begin with a simple and effective recursive construction that combines any two domination mappings \u03d51 and \u03d52 to produce another domination mapping \u03d5. Notably, if the mappings \u03d51 and \u03d52 attain the bound of Lemma 7, then so does the mapping \u03d5 obtained from this construction. Let \u03d51 : {0, 1}m1 \u2192B(n1, w1) and \u03d52 : {0, 1}m2 \u2192B(n2, w2) be arbitrary domination mappings. Then their product \u03d5 = \u03d51 \u00d7 \u03d52 is a mapping from {0, 1}m1+m2 into B(n1 + n2, w1 + w2) de\ufb01ned as follows: \u03d5(x1, x2) = \u0000\u03d51(x1), \u03d52(x2) \u0001 (7) where x1 \u2208{0, 1}m1, x2 \u2208{0, 1}m2, and (\u00b7, \u00b7) stands for string concatenation. That is, in order to \ufb01nd the image of a word x \u2208{0, 1}m1+m2 under \u03d5, we \ufb01rst parse x as (x1, x2), then apply \u03d51 and \u03d52 to the two parts. Theorem 9. If \u03d51 is an (m1, n1, w1)-domination mapping and \u03d52 is an (m2, n2, w2)-domination mapping, then their product \u03d5 = \u03d51 \u00d7 \u03d52 is an (m1 + m2, n1 + n2, w1 + w2)-domination mapping. Proof. The parameters of \u03d5 and the fact that \u03d5 is injective are obvious from (7). Thus it remains to show that every position j \u2208[n1 + n2] is dominated by some position i \u2208[m1 + m2]. But this is easy. Let G1 and G2 be the domination graphs for \u03d51 and \u03d52, respectively. If j \u2a7dn1, we follow the corresponding edge of G1 to \ufb01nd i. If j > n1, we follow the edge of G2 that corresponds to j \u2212n1 to \ufb01nd i\u2217, then set i = i\u2217+ m1. Observe that the proof of Theorem 9 relates the degree distribution (d1, d2, . . . , d\u2206) of (the domination graph for) the product \u03d5 = \u03d51 \u00d7 \u03d52 to the degree distributions (d\u2032 1, d\u2032 2, . . . , d\u2032 \u2206\u2032) and (d\u2217 1, d\u2217 2, . . . , d\u2217 \u2206\u2217) of the constituent mappings \u03d51 and \u03d52. Clearly \u2206= max{\u2206\u2032, \u2206\u2217} and di = d\u2032 i + d\u2217 i for all i. Corollary 10. For all w \u2a7e3, the product construction yields a (3w, 5w, w)-domination mapping. Proof. Using computer search, we found domination mappings \u03d53, \u03d54, and \u03d55 with parameters (9, 15, 3), (12, 20, 4), and (15, 25, 5), respectively. Since any w \u2a7e3 is an integer linear combination of 3, 4, and 5, a (3w, 5w, w)-domination mapping can be constructed as the appropriate product of \u03d53, \u03d54, and \u03d55. Theorem 11. \u03bd(m, w) = 2m \u2212w for all w \u2a7e3 and w \u2a7dm \u2a7d3w Proof. In view of Lemma 7, it suf\ufb01ces to show that an (m, 2m \u2212w, w)-domination mapping exists for the speci\ufb01ed ranges of m and w. We begin with the (3w, 5w, w) mapping constructed in Corollary 10. Note that the degree distribution of this mapping is (w, 2w) by Lemma 8. Thus for all i = 1, 2, . . . , 2w, we can invoke Lemma 4 to remove i vertices of degree 2, thereby producing a (3w \u2212i, 5w \u22122i, w)-domination mapping. We observe that optimal domination mappings that attain the bound of Lemma 7 also exist beyond the m = 3w threshold. For example, there exists an optimal (13, 22, 4)-domination mapping with degree distribution (4, 9). Taking this mapping as both \u03d51 and \u03d52 in the product construction, we obtain a (26, 44, 8)domination mapping with degree distribution (8, 18). This mapping is again optimal by Lemma 8. The product construction further implies the following immediate upper bound on \u03bd(m, w). Lemma 12. If \u03bd(m1, w1) \u2a7dn1 and \u03bd(m2, w2) \u2a7dn2, then \u03bd(m1 + m2, w1 + w2) \u2a7dn1 + n2. 7 \fUsing Lemma 12, we can \ufb01nd an upper bound on \u03bd(m, w) for all admissible parameters. We can write m = m1 + m2 and w = w1 + w2 and use the related upper bounds on \u03bd(m1, w1) and \u03bd(m2, w2) and apply Lemma 12. Hence, we can obtain recursively upper bounds on \u03bd(m, w) for any triple (m, n, w). 4.2. Domination mappings into the Hamming ball of small radius Perfect domination mappings for w = 1 are easily constructed. By Lemma 5, we have \u03bd(m, 1) \u2a7e2m \u22121 for w = 1. For a given m \u2a7e1, we will construct a perfect (m, 2m \u22121, 1)-domination mapping with degree sequence (\u03b41, \u03b42, . . . , \u03b4m), where \u03b4i = 2i for all i. It should be noted that we can shorten this domination mapping to obtain a perfect (m\u2032, 2m\u2032 \u22121, 1)-domination mapping for any m\u2032 < m. The (m, 2m \u22121, 1)domination mapping \u03d5 is quite simple; in fact, many such mappings exist. Clearly, \u03d5(0) = 0. Now, for an integer j \u2208[2m \u22121], let b(j) be the binary word of length m which forms the binary representation of j. We then de\ufb01ne \u03d5(b(j)) = y, where y is the binary word of length 2m \u22121 and weight one, with the single nonzero in its j-th position. The proof of correctness for this construction can be given by induction on m or a related argument. This is summarized with the following theorem. Theorem 13. There exists a perfect (m, 2m \u22121, 1)-domination mapping for all m \u2a7e1. Hence \u03bd(m, 1) = 2m \u22121 and \u00b5(n, 1) = \u2308log2(n + 1)\u2309. A construction of domination mappings for w = 2 is given in Appendix A.2. This construction attains the bound of Lemma 5 when m is odd. In fact, using these results and Section 6, we determine \u03bd(m, 2) for all m. 5. Polynomial-Time Algorithm In this section, we present a polynomial-time algorithm that, given any m, n, and w, determines whether an (m, n, w)-domination mapping exists for a domination graph G with an equitable degree distribution. To do so, we introduce a certain bipartite graph associated to G, which we call the compatibility graph. De\ufb01nition 2. Given a domination graph G = ([m] \u222a[n], E), the compatibility graph de\ufb01ned by G is the bipartite graph whose vertex set is {0, 1}m \u222aB(n, w). There is an edge from x \u2208{0, 1}m to y \u2208B(n, w) if and only if for (i, j) \u2208E, we have that xi = 0 implies yj = 0. Fix a domination graph G = ([m] \u222a[n], E). For brevity, we write S def = {0, 1}m and R def = B(n, w). Also, for x \u2208S and y \u2208R, we say that x dominates y if and only if for (i, j) \u2208E, we have that xi = 0 implies yj = 0. Hence, the compatibility graph de\ufb01ned by G has vertex set S \u222aR and x \u2208S is adjacent to y \u2208R if and only if x dominates y. The next theorem states that the existence of a G-domination mapping is equivalent to a certain graph theoretic property of the compatibility graph de\ufb01ned by G. Theorem 14. Let H be the compatibility graph de\ufb01ned by domination graph G = ([m] \u222a[n], E). There exists a G-domination mapping if and only if there exists a subgraph of H such the degree of each vertex in S is exactly one and the degree of each vertex in R is at most one. Proof. Let \u03d5 : S \u2192R be a G-dominating mapping. We de\ufb01ne the subgraph H\u2032 = (S \u222aR, E\u2032) where E\u2032 = {(x, \u03d5(x)) : x \u2208S}. Since x dominates \u03d5(x), H\u2032 is a subgraph of H. Clearly, the degree of every vertex in S is exactly one. Since \u03d5 is an injection, the degree of every vertex in R is at most one. Conversely, suppose that there exists such a subgraph H\u2032 = (S \u222aR, E\u2032). For each x \u2208S, de\ufb01ne \u03d5(x) to be the unique vertex in R adjacent to x. Then it is readily veri\ufb01ed that \u03d5 is a G-domination mapping. 8 \fTherefore, following Theorem 14, our task is reduced to determining the existence of a perfect matching in the bipartite compatibility graph de\ufb01ned by G. To this end, we recall the famous K\u00f6nig-Egev\u00e1ry theorem [7] for general graphs. Let G = (V, E) be a graph and we de\ufb01ne the following quantities via certain integer optimization problems. M(G) def = max ( \u2211 e\u2208E Xe : Xe \u2208{0, 1} for all e \u2208E, \u2211 e incident to v Xe \u2a7d1 for all v \u2208V ) , (8) C(G) def = min ( \u2211 v\u2208V Yv : Yv \u2208{0, 1} for all v \u2208V, \u2211 v incident to e Yv \u2a7e1 for all e \u2208E ) . (9) Then M(G) and C(G) correspond to the sizes of the maximum matching and minimum vertex cover, respectively. Via weak duality, we have that M(G) \u2a7dC(G). Next, we consider the relaxed or fractional versions of the optimization problems. M\u2217(G) def = max ( \u2211 e\u2208E Xe : 0 \u2a7dXe \u2a7d1 for all e \u2208E, \u2211 e incident to v Xe \u2a7d1 for all v \u2208V ) , (10) C\u2217(G) def = min ( \u2211 v\u2208V Yv : 0 \u2a7dYv \u2a7d1 for all v \u2208V, \u2211 v incident to e Yv \u2a7e1 for all e \u2208E ) . (11) We refer to M\u2217(G) and C\u2217(G) as the maximum fractional matching and minimum fractional vertex cover, respectively. Then we have the following inequality, M(G) \u2a7dM\u2217(G) \u2a7dC\u2217(G) \u2a7dC(G). For bipartite graphs, K\u00f6nig-Egev\u00e1ry theorem [7] states that M(G) = C(G). In other words, the maximum matching and the maximum fractional matching in a bipartite graph have the same size. Speci\ufb01cally, M(G) = M\u2217(G) = C\u2217(G) = C(G). 5.1. Maximum matching in compatibility graphs Recall that the vertex set in our compatibility graph is V = S \u222aR while the edge set is E = {(u, v) \u2208 S \u00d7 R : u dominates v}. Then the size of maximum matching is given by the following linear program max ( \u2211 e\u2208E Xe : AX \u2a7d1, X \u2a7e0 ) , (12) where A is a matrix whose rows are indexed by V and columns are indexed by E. Here, A(v, e) = 1 if v is incident to e, and A(v, e) = 0, otherwise. Example 1. Set m = 2, n = 4, and w = 1. Then R = {00, 01, 10, 11} and S = {0000, 0001, 0010, 0100, 1000}. Here, V = S \u222aR, and E = {ei : i \u2208[12]}, where e1 def = (00, 0000), e2 def = (01, 0000), e3 def = (01, 0001), e4 def = (01, 0010), e5 def = (10, 0000), e6 def = (10, 0100), e7 def = (10, 1000), e8 def = (11, 0000), e9 def = (11, 0001), e10 def = (11, 0010), e11 def = (11, 0100), e12 def = (11, 1000). 9 \fThen the matrix A is given by A = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . Set X = \u0012 1, 0, 1 2, 1 2, 0, 1 2, 1 2, 0, 1 4, 1 4, 1 4, 1 4 \u0013 . We check that X is indeed a feasible vector whose objective value is four. Since the size of a matching in the compatibility graph is at most four, we \ufb01nd that size of maximum matching in this graph is four. Unfortunately, the number of variables in the program (12) is given by |E| = \u0398(22m\u22121) when m \u2a7e2w. Indeed, for u \u2208{0, 1}m, when wt(u) \u2a7ew, u dominates all words in B(n, w). Since |R| = |B(n, w)| \u2a7e |{0, 1}m| = 2m, we have that |E| \u2a7e(\u2211m j=w (m j ))|R| \u2a7e2m\u22121 \u00b7 2m = 22m\u22121. In other words, to determine the existence of a mapping by solving this linear programme requires time exponential in m. In what follows, we reduce the running time to polynomial in m and w. In do so, we notice that many entries in the optimal solution of Example 1 are identical. This hints that we may reduce the number of variables in the program and we do so by exploiting certain symmetries of the linear program. 5.2. Symmetries of the linear program Let M and N be integers and let A be an M \u00d7 N matrix. Consider a linear program of the form max ( \u2211 i\u2208[N] xi : Ax \u2a7d1, x \u2a7e0 ) . Let \u03c0 : [N] \u2192[N] be a permutation on the set of N variables. For any vector x \u2208RN, let x\u03c0 denote the vector whose ith component is given by x\u03c0(i). Let P\u03c0 denote the binary matrix that represents the permutation \u03c0 and hence, by de\ufb01nition, we have that P\u03c0x = x\u03c0. A permutation on the N variables can be regarded as a permutation on the N columns of A. We also consider a permutation \u03c0row on the M rows of A and similarly let P\u03c0row denote the binary matrix that represents the permutation \u03c0row. De\ufb01nition 3. A permutation \u03c0 : [N] \u2192[N] is A-preserving if P\u03c0row AP\u03c0 = A for some permutation \u03c0row : [M] \u2192[M]. This de\ufb01nition of A-preserving permutations can be found in linear programming literature (see Margot [9], and B\u02dd odi and Herr [1]), where the authors exploit symmetries to reduce the dimension of their linear programs. In particular, B\u02dd odi and Herr demonstrated Proposition 15 in a more general setting. Independently, Fazelli et al. obtained Proposition 15 in the specialized setting of \ufb01nding a fractional transversal in hypergraphs. For completeness, we rederive Proposition 15 in Appendix B. 10 \fLet GA denote a subgroup of the group of all A-preserving permutations and let GA act on [N]. Suppose that the collection of orbits under this action is O = {O1, O2, . . . , On}. A vector x is de\ufb01ned to be Oregular if for all k \u2a7dN, xi = xj for all i, j \u2208Ok. Proposition 15. Suppose that Ax \u2a7d1 and \u2211N i=1 xi = \u03bb. Then there exists an O-regular vector x\u2217such that Ax\u03c0 \u2a7d1 and \u2211N i=1 x\u2217 i = \u03bb. Therefore, applying Proposition 15, we establish the following equality. max ( \u2211 i\u2208[N] xi : Ax \u2a7d1, x \u2a7e0 ) = max ( \u2211 i\u2208[N] xi : Ax \u2a7d1, x \u2a7e0, x is O-regular ) . In other words, we reduce the number of variables from N to n. We may then rewrite the linear program as follows: max ( \u2211 i\u2208[n] |Oi|x\u2217 i : A\u2217x\u2217\u2a7d1, x\u2217\u2a7e0 ) , where A\u2217is an M \u00d7 n matrix de\ufb01ned by A\u2217(i, k) = \u2211 j\u2208Ok A(i, j) for all i \u2208[M], k \u2208[n]. 5.3. Reducing the dimension of the linear program de\ufb01ned by (12) To exploit the symmetries of our linear program, we consider a domination graph with an equitable degree distribution. Speci\ufb01cally, for given values of m and n, we set \u03b4 = \u230an/m\u230b, and m1 = n mod m, m2 = m \u2212m1, n1 = m1(\u03b4 + 1), n2 = m2\u03b4. We partition [m] into I1 def = [m1] and I2 def = [m1 + 1, m] (here, [i, j] denotes the set {i, i + 1, . . . , j}) and assign the vertices in I1 and I2 the degrees \u03b4 + 1 and \u03b4, respectively. Hence, in this domination graph, there are m1 vertices of degree \u03b4 + 1 and m2 vertices of degree \u03b4. We also partition [n] into m groups: Ji def = ( [(i \u22121)(\u03b4 + 1) + 1, i(\u03b4 + 1)], if i \u2208[m1], [n1 + (i \u2212m1 \u22121)\u03b4 + 1, n1 + (i \u2212m1)\u03b4], otherwise. In other words, we partition [n1] into m1 groups of size \u03b4 + 1 and [n1 + 1, n] into m2 groups of size \u03b4. Next, we de\ufb01ne a group of A-preserving permutations. Here, let SX denote the set of permutations on the set X and we \ufb01rst produce permutations in SS and SR. Consider a permutation \u03b3 \u2208SI1 \u00d7 SI2. Since I1 and I2 partition [m], the permutation \u03b3 belongs to S[m] and we de\ufb01ne \u03c0S \u03b3 \u2208SS such that \u03c0S \u03b3 (u) = u\u03b3 for u \u2208S. We then consider the subset B = {\u03b2 \u2208SR : \u03b2(v)|Ji \u0338= 0 if and only if v|Ji \u0338= 0 for all i \u2208[m], v \u2208S}. It can be veri\ufb01ed that B is a subgroup of SR. Given \u03b3 \u2208SI1 \u00d7 SI2 and \u03b2 \u2208B, we de\ufb01ne \u03c0R \u03b3,\u03b2 \u2208SR to be the permutation such that \u03c0R \u03b3,\u03b2(v) is the word obtained rearranging the m subwords in \u03b2(v) in accordance to \u03b3. Finally, we obtain a permutation \u03c0\u03b3,\u03b2 on S \u00d7 R by simply setting \u03c0\u03b3,\u03b2(u, v) = \u0010 \u03c0S \u03b3 (u), \u03c0R \u03b3,\u03b2(v) \u0011 . Set G = {\u03c0\u03b3,\u03b2 : \u03b3 \u2208SI1 \u00d7 SI2, \u03b2 \u2208B}. Abusing notation, we simply write G = (SI1 \u00d7 SI2) \u00d7 B. 11 \fExample 2. (i) (Example 1 continued.) Consider m = 2, n = 4, and w = 1. Then \u03b4 = 2, m1 = n1 = 0, m2 = 2 and n2 = 4. Label the words in R as follow: v0 = 0000, v1 = 0001, v2 = 0010, v3 = 0100, and v4 = 1000. So, B = {(v0, v1, v2, v3, v4), (v0, v2, v1, v3, v4), (v0, v1, v2, v4, v3), (v0, v2, v1, v4, v3)}. Since m1 = 0, G = S[2] \u00d7 B. Let \u03b3 = (2, 1) and \u03b2 = (v0, v2, v1, v3, v4). Then \u03c0S \u03b3 (01) = 10, \u03c0R \u03b3,\u03b2(0001) = 1000 and so, \u03c0\u03b3,\u03b2(01, 0001) = (10, 1000). Consider the orbit of (01, 0001) which is {\u03c0(01, 0001) : \u03c0 \u2208G}. The orbit is given by {(01, 0001), (01, 0010), (10, 0100), (10, 1000)}. (ii) Consider m = 3, n = 4, and w = 2. Then \u03b4 = 1, m1 = 1, m2 = 2, n1 = 2 and n2 = 2. The orbit of (101, 0101) is given by {(101, 0101), (101, 1001), (110, 0110), (110, 1010)}, while the orbit of (100, 0100) is given by {(100, 0100), (100, 1000)}. Recall that the columns of A are indexed by E \u2286S \u00d7 R, while G is a subgroup of SS\u00d7R. In the next lemma, we show that G may be regarded as a subgroup of SE. We do this by showing that the image of E under the permutation \u03c0\u03b3,\u03b2 remains as E. Lemma 16. Let \u03c0\u03b3,\u03b2 \u2208G, u \u2208S and v \u2208R. Then u dominates v if and only if \u03c0S \u03b3 (u) dominates \u03c0R \u03b3,\u03b2(v). Proof. Since G is a group, it suf\ufb01ces to prove in one direction. Let supp\u03b4(v) def = {i \u2208[m] : v|Ji \u0338= 0}. Since u dominates v, we have that supp(u) \u2287supp\u03b4(v). Since \u03b2 \u2208B, we have that supp\u03b4(v) = supp\u03b4(\u03b2(v)). From the de\ufb01nition of \u03c0S \u03b3 and \u03c0R \u03b3,\u03b2, we have that supp(\u03c0S \u03b3 (u)) = \u03b3(supp(u)) and supp\u03b4(\u03c0R \u03b3,\u03b2(v)) = \u03b3(supp\u03b4(\u03b2(v))) = \u03b3(supp\u03b4(v)). Therefore, supp(\u03c0S \u03b3 (u)) \u2287supp\u03b4(\u03c0R \u03b3,\u03b2(v)) and so, \u03c0S \u03b3 (u) dominates \u03c0R \u03b3,\u03b2(v). Therefore, since E = {(u, v) \u2208S \u00d7 R : u dominates v}, we have that G is a subgroup of SE. Lemma 17. Every permutation in G is a A-preserving permutation. Proof. Let \u03c0 = \u03c0\u03b3,\u03b2 \u2208G. To show that \u03c0 is A-preserving, it suf\ufb01ces to provide a permutation \u03c0row on the rows of A such that P\u03c0rowAP\u03c0 = A, or AP\u03c0 = P\u22121 \u03c0rowA = P\u03c0\u22121 rowA. (13) Let (u, v) \u2208E and z \u2208S \u222aR. Let the entry of A at row z and column (u, v), or the (z, (u, v))th entry of A, be written as A(z, (u, v)). Then the (z, (u, v))th entry of AP\u03c0 is given by A(z, \u03c0\u22121(u, v)). We next consider the permutations on the rows. Note that \u03c0S \u03b3 \u2208SS and \u03c0R \u03b3,\u03b2 \u2208SR. So, let G act on S \u222aR by setting \u03c0(z) = \u03c0S \u03b3 (z) when z \u2208S and \u03c0(z) = \u03c0R \u03b3,\u03b2(z) when z \u2208R. Set \u03c0row = \u03c0\u22121, the inverse of \u03c0 in G. Then the (z, (u, v))th entry of P\u03c0\u22121 rowA is given by A(\u03c0\u22121 row(z), (u, v)) = A(\u03c0(z), (u, v)). Finally, to establish (13) we show that A(z, \u03c0\u22121(u, v)) = A(\u03c0(z), (u, v)). This then follows from the fact that A(z, (u, v)) = ( 1, if z = u or z = v, 0, otherwise. 12 \fWe next study the orbits of S \u00d7 R under this group action of G. To this end, we de\ufb01ne the (m1, m2)weights and the (n1, n2; \u03b4)-weights of words in S and R, respectively. For u \u2208S and v \u2208R, de\ufb01ne (m1, m2)wt(u) = (\u03c31, \u03c32), where \u03c3i = wt(u|Ii) for i \u2208[2], (n1, n2; \u03b4)wt(v) = (\u03c11, \u03c12), where \u03c1i = |{j \u2208Ij : v|Jj \u0338= 0}| for i \u2208[2]. Using these weights, we then characterise the orbits of S \u00d7 R under this group action of G. Lemma 18. Let (u, v), (u\u2032, v\u2032) \u2208E. (u, v) and (u\u2032, v\u2032) belong to the same orbit if and only if (m1, m2)wt(u) = (m1, m2)wt(u\u2032) and (n1, n2; \u03b4)wt(v) = (n1, n2; \u03b4)wt(v\u2032). Proof. If (u, v) and (u\u2032, v\u2032) belong to the same orbit, then \u03c0\u03b3,\u03b2(u, v) = (u\u2032, v\u2032) for some \u03c0\u03b3,\u03b2 \u2208G. In other words, \u03c0S \u03b3 (u) = u\u2032 and \u03c0R \u03b3,\u03b2(v) = v\u2032. Since \u03b3 \u2208SI1 \u00d7 SI2, we have wt(u|Ii) = wt(u\u2032|Ii) for i \u2208[2], and so, (m1, m2)wt(u) = (m1, m2)wt(u\u2032). Similarly, since \u03b2 \u2208B, we have |{j \u2208Ij : v|Jj \u0338= 0}| = |{j \u2208Ij : v\u2032|Jj \u0338= 0}| and hence, (n1, n2; \u03b4)wt(v) = (n1, n2; \u03b4)wt(v\u2032). Conversely, suppose that (m1, m2)wt(u) = (m1, m2)wt(u\u2032) = (\u03c31, \u03c32) and (n1, n2; \u03b4)wt(v) = (n1, n2; \u03b4)wt(v\u2032) = (\u03c11, \u03c12). Consider the words u\u2217= m1 z }| { 00 \u00b7 \u00b7 \u00b7 0 11 \u00b7 \u00b7 \u00b7 1 | {z } \u03c31 m2 z }| { 00 \u00b7 \u00b7 \u00b7 0 11 \u00b7 \u00b7 \u00b7 1 | {z } \u03c32 , v\u2217= \u03b4+1 z }| { 0 \u00b7 \u00b7 \u00b7 0 \u00b7 \u00b7 \u00b7 \u03b4+1 z }| { 0 \u00b7 \u00b7 \u00b7 0 | {z } m1\u2212\u03c11 \u03b4+1 z }| { 0 \u00b7 \u00b7 \u00b7 01 \u00b7 \u00b7 \u00b7 \u03b4+1 z }| { 0 \u00b7 \u00b7 \u00b7 01 | {z } \u03c11 \u03b4 z }| { 0 \u00b7 \u00b7 \u00b7 0 \u00b7 \u00b7 \u00b7 \u03b4 z }| { 0 \u00b7 \u00b7 \u00b7 0 | {z } m2\u2212\u03c12 \u03b4 z }| { 0 \u00b7 \u00b7 \u00b7 01 \u00b7 \u00b7 \u00b7 \u03b4 z }| { 0 \u00b7 \u00b7 \u00b7 01 | {z } \u03c12 . Then (m1, m2)wt(u\u2217) = (\u03c31, \u03c32) and (n1, n2; \u03b4)wt(v\u2217) = (\u03c11, \u03c12). We \ufb01nd a permutation in G that maps (u\u2217, v\u2217) to (u, v). Let \u03b3\u22121 be a permutation in \u03b3 \u2208SI1 \u00d7 SI1 that rearranges the coordinates the m coordinates of u and the m subwords of v such that (u\u03b3\u22121, \u02c6 v) is the lexicographical smallest amongst all permutations. Let \u03b2 \u2208B be a permutation that maps v\u2217to \u02c6 v. Then \u03c0\u03b3,\u03b2(u\u2217, v\u2217) = (u, v). Since G is a subgroup of SE, we can then \ufb01nd a permutation that maps (u, v) to (u\u2032, v\u2032). Following Lemma 18, we can then index each orbit with the quadruple (\u03c31, \u03c32, \u03c11, \u03c12), where (\u03c31, \u03c32) = (m1, m2)wt(u) and (\u03c11, \u03c12) = (n1, n2; \u03b4)wt(v) for some (u, v) in the orbit. In particular, the index set is given by \u2126def = {(\u03c31, \u03c32, \u03c11, \u03c12) : 0 \u2a7d\u03c3i \u2a7dmi for i \u2208[2], 0 \u2a7d\u03c11 \u2a7dmin{\u03c31, w}, 0 \u2a7d\u03c12 \u2a7dmin{\u03c32, w \u2212\u03c11}}. Hence, the number of variables is reduced to O(m2w2). Besides reducing the number of variables, the group action also identi\ufb01es certain constraints. In particular, the 2m + \u2211w j=0 (n j) constraints are reduced to O(m2 + w2) constraints. Then by the equivalence of the linear programs, we are able to determine the existence of the desired mapping in time polynomial in m and w. Next, we compute the size of the orbits and then state the reduced linear program. Let 0 \u2a7d\u03c11, \u03c12 \u2a7dw and 0 \u2a7d\u03c11 + \u03c12 \u2a7dw and de\ufb01ne the quantity C\u03c11\u03c12 def = \u2211 w(1) 1 +\u00b7\u00b7\u00b7+w(1) \u03c11 +w(2) 1 +\u00b7\u00b7\u00b7+w(2) \u03c12 \u2a7dw w(i) j \u2a7e1 \u03c11 \u220f j=1 \u0012\u03b4 + 1 w(1) j \u0013 \u03c12 \u220f j=1 \u0012 \u03b4 w(2) j \u0013 . Here, C\u03c11\u03c12 computes the number of words v in R such that {j \u2208I1 : v|Jj \u0338= 0} = L1 and {j \u2208I2 : v|Jj \u0338= 0} = L2 for some \u03c11-subset L1 of I1 and \u03c12-subset L2 of I2. 13 \fLemma 19. Fix (\u03c31, \u03c32, \u03c11, \u03c12) \u2208\u2126. (i) The number of pairs (u, v) in E with (m1, m2)wt(u) = (\u03c31, \u03c32) and (n1, n2; \u03b4)wt(v) = (\u03c11, \u03c12) is (m1 \u03c31 )(m2 \u03c32 )(\u03c31 \u03c11)(\u03c32 \u03c12)C\u03c11\u03c12. (ii) Fix u \u2208S with (m1, m2)wt(u) = (\u03c31, \u03c32). The number of words v \u2208R that are dominated by u with (n1, n2; \u03b4)wt(v) = (\u03c11, \u03c12) is (\u03c31 \u03c11)(\u03c32 \u03c12)C\u03c11\u03c12. (iii) Fix v \u2208R with (n1, n2; \u03b4)wt(v) = (\u03c31, \u03c32). The number of words u \u2208S that dominates v with (m1, m2)wt(u) = (\u03c31, \u03c32) is (m1\u2212\u03c11 \u03c31\u2212\u03c11 )(m2\u2212\u03c32 \u03c32\u2212\u03c12 ). Proof. For (u, v) \u2208E, we set Ki to be the support of u|Ii and Li = {j \u2208Ii : v|Jj \u0338= 0} for i \u2208[2]. Note that if (m1, m2)wt(u) = (\u03c31, \u03c32) and (n1, n2; \u03b4)wt(v) = (\u03c11, \u03c12), then |Ki| = \u03c3i and |Li| = \u03c1i for i \u2208[2]. (i) There are (m1 \u03c31 )(m2 \u03c32 ) to choose words u with (m1, m2)wt(u) = (\u03c31, \u03c32). Since u dominates v, we have (\u03c31 \u03c11)(\u03c32 \u03c12) choices for L1 and L2. For \ufb01xed L1 and L2, there are C\u03c11\u03c12 choices for v. Therefore, the total number of pairs is (s1 \u03c31)(s2 \u03c32)(\u03c31 \u03c11)(\u03c32 \u03c12)C\u03c11\u03c12. (ii) When K1 and K2 are \ufb01xed, the previous argument demonstrates that there are (\u03c31 \u03c11)(\u03c32 \u03c12)C\u03c11\u03c12 choices for v. (iii) When L1 and L2 are \ufb01xed, there are (m1\u2212\u03c11 \u03c31\u2212\u03c11 ) and (m2\u2212\u03c12 \u03c32\u2212\u03c12 ) choices for K1 and K2, respectively. Therefore, the desired number is (s1\u2212\u03c11 \u03c31\u2212\u03c11)(s2\u2212\u03c32 \u03c32\u2212\u03c12). Finally, we state the reduced linear program. max \u2211 (\u03c31,\u03c32,\u03c11,\u03c12)\u2208\u2126 \u0012m1 \u03c31 \u0013\u0012m2 \u03c32 \u0013\u0012\u03c31 \u03c11 \u0013\u0012\u03c32 \u03c12 \u0013 C\u03c11\u03c12X\u03c31\u03c32\u03c11\u03c12 (14) subject to the following constraints. (I) S-side constraints: min{\u03c31,w} \u2211 \u03c11=0 min{\u03c32,w\u2212\u03c11} \u2211 \u03c12=0 \u0012\u03c31 \u03c11 \u0013\u0012\u03c32 \u03c12 \u0013 C\u03c11\u03c12X\u03c31\u03c32\u03c11\u03c12 \u2a7d1 for all 0 \u2a7d\u03c31 \u2a7dm1, 0 \u2a7d\u03c32 \u2a7dm2. (II) R-side constraints: m1 \u2211 \u03c31=\u03c11 m2 \u2211 \u03c32=\u03c12 \u0012m1 \u2212\u03c11 \u03c31 \u2212\u03c11 \u0013\u0012m2 \u2212\u03c12 \u03c32 \u2212\u03c12 \u0013 X\u03c31\u03c32\u03c11\u03c12 \u2a7d1 for all 0 \u2a7d\u03c11 \u2a7dm1, 0 \u2a7d\u03c12 \u2a7dm2, 0 \u2a7d\u03c11 + \u03c12 \u2a7dw. (III) Variable constraints: 0 \u2a7dX\u03c31\u03c32\u03c11\u03c12 \u2a7d1 for all (\u03c31, \u03c32, \u03c11, \u03c12) \u2208\u2126. There exists a mapping if and only if the objective value achieves 2m. Furthermore, we know that the objective value attains 2m if and only if the S-side constraints are active, or met with equality. Example 3. 14 \f(i) (Example 1 continued.) Consider m = 2, n = 4, and w = 1. Then \u03b4 = 2, m1 = n1 = 0, m2 = 2, n2 = 4. Hence, E is partitioned into the orbits O = {O(0,0) = {e1}, O(1,0) = {e2, e5}, O(1,1) = {e3, e4, e6, e7}, O(2,0) = {e8}, O(2,1) = {e9, e10, e11, e12}}, with \u2126= {(0, 0), (1, 0), (1, 1), (2, 0), (2, 1)}. Then A\u2217is given by the 5 \u00d7 5 matrix A\u2217= \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 0 0 1 2 0 0 0 0 0 1 4 1 2 0 1 0 0 0 1 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . (ii) Consider m = 3, n = 4, and w = 2. Then \u03b4 = 1, m1 = 1, m2 = 2, n1 = 2 and n2 = 2. Here, \u2126= {(0, 0, 0, 0), (0, 1, 0, 1), (0, 1, 0, 1), (0, 2, 0, 0), (0, 2, 0, 1), (0, 2, 0, 2), (1, 0, 0, 0), (1, 0, 1, 0), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1), (1, 2, 0, 0), (1, 2, 0, 1), (1, 2, 0, 2), (1, 2, 1, 0), (1, 2, 1, 1)}. Then A\u2217is given by the 11 \u00d7 17 matrix A\u2217= \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 3 4 1 2 0 1 0 0 1 0 2 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . We summarize the results in this section with the following theorem. Theorem 20. Let G = ([m] \u222a[n], E) be an equitable domination graph. The linear program de\ufb01ned by (14) is equivalent to (12) and has O(m2w2) variables and O(m2 + w2) constraints. Therefore, we can determine the existence of an G-domination mapping in time polynomial in m and w. 6. Existence Proof In this section, we prove Theorem 1. In particular, we show that for suf\ufb01ciently large m, an (m, n, w)domination mapping exists whenever |B(n, w)| \u2a7e2m. From Theorem 14, this is equivalent to \ufb01nding a perfect matching in the associated compatibility graph. To this end, we invoke the celebrated Hall\u2019s marriage theorem [5]. 15 \fTheorem 21. Consider a bipartite graph with V1 and V2 as its left and right vertices, and |V1| \u2a7d|V2|. There is a matching of size |V1| (i.e. a perfect matching) if and only if for each subset X of V1, the number of vertices in the neighborhood of X has at least |X| vertices. In what follows, we study the suffciency condition of Theorem 21 for our compatibility graph. In particular, a set of vertices X \u2282S is called a bad set if its neighbourhood N(X) in R has size less than |X|. In other words, |X| > |N(X)|. We examine properties of bad sets and \ufb01nd compatibility graphs that contain no bad sets. Therefore, in such compatibility graphs, the condition of Theorem 21 is satis\ufb01ed and hence the corresponding (m, n, w)-domination mapping exists. For our exposition, we de\ufb01ne certain quantities and properties for sets of vertices in S. First, we de\ufb01ne the notion of descendant-closed. De\ufb01nition 4. Let u, v \u2208S. Recall that v \u227au if supp(v) \u2286supp(u) and call v a descendant of u. A set X is descendant-closed or d-closed if for all u \u2208X, v \u227au implies that v \u2208X. Given any bad set, we are always able to construct a (possibly different) d-closed bad set with the same cardinality. Lemma 22 (Closure Lemma). If X is a bad set, then there exists a d-closed bad set X\u2032 for which |X| = |X\u2032|. Proof. Suppose X is not d-closed. Then there exists u \u2208X and v \u2208S such that v \u227au but v / \u2208X. Set X(1) = X \u222a{v} \\ {u}. Then |X(1)| = |X| and |N(X(1))| \u2a7d|N(X)| < |X| = |X(1)|. Hence, X(1) is a bad set with the same size as X. If X(1) is not d-closed, then we similarly construct X(2) such that |X(2)| = |X| and X(2) is bad. Since this process is \ufb01nite, we eventually obtain a d-closed set X(m) such that |X(m)| = |X| and X(m) is bad. Next, we de\ufb01ne the notion of balanced sets. De\ufb01nition 5. Let i \u2208[m]. We say that X is i-balanced if the following holds. If ab \u2208X with |a| = i, then a\u2032b \u2208X for all a\u2032 \u2208{0, 1}i. Observe that if X is i-balanced, then X is j-balanced for all j \u2a7di. As a convention, we say that all sets X are 0-balanced. Finally, we look at certain collection of bad sets that are both d-closed and balanced. De\ufb01nition 6. Let 0 \u2a7di \u2a7ds. Let Xi be the collection of all bad sets that are d-closed and i-balanced. Our proof on the non-existence of bad sets follows an induction strategy. (I) Assume that there exists a bad set X0. (II) Base case. Using the closure lemma, we may modify X0 so that X\u2032 0 is both d-closed and 0-balanced. This implies that X0 is nonempty. (III) Induction step. Let 1 \u2a7di \u2a7dm. By induction hypothesis, we have that Xi\u22121 is nonempty. Pick X \u2208Xi\u22121 with the smallest cardinality. We then construct a set X\u2032 such that X\u2032 \u2208Xi. This implies that Xi is nonempty. (IV) Therefore, Xm is nonempty and X \u2208Xm implies that X = S. This contradicts the fact that |S| \u2a7d |R| = |N(S)|. 16 \fIn what follows, we focus on the induction step. We \ufb01x X \u2208Xi\u22121 and augment X so that the resulting X\u2032 is in Xi. To ensure that X\u2032 is bad, we need to compute the increase in the size of the neighbourhood. Formally, we have the following de\ufb01nition. De\ufb01nition 7. Let U, V \u2286S. Set \u039eU(V) def = |N(U \u222aV)| \u2212|N(U)|. In other words, \u039eU(V) is the number of additional neighbours that V adds when adjoined to U. When X is the smallest set in Xi\u22121, we have the following property of \u039eX\\Y(Y) for certain subset Y \u2286X. Lemma 23 (Removal Lemma). Let X be the smallest set in Xi. Let Y \u2286X such that X \\ Y is both d-closed and i-balanced. Then |Y| > \u039eX\\Y(Y). Proof. Since X is a bad set, ww have that |X| > |N(X)|. By the minimality of X, we have that X \\ Y is not a bad set, because X \\ Y is both d-closed and i-balanced and |X \\ Y| < |X|. Therefore, |X \\ Y| \u2a7d |N(X \\ Y)|. Hence, |Y| = |X| \u2212|X \\ Y| > |N(X)| \u2212|N(X \\ Y)| = \u039eX\\Y(Y). Next, we describe how we augment X. De\ufb01ne Y = {u \u2208X : ui = 0, u + ei / \u2208X}, and Z = Y + ei. We have the following lemmas on X \u222aZ and X \\ Y. Lemma 24. If X is d-closed and (i \u22121)-balanced, then X \u222aZ is both d-closed and i-balanced. Proof. First, we show that X \u222aZ is d-closed. Let u \u2208X \u222aZ and v \u227au. If u \u2208X, then v \u2208X \u2286X \u222aZ since X is d-closed. Otherwise, u \u2208Z and hence, u = u\u2032 + ei with u\u2032 \u2208X. If vi = 0, then v \u227au\u2032 and so, v \u2208X since X is d-closed. Hence, v \u2208X \u222aZ. If vi = 1, then v \u2212ei \u227au\u2032 and so, v \u2212ei \u2208X. Hence, v = (v \u2212ei) + ei \u2208Z \u2286X \u222aZ. Thus, X \u222aZ is d-closed. Next, we show that X \u222aZ is i-balanced. Let ab \u2208X \u222aZ with |a| = i and let c \u2208{0, 1}i. Let a\u2032 and c\u2032 be the \ufb01rst i \u22121 symbols of a and c, respectively. So, a = a\u2032ai and c = c\u2032ci. Since X \u222aZ is d-closed, it suf\ufb01ces to consider the case, ci = 1. In other words, we want to show that c\u20321b \u2208X \u222aZ. We distinguish between two cases: (a) a\u20321b \u2208X. Since X is (i \u22121)-balanced, we have c\u20321b \u2208X \u2286X \u222aZ. (b) a\u20321b / \u2208X. This implies that a\u20320b \u2208X by the de\ufb01nition of Y and Z. Again, since X is (i \u22121)-balanced, we have c\u20320b \u2208X and so, c\u20321b = c\u20320b + ei \u2208X \u222aZ. Thus, X \u222aZ is i-balanced. Lemma 25. If X is d-closed and (i \u22121)-balanced, then X \\ Y is both d-closed and (i \u22121)-balanced. Proof. First, we show that X \\ Y is d-closed. Let u \u2208X \\ Y and v \u227au. Since u / \u2208Y, either ui = 1 or u + ei \u2208X. Hence, set u\u2032 = u if ui = 1 and u\u2032 = u + ei otherwise. If vi = 0, then v + ei \u227au\u2032 \u2208X, and so, v + ei \u2208X. Hence, v / \u2208Y. If vi = 1, then v / \u2208Y by de\ufb01nition. In both cases, v / \u2208Y. Since X is d-closed, v \u2208X and therefore, v \u2208X \\ Y. Thus, X \\ Y is d-closed. Next, we show that X \\ Y is (i \u22121)-balanced. Let ab \u2208X \\ Y with |a| = i \u22121 and let c \u2208{0, 1}i\u22121. Since X is (i \u22121)-balanced, we have that cb \u2208X. When b1 = 0, we have by de\ufb01nition of Y that ab + ei \u2208 X since ab / \u2208Y. Since X is (i \u22121)-balanced and ab + ei \u2208X, we have that cb + ei \u2208X and so, cb / \u2208Y. When b1 = 1, we have that cb / \u2208Y by de\ufb01nition. Therefore, since cb \u2208X, it follows that cb \u2208X \\ Y. Thus, X \\ Y is (i \u22121)-balanced. 17 \fFinally, we have the following lemma that reduces the induction step to demonstrating inequality (15). Lemma 26. If X is an d-closed, (i \u22121)-balanced and bad set, and \u039eX(Z) \u2a7d\u039eX\\Y(Y), (15) then X \u222aZ is a d-closed, i-balanced and bad set. In other words, X \u222aZ \u2208Xi. Proof. By Lemma 24, it remains to show that X \u222aZ is a bad set. Since X \\ Y is both d-closed and (i \u22121)-balanced by Lemma 25, we apply the removal lemma to have that |Y| > \u039eX\\Y(Y). Futhermore, we have that |X| + |Z| = |X \u222aZ| and |Z| = |Y|. Combining these inequalities and equalities, we have that |X \u222aZ| = |X| + |Z| = |X| + |Y| > |N(X)| + \u039eX\\Y(Y) \u2a7e|N(X)| + \u039eX(Z) = |N(X \u222aZ)|. Thus, X \u222aZ is a bad set. In view of Lemma 26, our goal is to show that (15) holds. To this end, let us \ufb01rst introduce more notation. De\ufb01nition 8. For any vector v \u2208S, we let \u03a8(v) denote the additional neighbourhood of v, i.e. \u03a8(v) def = \f \f{u \u2208N(v) : u / \u2208N(v\u2032) for all proper descendant v\u2032 \u227av} \f \f. For a set V \u2286S, de\ufb01ne \u03a8(V) = \u2211v\u2208V \u03a8(v). The de\ufb01nition of \u03a8 is useful for computing the size of certain neighbourhoods. Lemma 27. For any d-closed set, we have that |N(V)| = \u03a8(V). Proof. Suppose V = {v1, v2, . . . , vm}, where wt(vj) \u2a7dwt(vj+1) for j = 1, 2, . . . , m \u22121. That is, we have ordered the vectors in V by weight. Let Vj def = {v1, v2, . . . , vj} for j = 1, 2, . . . , m with the convention that V0 = \u2205. Then |N(V)| = m \u2211 j=1 \u039eVj\u22121(Vj) = m \u2211 j=1 \u03a8(vj) = \u03a8(V). \u039eVj\u22121(Vj) = \u03a8(vj) follows from the fact that any proper descendant of vj is contained in Vj\u22121. Lemma 28. Suppose a subset U \u222aV of S is d-closed. Then \u039eU(V) = \u03a8(V), provided that the sets U and V are disjoint, and U does not contain an ancestor of any vector in V. Hence, if U \u222aV and U are d-closed and U and V are disjoint, then \u039eU(V) = \u03a8(V). Proof. Similar to Lemma 27, we adjoin the vectors in V to U in the order of their weight and de\ufb01ne V0 = U by convention. Then \u039eU(V) = m \u2211 j=1 \u039eVj\u22121(Vj) = m \u2211 j=1 \u03a8(vj) = \u03a8(V). as before. To see that \u039eVj\u22121(Vj) = \u03a8(vj), we note the following. First, when vj is adjoined to Vj\u22121, all of its descendants of vj are already in Vj\u22121 since U \u222aV is d-closed. Moreover, vj is not in Vj\u22121 and none of its ancestors are in Vj\u22121. 18 \fLemma 28 then allows us to compute the \u039eX\\Y(Y) and \u039eX(Z) by simply evaluating \u03a8. Corollary 29. \u039eX\\Y(Y) = \u03a8(Y), and \u039eX(Z) = \u03a8(Z). Combining Lemma 26 and Corollary 29, to complete the induction step, it is suf\ufb01cient to show that \u03a8(Z) \u2a7d\u03a8(Y). (16) 6.1. Equitable domination graphs We complete the proof of Theorem 1. To this end, we consider an equitable domination graph. If we set \u03b4 = \u230an/m\u230b, then the left vertices in the domination graph have degrees \u03b4 and \u03b4 + 1. We have the following characterization of \u03a8(v) in terms of its weight. Lemma 30. For v \u2208{0, 1}m, let the weight of v be \u2113. Then for suf\ufb01ciently large m (which implies suf\ufb01ciently large \u03b4), there exist constants c1, c2, . . . , cw such that \u03a8(v) = ( c\u2113\u03b4w + O(\u03b4w\u22121), if \u2113\u2a7dw, 0, otherwise. Furthermore, when wt(v) = w, we have that \u03a8(v) = \u03b4j(\u03b4 + 1)w\u2212j \u2a7e\u03b4w for some 0 \u2a7dj \u2a7dw . Proof. It suf\ufb01ces to consider the case where \u2113\u2a7dw. Let j be the number of vertices in supp(v) with degree \u03b4. Then the number of vertices in supp(v) with degree \u03b4 + 1 is \u2113\u2212j and we have that \u03a8(v) = \u2211 i1+i2+\u00b7\u00b7\u00b7+i\u2113\u2a7dw i1,i2,...,i\u2113>0 \u0012\u03b4 i1 \u0013\u0012\u03b4 i2 \u0013 \u00b7 \u00b7 \u00b7 \u0012\u03b4 ij \u0013\u0012\u03b4 + 1 ij+1 \u0013\u0012\u03b4 + 1 ij+2 \u0013\u0012\u03b4 + 1 i\u2113 \u0013 = \u2211 i1+i2+\u00b7\u00b7\u00b7+i\u2113=w i1,i2,...,i\u2113>0 \u0010 \u03bbi1,i2,...,i\u2113\u03b4w + O(\u03b4w\u22121) \u0011 + O(\u03b4w\u22121) = c\u2113\u03b4w + O(\u03b4w\u22121), where c\u2113= \u2211 i1+i2+\u00b7\u00b7\u00b7+i\u2113=w i1,i2,...,i\u2113>0 \u03bbi1,i2,...,i\u2113. When \u2113= w, the only index for the summand is i1 = i2 = \u00b7 \u00b7 \u00b7 = i\u2113= 1 and hence, \u03a8(v) = \u03b4j(\u03b4 + 1)w\u2212j \u2a7e\u03b4w. Thus, the proof is completed. In addition, we introduce the notion of maximal-support words. De\ufb01nition 9. Given V \u2286S, we say that v \u2208V is maximal-support in V (or a maximal-support word of V) if V does not contain a proper ancestor of v. Recall that X is the smallest set in Xi\u22121. We have the following technical lemmas. Lemma 31. Any maximal-support word in X is of the form 1b, where 1 is the all-ones vector of length i \u22121. Lemma 32. A maximal-support word in X has weight at least w + 1. 19 \fProof. It suf\ufb01ces to consider the case i \u22121 \u2a7dw. We prove by contradiction and assume that u = 1b is a maximal-support word of weight at most w. and where 1 is the all-ones vector of length i \u22121. Consider the set U = {ab : a \u2208{0, 1}i\u22121}. We claim that X \\ U is both (i \u22121)-balanced and d-closed. Since both U and X are (i \u22121)-balanced, it follows that X \\ U is (i \u22121)-balanced. On the other hand, consider any word ab in U and suppose a\u2032b\u2032 \u2208X is an ancestor of ab. In other words, b \u227ab\u2032. Since 1b is a maximal-support word, we have that b\u2032 = b and so, a\u2032b\u2032 \u2208U. Hence, X \\ U is d-closed. Since all words in U have weight at most w, \u03a8(v) \u2a7e1 for all v \u2208U. Therefore, \u039eX\\U(U) = \u03a8(U) \u2a7e |U|, contradicting the removal lemma. Lemma 33. Given w and m \u2a7e2w, if \u03b4w \u2a7e22w\u22121 w\u22121 \u2211 j=0 \u0012m \u2212w j \u0013 (17) then any word u of weight at most w is contained in some maximal-support word of weight at least 2w. Proof. We prove by contradication, i.e. suppose that all maximal-support words have weight at most 2w \u2212 1. From Lemma 32, we have that u \u227av for some word v with weight w. Write v = vavb with |va| = i \u22121. As before, we set U = {ab \u2208X : a \u2208{0, 1}i\u22121, vb \u227ab}. We claim that X \\ U is both (i \u22121)-balanced and d-closed. Since both U and X are (i \u22121)-balanced, it follows that X \\ U is (i \u22121)-balanced. On the other hand, consider any word ab in U and suppose a\u2032b\u2032 \u2208X is an ancestor of ab. In other words, b \u227ab\u2032 and so, vb \u227ab\u2032. Therfore, a\u2032b\u2032 \u2208U, and hence, X \\ U is d-closed. Next, we provide an upper bound for the set |U|. Consider ab \u2208U. Since all maximal-support words of v has weight at most 2w \u22121, the weight of b is at most 2w \u2212i. Let the weight of vb be wb and so, wb \u2a7ew \u2212i + 1. Since vb \u227ab, the number of choices for b is 2w\u2212i\u2212wb \u2211 j=0 \u0012m \u2212wb \u2212i + 1 j \u0013 \u2a7d w\u22121 \u2211 j=0 \u0012m \u2212w j \u0013 . Hence, |U| \u2a7d2i\u22121 \u2211w\u22121 j=0 (m\u2212w j ) \u2a7d22w\u22121 \u2211w\u22121 j=0 (m\u2212w j ). On the other hand, since X \\ U is d-closed, we have \u039eX\\U(U) = \u03a8(U) \u2a7e\u03a8(v) \u2a7e\u03b4w. Then (17) contradicts the removal lemma that states |U| > \u039eX\\U(U). Given the existence of such maximal-support words, we now make estimates on \u03a8(Y) and \u03a8(Z). For convenience, we partition Y = Sm\u22121 \u2113=0 Y\u2113and Z = Sm \u2113=1 Z\u2113such that Y\u2113and Z\u2113are words of weight \u2113in Y and Z, respectively. The following lemma is an immediate consequence from the de\ufb01nition of Y and Z. Lemma 34. For 0 \u2a7d\u2113\u2a7dm \u22121, |Y\u2113| = |Z\u2113+1|. The number of words with weight \u2113in Y is equal to the number of words with weight \u2113+ 1 in Z. The following lemma is a consequence from Lemma 33. Lemma 35. If (17) holds, then for 1 \u2a7d\u2113\u2a7dw, we have that |Y\u2113| > |Y\u2113\u22121|. 20 \fProof. We form a bipartite graph G = (Y\u2113\u222aY\u2113\u22121, E). Two vertices v1 \u2208Y\u2113and v2 \u2208Y\u2113\u22121 are connected by an edge if v2 \u227av1. Observe that the degree of a vertex in Y\u2113is at most \u2113. On the other hand, the degree of a vertex u in Y\u2113\u22121 is at least 2w \u2212\u2113+ 1 > \u2113since Lemma 33 provides a maximal support word v of weight at least 2w such that u \u227av. Since the sum of degrees in Y\u2113is equal the sum of degrees in Y\u2113\u22121 and each vertex in Y\u2113has a smaller degree than a vertex in Y\u2113\u22121, it follows that |Y\u2113| > |Y\u2113\u22121|. Corollary 36. If (17) holds, then for 1 \u2a7d\u2113\u2a7dw, we have that |Y\u2113| > |Z\u2113|. We now prove the main result on asymptotic existence. Proof of Theorem 1. Applying Lemma 30, we have that \u03a8(Y) = m \u2211 \u2113=0 \u2211 v\u2208Y\u2113 \u03a8(v) = w \u2211 \u2113=0 |Y\u2113| \u0010 c\u2113\u03b4w + O(\u03b4w\u22121) \u0011 \u2a7e w \u2211 \u2113=1 |Y\u2113| \u0010 c\u2113\u03b4w + O(\u03b4w\u22121) \u0011 . Similar manipulations yield \u03a8(Z) = w \u2211 \u2113=1 |Z\u2113| \u0010 c\u2113\u03b4w + O(\u03b4w\u22121) \u0011 . Since |Y\u2113| \u2212|Z\u2113| \u2a7e1, we estimate the difference \u03a8(Y) \u2212\u03a8(Z) by \u03a8(Y) \u2212\u03a8(Z) \u2a7e w \u2211 \u2113=1 c\u2113\u03b4w + O(\u03b4w\u22121). (18) Therefore, for \ufb01xed values of w, if we choose m suf\ufb01ciently large such that (17) holds and the right hand side(18) is nonnegative, we have that \u03a8(Z) \u2a7d\u03a8(Y). In other words, we establish (16) and complete our induction argument. Hence, no bad sets exist in the associated compatibility graph and a perfect matching or an (m, n, w)-domination mapping exists. Unfortunately, estimating the values of c\u2113is dif\ufb01cult and hence, we are unable to estimate a lower bound of m for which domination mapping exists. Nevertheless, for the remaining part of the section, we consider the case when m divides n and demonstrate that the requirement de\ufb01ned by (17) is mild. In this case, the domination graph is regular where each vertex in [m] has degree exactly \u03b4. Then it follows from symmetry that \u03a8(v) is dependent only on the weight of v. In other words, for any word v of weight \u2113, we can write \u03a8(v) as \u03a8\u2113. Therefore, \u03a8(Y) = w \u2211 \u2113=0 |Y\u2113|\u03a8\u2113\u2a7e w \u2211 \u2113=1 |Y\u2113|\u03a8\u2113> w \u2211 \u2113=1 |Z\u2113|\u03a8\u2113= \u03a8(Z). Hence, we have the following corollary. Corollary 37. If (17) holds, then \u03a8(Z) < \u03a8(Y). The next theorem provide certain suf\ufb01cient numerical conditions for the existence of (m, \u03b4m, w)-domination mappings and imply Theorem 1. Theorem 38. Let w \u2a7e3. Let \u03f5 > 0 and set N\u03f5 such that m\u03f5/(1+\u03f5) \u2a7e1 + log m for m \u2a7eN\u03f5. If m \u2a7emax n (2w)1+\u03f5, N\u03f5 o , (19) 2m \u2a7d w \u2211 j=0 \u0012\u03b4m j \u0013 , (20) then (17) holds and therefore, an (m, \u03b4m, w)-domination mapping exists. 21 \fProof. Since m \u2a7e2w, observe that \u2211w j=0 (\u03b4m j ) \u2a7d(w + 1)(\u03b4m w ) = (w + 1)(\u03b4m)w/w! . Hence, (20) implies that \u03b4w \u2a7e2m (w + 1)! wmw . (21) On the other hand, (19) implies that m = m 1 1+\u03f5 m \u03f5 1+\u03f5 \u2a7e2w(1 + log m) \u2a7e2w log(2m). Hence, 2m \u2a7e(2m)2w. Together with (21), we have \u03b4w \u2a7e22w w!mw (w + 1) \u2a7e22w\u22121 wmw (w \u22121)! \u2a7e22w\u22121w \u0012m \u2212w w \u22121 \u0013 \u2a7e22w\u22121 w\u22121 \u2211 j=0 \u0012m \u2212w j \u0013 . Note that 2w!/(w + 1) \u2a7ew/(w \u22121)! for w \u2a7e3. Therefore, (17) holds as desired. Corollary 37 then yields (16), which in turn completes our induction argument. A. Constructions and Descendant Arrays A.1. Descendant arrays We will now make use of a different representation of (m, n, w)-domination mappings. For two vectors of the same length v and u we say that v covers u (or v dominates u; or u is a descendant of v) and denote it by u \u227av if u has a zero in each entry in which v has a zero. In other words, the value of u in each position is less or equal from the value of v in the same position. Let A be an r \u00d7 \u21131 binary array and B be an r \u00d7 \u21132 binary array, \u21132 \u2a7e\u21131. If for each column in B there exists a column in A which dominates it, then (A, B) is called a pair of (\u21131, \u21132)-descendant arrays. Let A be a 2m \u00d7 m matrix which contains all the binary words of length m (in lexicographic order) and let B be a 2m \u00d7 n matrix which contains distinct binary words of length n and weight at most w. This pair of arrays is called a pair of (n, m, w)-descendant arrays if (A, B) is a pair of (n, m)-descendant arrays. By de\ufb01nition we infer that Lemma 39. An (m, n, w)-domination mapping exists if and only if there exists a pair of (m, n, w)-descendant arrays. Proof. This is an immediate observation from the de\ufb01nition of the domination graph and de\ufb01ning \u03d5(v1, v2, . . . , vm) = (u1, u2, . . . , un), where (v1, v2, . . . , vm) and (u1, u2, . . . , un) are the jth words of the matrices A and B, respectively. The description which leads to Lemma 39 can be used as an alternative way to de\ufb01ne the injective mapping \u03d5(m, n, w) and to verify whether such a mapping is a domination mapping. A.2. Construction of domination mappings for w = 2 In this appendix we consider (m, n, 2)-domination mappings for which n = \u03bd(m, w) if m is odd and discuss the case of even m. We start with odd m. We claim that for odd m = 2\u2113+ 1 there exists a (2\u2113+ 1, 2\u2113+1, 2)domination mapping. Note, that \u22112 i=0 (2\u2113+1 i ) = 22\u2113+1 + 2\u2113+ 1 and \u22112 i=0 (2\u2113+1\u22121 i ) = 22\u2113+1 \u22122\u2113+ 1, and hence by Lemma 5 we have that a (2\u2113+ 1, 2\u2113+1, 2)-domination mapping is optimal. A recursive construction for such a domination mapping will be given. We start with the (3, 4, 2)-domination mapping from Example 1. 22 \fAssume now that there exists a (2\u2113\u22121, 2\u2113, 2)-domination mapping \u03d5 with the degree sequence (1, 1, 2, 2, 2, 4, 4, 8, 8, . . . , 2\u2113\u22123, 2\u2113\u22123, 2\u2113\u22122, 2\u2113\u22122). We will construct a (2\u2113+ 1, 2\u2113+1, 2)-domination mapping \u03d5\u2032 with the degree sequence (1, 1, 2, 2, 2, 4, 4, 8, 8, . . . , 2\u2113\u22122, 2\u2113\u22122, 2\u2113\u22121, 2\u2113\u22121). Let (A(\u03d5), B(\u03d5)) be a pair of (2\u2113\u22121, 2\u2113, 2)-descendant arrays related to the domination mapping \u03d5. We will describe now a construction for a pair of (2\u2113+ 1, 2\u2113+1, 2)-descendant arrays (A(\u03d5\u2032), B(\u03d5\u2032)) from which \u03d5\u2032 can be derived. Let A1A2 be a 22\u2113\u22121 \u00d7 (2\u2113+ 1) matrix, where A1 has two columns and A2 has 2\u2113\u22121 columns; A1A2 represents a quarter of the matrix A(\u03d5\u2032), i.e. A1 has one of the for values 00, 01, 10, or 11, and A2 has 22\u2113\u22121 rows with exactly all the words in F2\u2113\u22121 2 in the lexicographic order. Let B1B2 be a 22\u2113\u22121 \u00d7 2\u2113+1 matrix, where B1 has 2\u2113columns and B2 has 2\u2113columns; B1B2 represents a quarter of the matrix B(\u03d5\u2032). Furthermore, let B1 = B0 1B1 1, where B0 1 and B1 1 are 22\u2113\u22121 \u00d7 2\u2113\u22121 matrices. The ith row of A1A2 will be mapped by \u03d5\u2032 to the ith row of B1B2. We distinguish now between four cases related to the values of the two columns of A1. [D1] If the two columns of A1 are 00, then B1 is the all-zeroes matrix and (A2, B2) is a pair of (2\u2113\u2212 1, 2\u2113, 2)-descendant arrays with the degree sequence (1, 1, 2, 2, 2, 4, 4, 8, 8, . . . , 2\u2113\u22123, 2\u2113\u22123, 2\u2113\u22122, 2\u2113\u22122). [D2] If the two columns of A1 are 11, then B2 is the all-zeroes matrix, and B1 can be any matrix whose rows are different, 22\u2113\u22121 \u22122\u2113\u22121 rows have weight two, 2\u2113\u22121 rows have weight one (2\u2113\u22122 of these ones in the last 2\u2113\u22122 columns of B0 1 and the other 2\u2113\u22122 ones in the last 2\u2113\u22122 columns, which are the most signi\ufb01cant bits, of B1 1). Simple enumeration yields that there are 22\u2113\u22121 such possible different rows as required. [D3] If the two columns of A1 are 01 then the matrix B2 is chosen in a way that each column, except for the \ufb01rst one (least signi\ufb01cant bit) has exactly 2\u2113\u22121 ones and the \ufb01rst column has 2\u2113\u22122 ones; each row, except for 2\u2113\u22122 rows of B2, has exactly one one and these 2\u2113\u22122 rows are all-zeroes rows. The distribution of the ones is done in a way that the requirements of the related domination graph, i.e. the degree sequence of (A2, B2) as a pair of (2\u2113\u22121, 2\u2113)-descendant arrays is (1, 1, 2, 2, 2, 4, 4, 8, 8, . . . , 2\u2113\u22123, 2\u2113\u22123, 2\u2113\u22122, 2\u2113\u22122). For each 2\u2113\u22121 ones in the same column of B2, the corresponding 2\u2113\u22121 rows in B1 have unique ones in different 2\u2113\u22121 columns in the last 2\u2113\u22121 columns of B1. The same is done for the \ufb01rst column of B2 for which the all-zeroes rows are added. The concrete de\ufb01nition will be left as an exercise (not completely trivial). [D4] If the two columns of A1 are 10 then B2 is exactly as in the case where the two columns of A1 are 01, In B1 the \ufb01rst 2\u2113\u22121 columns are swapped with the last 2\u2113\u22121 columns compared to the matrix B1 in the case where the \ufb01rst two columns of A1 are 01, i.e. B0 1 is swapped with B1 1. Finally, A(\u03d5\u2032) consists of the four matrices A1A2 of these four cases and B(\u03d5\u2032) are the related four matrices B1B2. The construction leads to the following result. Theorem 40. If (A(\u03d5), B(\u03d5)) is a pair of (2\u2113\u22121, 2\u2113, 2)-descendant arrays then (A(\u03d5\u2032), B(\u03d5\u2032)) are two (2\u2113+ 1, 2\u2113+1, 2)-descendant arrays. Theorem 40 implies the existence of a (2\u2113+ 1, 2\u2113+1, 2)-domination mapping for odd m. What about an (m, n, 2)-domination mapping for even m. The following (m, n) pairs were found by computer search to form optimal (m, n, 2)-domination mappings: (4, 6), (6, 11), (8, 23), (10, 45), (12, 90), (14, 181), (16, 362), (18, 724), (20, 1448), (22, 2896), (24, 5793), (26, 11585), (28, 23170), and so on. For m \u2a7e6, these mappings attains the bound of Lemma 5. In this sequence of optimal domination mappings one can observe that 23 \fthere is no obvious rule and a hence a recursive construction for optimal domination mapping won\u2019t be an easy task. But, optimal domination mappings exist and an existence proof for such mappings can be given similarly (but with a simpler proof) to the existence proof in Section 6. B. Proof of Proposition 15 We provide a detailed proof of Proposition 15. To this end, we need the following lemmas. Lemma 41. The set of all A-preserving permutations is a subgroup of the set of permutations on [N]. Lemma 42. Suppose that \u03c0 is A-preserving. Then Ax\u03c0 \u2a7d1 if and only if Ax \u2a7d1. Proof. Since the set of all A-preserving permutations form a group, it suf\ufb01ces to show one direction. Suppose that Ax \u2a7d1. Since P\u03c0row AP\u03c0 = A for some permutation \u03c0row : [M] \u2192[M], we have that 1 \u2a7eAx = P\u03c0row AP\u03c0x = P\u03c0row Ax\u03c0. Hence, Ax\u03c0 \u2a7dP\u22121 \u03c0row1 = 1. Proof of Proposition 15 . Set x\u2217= \u2211\u03c0\u2208GA x\u03c0 |GA| . By Lemma 42, we have that Ax\u03c0 \u2a7d1 for all \u03c0 \u2208GA. Therefore, Ax\u2217\u2a7d1 . It is also readily veri\ufb01ed that \u2211N i=1 x\u2217 i = \u03bb Finally, we show that x\u2217is O-regular. For all k \u2a7dN and i, j \u2208Ok, we have that x\u2217 i = \u2211\u03c0\u2208GA x\u03c0 i |GA| = \u2211\u03c0\u2208GA x\u03c0(i) |GA| . On the other hand, since i, j \u2208Ok, there exists a \u03c0\u2217such that \u03c0\u2217(j) = i. Hence, x\u2217 j = \u2211\u03c0\u2208GA x\u03c0 j |GA| = \u2211\u03c0\u2208GA x\u03c0(j) |GA| = \u2211\u03c0\u2208GA x\u03c0\u25e6\u03c0\u2217(j) |GA| == \u2211\u03c0\u2208GA x\u03c0(i) |GA| . Therefore, x\u2217 i = x\u2217 j . 24"
+                },
+                {
+                    "url": "http://arxiv.org/abs/1707.03956v2",
+                    "title": "Deciding the Confusability of Words under Tandem Repeats",
+                    "abstract": "Tandem duplication in DNA is the process of inserting a copy of a segment of\nDNA adjacent to the original position. Motivated by applications that store\ndata in living organisms, Jain {\\em et al.} (2016) proposed the study of codes\nthat correct tandem duplications to improve the reliability of data storage. We\ninvestigate algorithms associated with the study of these codes.\n  Two words are said to be ${\\le}k$-confusable if there exists two sequences of\ntandem duplications of lengths at most $k$ such that the resulting words are\nequal. We demonstrate that the problem of deciding whether two words is\n${\\le}k$-confusable is linear-time solvable through a characterisation that can\nbe checked efficiently for $k=3$. Combining with previous results, the decision\nproblem is linear-time solvable for $k\\le 3$. We conjecture that this problem\nis undecidable for $k>3$.\n  Using insights gained from the algorithm, we study the size of\ntandem-duplication codes. We improve the previous known upper bound and then\nconstruct codes with larger sizes as compared to the previous constructions. We\ndetermine the sizes of optimal tandem-duplication codes for lengths up to\ntwenty, develop recursive methods to construct tandem-duplication codes for all\nword lengths, and compute explicit lower bounds for the size of optimal\ntandem-duplication codes for lengths from 21 to 30.",
+                    "authors": "Yeow Meng Chee, Johan Chrisnata, Han Mao Kiah, Tuan Thanh Nguyen",
+                    "published": "2017-07-13",
+                    "updated": "2017-11-17",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "INTRODUCTION At the beginning of the millenium, Lander et al. [1] published a draft sequence of the human genome and reported that more than 50% of the human genome consists of repeated substrings [1]. There are two types of common repeats: interspersed repeats and tandem repeats. Interspersed repeats are caused by transposons when a segment of DNA is copied and pasted into new positions of the genome. In contrast, tandem repeats are caused by slipped-strand mispairings [2], and they occur when a pattern of one or more nucleotides is repeated and the repetitions are adjacent to each other. For example, consider the word AGTAGTCTGC. The substring AGTAGT is a tandem repeat, and we say that AGTAGTCTGC is generated from AGTCTGC by a tandem duplication of length three. Tandem repeats are believed to be the cause of several genetic disorders [3]\u2013[5] and this motivated the study of tandem duplication mechanisms in a variety of contexts. \u2022 Formal languages: Leopold et al. [6], [7] de\ufb01ned the unbounded duplication language and the k-bounded duplication language to be the set of words generated by seed word under tandem duplications of unbounded length and tandem duplications of length up to k, respectively. They investigated certain decidability problems involving unbounded duplication languages and showed that all k-bounded duplication languages are context free. Furthermore, they showed that k-bounded duplication language is always regular for any binary seed and k \u2a7e1. On the other hand, the k-bounded duplication language is not regular for any square-free seed word over an alphabet of at least three letters and k \u2a7e4. More recently, Jain et al. [8] completed this characterization and proved that k-bounded duplication languages are regular for k \u2a7d3. \u2022 Information theory: Farnoud et al. [9] introduced the concept of capacity to determine average information content of a k-bounded duplication language. Later, Jain et al. [8] introduced the notion of expressiveness to measure a language\u2019s capability to generate words that contain certain desired substrings. A complete characterization of fully expressive k-bounded duplication languages was provided by Jain et al. for all alphabet sizes and all k. \u2022 Codes correcting tandem duplications: Motivated by applications that store data in living organisms [10]\u2013 [12], Jain et al. [13] proposed the study of codes that correct tandem duplications to improve the reliability of data storage. They investigated various types of tandem duplications and provided optimal code construction in the case where duplication length is at most two. In this paper, we study the last problem and investigate algorithms associated with these codes. In particular, given two words x and y, we look for ef\ufb01cient algorithms that answer the following question: When are the words x and y confusable under tandem repeats? In other words, are there two sequences of tandem duplications such that the resulting words x\u2032 and y\u2032 are equal? Interestingly, as we demonstrate, even for small duplication lengths, the solutions to this question are nontrivial. This is surprising as ef\ufb01cient algorithms are well known for analogous questions in other problems in string matching [14]. Before we give an account of these results, we introduce some necessary notations and provide a formal statement of our problem. II. PRELIMINARIES Let \u03a3q = {0, 1, \u00b7 \u00b7 \u00b7 , q \u22121} be an alphabet of q \u2a7e2 symbols. For a positive integer n, let \u03a3n q denote the set of all strings or words of length n over \u03a3q, and let \u03a3\u2217 q denote the set of all \ufb01nite words over \u03a3q, or the Kleene closure of \u03a3q. Given two words x, y \u2208\u03a3\u2217 q, we denote their concatenation by xy. We state the tandem duplication rules. For nonnegative integers k \u2a7dn and i \u2a7dn\u2212k, we de\ufb01ne Ti,k : \u03a3n q \u2192\u03a3n+k q such that Ti,k(x) = uvvw, where x = uvw, |u| = i, |v| = k. If a \ufb01nite sequence of tandem duplications of length k is performed to obtain y from x, then we say that y is a k-descendant of x, or x is a k-ancestor of y, and denote this relation by x \u2217 = \u21d2 k y. Formally, x \u2217 = \u21d2 k y means that for some positive t, there exist t non-negative integers i1, i2, . . . it such that y = Tit,k \u25e6Tit\u22121,k \u25e6\u00b7 \u00b7 \u00b7 \u25e6Ti1,k(x). Given a sequence x and integer k, we consider the set of words that may be obtained from x via a \ufb01nite number of tandem duplications of length k. We de\ufb01ne the k-descendant cone of x to be the set of all k-descendants of x and denote this cone by D\u2217 k(x). In other words, D\u2217 k(x) \u225c \u001a y \u2208\u03a3\u2217 q | x \u2217 = \u21d2 k y \u001b . 2 \fOur work studies tandem duplications whose lengths are upper bounded by an integer k and we extend the previous de\ufb01nitions. Formally, if a \ufb01nite sequence of tandem duplications of length up to k is performed to obtain y from x, then we say that y is a \u2a7dk-descendant of x and denote this relation by x \u2217 = \u21d2 \u2a7dk y. We have analogous de\ufb01nitions of \u2a7dk-descendant, \u2a7dk-ancestor, and \u2a7dk-descendant cone D\u2217 \u2a7dk(x). Example 1. Consider x = 01210 over \u03a33. Since T1,3(x) = 01211210 and T0,2(01211210) = 0101211210, we write 01210 \u2217 = \u21d2 3 01211210 and 01210 \u2217 = \u21d2 \u2a7d3 0101211210. Alternatively, we have that 01211210 \u2208D\u2217 3(x), and 0101211210 \u2208D\u2217 \u2a7d3(x). A. Problem Formulation Motivated by applications that store data on living organisms, Jain et al. [13] looked at the \u2a7dk-descendant cones of a pair of words and asked whether the two cones have a nontrivial intersection. Speci\ufb01cally, we introduce the notion of confusability. De\ufb01nition 1 (Confusability). Two words x and y, are said to be k-confusable if D\u2217 k(x) \u2229D\u2217 k(y) \u0338= \u2205. Similarly, they are said to be \u2a7dk-confusable if D\u2217 \u2a7dk(x) \u2229D\u2217 \u2a7dk(y) \u0338= \u2205. To design error-correcting codes that store information in the DNA of living organisms, Jain et al. then proposed the use of codewords that are not pairwise confusable. De\ufb01nition 2 (\u2a7dk-Tandem-Duplication Codes). A subset C \u2286\u03a3n q is a \u2a7dk-tandem-duplication code if for all x, y \u2208C and x \u0338= y, we have that x and y are not \u2a7dk-confusable. We say that C is an (n, \u2a7dk; q)-tandem-duplication code or (n, \u2a7dk; q)-TD code. Therefore, to determine if a set of words is a tandem-duplication code, we need to verify that all pairs of distinct words are not confusable. Hence, we state our problem of interest. CONFUSABILITY PROBLEM Instance: Two words x and y over \u03a3q, and an integer k Question: Are x and y \u2a7dk-confusable? While the confusability problem is a natural question, ef\ufb01cient algorithms are only known for the case where k \u2208{1, 2}. We review these results in the next subsection. B. Previous Work Confusability. We summarize known ef\ufb01cient algorithms that determine whether two words are k-confusable for all k, and whether they are \u2a7dk-confusable for k \u2208{1, 2}. We then highlight why these methods cannot be extended for the case k \u2a7e3. Crucial to the algorithms is the concept of irreducible words and roots. De\ufb01nition 3. A word x is said to be k-irreducible if x cannot be deduplicated into shorter words with deduplication of length k. In other words, if y \u2217 = \u21d2 k x, then y = x. The set of k-irreducible q-ary words is denoted by Irrk(q) and the set of k-irreducible words of length n is denoted by Irrk(n, q). The k-ancestors of x \u2208\u03a3\u2217 q that are k-irreducible words are called the k-roots of x. The set of k-roots of x is denoted by Rk(x). In a similar fashion, we have the de\ufb01nitions of \u2a7dk-irreducible words and \u2a7dk-roots of x, and the notation Irr\u2a7dk(q), Irr\u2a7dk(n, q), and R\u2a7dk(x). Example 2. For alphabet \u03a33, we have R\u2a7d3(01012012) = {012}, and R\u2a7d2(012012) = {012012}. On the other hand, R\u2a7d4(012101212) = {012, 0121012}. As we see, it is possible for a word to have more than one root. Jain et al. determined when a word has exactly one root and provided ef\ufb01cient algorithms to compute this unique root. Proposition 1 (Jain et al. [13]). For any x \u2208\u03a3\u2217 q, we have that |Rk(x)| = 1 for all k, and |R\u2a7dk(x)| = 1 for all k \u2208{1, 2, 3}. Furthermore, there exist linear-time algorithms to compute Rk(x) for all k. One may easily derive linear-time algorithms to compute R\u2a7dk(x) for all for k \u2208{1, 2, 3}. For completeness, we formally describe these algorithms in Appendix A. As it turns out, for certain cases, determining the confusability of two words is equivalent to computing the roots for the words. 3 \fProposition 2 (Jain et al. [13]). For all x, y \u2208\u03a3\u2217 q, we have that (i) x and y are k-confusable if and only if Rk(x) = Rk(y) for all k; (ii) x and y are \u2a7dk-confusable if and only if R\u2a7dk(x) = R\u2a7dk(y) for k \u2208{1, 2}. (iii) If R\u2a7d3(x) \u0338= R\u2a7d3(y), then x and y are not \u2a7d3-confusable. In other words, when k \u2208{1, 2}, to determine x and y are \u2a7dk-confusable, it is both necessary and suf\ufb01cient to compute the \u2a7dk-roots of x and y. Therefore, applying Proposition 1 and the algorithms in Appendix A, we are able to determine whether two words are \u2a7dk-confusable in linear time for k \u2208{1, 2}. Unfortunately, when k = 3, it is no longer suf\ufb01cient to compute the \u2a7d3-roots of x and y. Speci\ufb01cally, even though R\u2a7d3(x) = R\u2a7d3(y), it is possible that x and y are not \u2a7d3-confusable. We illustrate this in the next example. Example 3. Consider x = 012012 and y = 011112 over \u03a33. The words have the same root as R\u2a7d3(x) = R\u2a7d3(y) = {012}. However, x and y are not \u2a7d3-confusable because any \u2a7d3-descendant of x has a 2 to the left of a 0, whereas any \u2a7d3-descendant of y does not. Therefore, the next smallest open case is where k = 3. A polynomial-time algorithm is implied by the results of Leupold et al. [6] and Jain et al. [8]. Proposition 3 (Leupold et al. [6], Jain et al. [8]). Let k \u2208{2, 3}. Let x \u2208\u03a3\u2217 q. Then D\u2217 \u2a7dk(x) is a regular language. Furthermore, if |x| = m, then the deterministic \ufb01nite automaton that generates D\u2217 \u2a7dk(x) has O(m) vertices and O(m) edges. The number of vertices and edges is independent of the alphabet size. Therefore, for two words x and y with |x| = m and |y| = n, the language D\u2217 \u2a7d3(x) \u2229D\u2217 \u2a7d3(y) is regular and the corresponding \ufb01nite automaton has O(mn) vertices and O(mn) edges. Hence, to determine if x and y are confusable, it suf\ufb01ces to determine if the language D\u2217 \u2a7d3(x) \u2229D\u2217 \u2a7d3(y) is nonempty. The latter can be done in O(mn) time. We improve this running time by providing an algorithm that runs in O(max{m, n}) time. Code Construction. We are interested in determining the maximum possible size of an (n, \u2a7dk; q)-TD code. An (n, \u2a7dk; q)-TD code that achieves this maximum is said to be optimal. Motivated by Proposition 2, Jain et al. used irreducible words to construct tandem-duplication codes. Construction 1. For k \u2208{2, 3} and n \u2a7ek. An (n, \u2a7dk; q)-TD code CI(n, \u2a7dk; q) is given by CI(n, \u2a7dk; 3) = n [ i=1 {\u03ben\u2212i(x) | x \u2208Irr\u2a7dk(i, q)} . Here, \u03bei(x) = xzi, where z is the last symbol of x. In other words, \u03bei(x) is the sequence x with its last symbol z repeated i more times. Furthermore, the size of CI(n, \u2a7dk; q) is Pn i=1 |Irr\u2a7dk(i, q)|. It then follows from Proposition 2 that CI(n, \u2a7d2; q) is an optimal (n, \u2a7d2; q)-TD code. Unfortunately, the TD code CI(n, \u2a7d3; q) is not an optimal for n \u2a7e6, and we illustrate this via the following example. Example 4. Consider n = 6, q = 3. The code CI(6, \u2a7d3; 3) from Construction 1 is given by CI(6, \u2a7d3; q) = {aaaaaa | a \u2208\u03a33} \u222a{abbbbb | a, b \u2208\u03a33, a \u0338= b} \u222a{abaaaa, abcccc | a, b, c \u2208\u03a33, a \u0338= b, b \u0338= c, a \u0338= c} \u222a{abaccc, abcaaa, abcbbb | a, b, c \u2208\u03a33, a \u0338= b, b \u0338= c, a \u0338= c} \u222a{abacaa, abacbb, abcabb, abcacc, abcbaa | a, b, c \u2208\u03a33, a \u0338= b, b \u0338= c, a \u0338= c} \u222a{abacab, abacba, abacbc, abcaba, abcacb, abcbab, abcbac | a, b, c \u2208\u03a33, a \u0338= b, b \u0338= c, a \u0338= c}. Hence, |CI(6, \u2a7d3; 3)| = 111. However, we may remove from CI(6, \u2a7d3; 3) the six codewords {abcccc | a, b, c \u2208 \u03a33, a \u0338= b, b \u0338= c, a \u0338= c} and augment the code with twelve more codewords {abcabc, abbbbc | a, b, c \u2208\u03a33, a \u0338= b, b \u0338= c, a \u0338= c}. Then we can check that the new code has size 117, and we later verify using Proposition 4 that the new code is indeed optimal. For lengths at least six, we construct TD codes with strictly larger sizes. Upper Bound. We also study upper bounds on the size of an optimal (n, \u2a7d3; q)-TD code. By de\ufb01nition, an (n, \u2a7d3; q)-TD code is also an (n, \u2a7d2; q)-TD code. Since an optimal (n, \u2a7d2; q)-TD code is provided by Construction 1, we have the following upper bound on the size of an optimal (n, \u2a7d3; q)-TD code. Proposition 4. The size of an (n, \u2a7d3; q)-TD code is at most Pn i=1 |Irr\u2a7d2(i, q)|. 4 \fProposition 4 implies that Construction 1 is tight for k = 3 and n \u2a7d5. Using a combinatorial characterization implied by our algorithm, we improve this upper bound for longer lengths in Section IV. C. Our Contributions \u2022 Confusability. In Section III, we present suf\ufb01cient and necessary conditions for two words to be \u2a7d3-confusable and propose a linear-time algorithm to solve the \u2a7d3-confusability problem. \u2022 Estimates on code sizes. Using insights gained from Section III, we study the size of tandem-duplication codes in Section IV. We \ufb01rst improve the upper bound given by Proposition 4 and then construct codes with larger sizes as compared to those given by Construction 1. We also provide certain explicit constructions and recursive constructions for tandem-duplication codes. Furthermore, we determine the sizes of optimal \u2a7d3-tandem-duplication codes for lengths up to twenty. III. DETERMINING \u2a7d3-CONFUSABILITY We derive a linear-time algorithm to determine the \u2a7d3-confusability of two words x and y. For the sake of simplicity, we omit the use of \u201c\u2a7d3\u201d and assume confusable, descendant and root to mean \u2a7d3-confusable, \u2a7d3descendant and \u2a7d3-root, respectively. Prior to stating the algorithm, we make some technical observations on tandem duplications of length at most three. The proofs of the lemmas are deferred to Appendix B. The \ufb01rst lemma states that given a sequence of tandem duplications, we may reorder the tandem duplications such that the lengths of the repeats are nonincreasing. Lemma 5. Let k1 < k2 \u2a7d3 and x \u2208\u03a3\u2217 q. Suppose that Ti2,k2 \u25e6Ti1,k1(x) = x\u2032 for some i1, i2. Then there exist integers 3 \u2a7e\u21131 \u2a7e\u21132 \u2a7e\u00b7 \u00b7 \u00b7 \u2a7e\u2113t and j1, j2, . . . , jt such that Tjt,\u2113t \u25e6Tjt\u22121,\u2113t\u22121 \u25e6\u00b7 \u00b7 \u00b7 \u25e6Tj1,\u21131(x) = x\u2032. The next lemma states that we may assume the duplications of length two and three are performed on segments whose symbols are distinct. This is because if a duplication is performed on a segment whose symbols are not distinct, we may \ufb01nd an equivalent sequence of duplications of strictly shorter lengths. Lemma 6. Let k \u2208{2, 3}. Suppose Ti,k(x) = uvvw, where x = uvw, |u| = i and |v| = k. If the symbols in v are not pairwise distinct, then there exist integers k > \u21131 \u2a7e\u21132 and j1, j2 such that Tj2,\u21132 \u25e6Tj1,\u21131(x) = Ti,k(x). Henceforth, we use the notation x \u2217 = \u21d2 kd y to denote that there is a sequence of tandem duplications of length k of distinct symbols from x to y. It follows from de\ufb01nition that x \u2217 = \u21d2 1 y is equivalent to x \u2217 = \u21d2 1d y. The next lemma is immediate from Lemmas 5 and 6. Lemma 7. Suppose that x \u2217 = \u21d2 \u2a7d3 x\u2032. Then there exist x1 and x2 such that x \u2217 = \u21d2 3d x2 \u2217 = \u21d2 2d x1 \u2217 = \u21d2 1d x\u2032. Furthermore, x1 and x2 are uniquely determined by x1 = R1(x\u2032) and x2 = R\u2a7d2(x\u2032). Equipped with this lemma, we state the following theorem that provides a necessary and suf\ufb01ciency condition for two words to be confusable. Theorem 8. Two words x and y are \u2a7d3-confusable if and only if there exist x\u2032 and y\u2032 such that x \u2217 = \u21d2 3d x\u2032, y \u2217 = \u21d2 3d y\u2032 and R\u2a7d2(x\u2032) = R\u2a7d2(y\u2032). Proof. Suppose that x and y are confusable. Then there exists z such that x \u2217 = \u21d2 \u2a7d3 z and y \u2217 = \u21d2 \u2a7d3 z. By Lemma 7, there exist x1, x2, y1, y2 such that x \u2217 = \u21d2 3d x2 \u2217 = \u21d2 2d x1 \u2217 = \u21d2 1d z and y \u2217 = \u21d2 3d y2 \u2217 = \u21d2 2d y1 \u2217 = \u21d2 1d z. Hence we set x\u2032 = x2 and y\u2032 = y2. Note that by Proposition 2, R\u2a7d2(x\u2032) = R\u2a7d2(y\u2032) since x\u2032 and y\u2032 are \u2a7d2-confusable. Conversely, suppose that there exist x\u2032 and y\u2032 such that x \u2217 = \u21d2 3d x\u2032, y \u2217 = \u21d2 3d y\u2032, and R\u2a7d2(x\u2032) = R\u2a7d2(y\u2032). Since x\u2032 and y\u2032 have a common \u2a7d2-root, it follows from Proposition 2 that x\u2032 and y\u2032 are \u2a7d2-confusable. Thus, x and y are confusable. \u25a0 In conclusion, for two words x and y to be confusable, it is equivalent to checking if there are tandem duplications of length three that make their descendants x\u2032 and y\u2032 have the same \u2a7d2-root. We make use of this important fact in the next section. 5 \fA. Strategy behind Algorithm We outline our strategy to determine the confusability of two words x and y. Recall from Proposition 2(iii), if x and y have different roots, then they are not confusable. Hence, our \ufb01rst step is to determine R\u2a7d3(x) and R\u2a7d3(y) and see if the roots are equal. If the roots are not equal, we immediately conclude that x and y are not confusable. Therefore, the nontrivial task is to determine confusability when R\u2a7d3(x) = R\u2a7d3(y). Henceforth, we assume that the common root of x and y is r. Suppose that r contains at most two distinct symbols. Then from Lemma 7, we have that r \u2217 = \u21d2 \u2a7d2 x and r \u2217 = \u21d2 \u2a7d2 y. In other words, r is also the \u2a7d2-root of both x and y. Applying Proposition 2, we have that x and y are \u2a7d2confusable and so, \u2a7d3-confusable. Therefore, it remains to consider the case where r contains at least three distinct symbols. Given r, x and y, our next step is to compute certain proper pre\ufb01xes \u2217Pref(r, x) and \u2217Pref(r, y) of x and y, respectively, such that under certain conditions the following holds: (x and y are confusable) if and only if (x \\ \u2217Pref(r, x) and y \\ \u2217Pref(r, y) are confusable) . Here, x \\ z denotes the word we obtain by removing the pre\ufb01x z from x. Since the words on the righthand side are strictly shorter than the words on the lefthand side, we may repeat this process for the shorter words until the common root of the words has less than three distinct symbols. To de\ufb01ne the pre\ufb01xes \u2217Pref(r, x) and \u2217Pref(r, y), we introduce more notation. Now, r contains at least three distinct symbols. Since r \u2208Irr\u2a7d3(q), we easily check that either r1r2r3 or r2r3r4 is a substring with three distinct symbols. De\ufb01ne main(r) to be the \ufb01rst substring in r with three distinct symbols and we de\ufb01ne the region of r, denoted by Reg(r), to be a certain pre\ufb01x of r according to the following rule. If r1 = r3, we set main(r) \u225cr2r3r4, and Reg(r) \u225c \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 r1r2r3r4 = r1r2r1r4, if r2 \u0338= r5, r1r2r3r4r5 = r1r2r1r4r2, if r2 = r5, r3 \u0338= r6, r1r2r3r4r5r6 = r1r2r1r4r2r1, otherwise. If r1 \u0338= r3, we set main(r) \u225cr1r2r3, and Reg(r) \u225c \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 r1r2r3, if r1 \u0338= r4, r1r2r3r4 = r1r2r3r1, if r1 = r4, r2 \u0338= r5, r1r2r3r4r5 = r1r2r3r1r2, otherwise. Intuitively, the region Reg(r) of r is de\ufb01ned as above so that the following properties of the region hold. Lemma 9. Suppose the root r has three distinct symbols. De\ufb01ne Reg(r) as above. Then Reg(r) can be written as w(abc)\u2113ab, where a, b, c are distinct symbols, \u2113\u2208{0, 1} and w is pre\ufb01x of length at most three over the alphabet {a, b, c}. Furthermore, if Reg(r) \u2217 = \u21d2 3d z, then z can be written as w(abc)mab with m \u2a7e\u2113. Also, the symbol in r after the pre\ufb01x Reg(r) is not c. Example 5. Suppose that r = 010201. Then main(r) = 102 and Reg(r) = 0102. Indeed, we can write Reg(r) as w(abc)\u2113ab with w = 01, \u2113= 0, and abc = 021. We also check that the next symbol after Reg(r) in r is 0, which is not c = 1. Given Reg(r) = w(abc)\u2113ab, we consider the longest pre\ufb01x of x that is generated from Reg(r) through a sequence of tandem duplications of length at most three. We call this pre\ufb01x the extended region of r in x and denote it with Ext(Reg(r), x). Finally, we de\ufb01ne our special pre\ufb01x \u2217Pref(r, x). Suppose that Ext(Reg(r), x) = p1p2 \u00b7 \u00b7 \u00b7 pt = p and pi is the last appearance of a in p. The special pre\ufb01x \u2217Pref(r, x) is given by p1p2 \u00b7 \u00b7 \u00b7 pi\u22121. Example 6. Let r = 010201 and x = 01102021020120111. Recall that Reg(r) = 0102 and a = 0. Then Ext(0102, x) = 0110202102 and \u2217Pref(r, x) = 01102021. We summarize the relationships of r, x, Reg(r), Ext(Reg(r), x), and \u2217Pref(r, x) in Figure 1 and we are ready to state the main characterization theorem. 6 \fFig. 1: Pre\ufb01xes Used to Determine Confusability Theorem 10. Suppose that x and y are two words such that r = R\u2a7d3(x) = R\u2a7d3(y) and r contains at least three distinct symbols. Set p = Ext(Reg(r), x), q = Ext(Reg(r), y) u = x \\ \u2217Pref(r, x) v = y \\ \u2217Pref(r, y). Then x and y are confusable if and only if p and q are confusable, and u and v are confusable. Proof. From Lemma 9, we may set Reg(r) = w(abc)\u2113ab. Since Reg(r) \u2217 = \u21d2 \u2a7d3 p, Lemma 7 implies that Reg(r) \u2217 = \u21d2 3d R\u2a7d2(p). So, R\u2a7d2(p) = w(abc)mab for some m \u2a7e\u2113, and R\u2a7d2(q) = w(abc)jab for some j \u2a7e\u2113. Suppose that x and y are confusable. By Theorem 8, there exist 3d-descendants of x and y namely x\u2032 and y\u2032, such that R\u2a7d2(x\u2032) = R\u2a7d2(y\u2032). Set p\u2032 = Ext(Reg(r), x\u2032), q\u2032 = Ext(Reg(r), y\u2032) u\u2032 = x\u2032 \\ \u2217Pref(r, x\u2032) v\u2032 = y\u2032 \\ \u2217Pref(r, y\u2032). Since the overlapping region between p and u must be of the form ab \u00b7 \u00b7 \u00b7 b, each duplication of distinct triplets from x to x\u2032 must be either entirely in p or in u. In other words, p \u2217 = \u21d2 3d p\u2032 and u \u2217 = \u21d2 3d u\u2032, and likewise q \u2217 = \u21d2 3d q\u2032 and v \u2217 = \u21d2 3d v\u2032. Looking at the overlapping region between p\u2032 and u\u2032, we can observe that R\u2a7d2(x\u2032) = R\u2a7d2(p\u2032u\u2032). Then Reg(r) \u2217 = \u21d2 \u2a7d3 p\u2032 and Lemma 7 imply that Reg(r) \u2217 = \u21d2 3d R\u2a7d2(p) and so, R\u2a7d2(p\u2032) = w(abc)iab for some i \u2a7em. Next, w(abc)iR\u2a7d2(u\u2032) is a \u2a7d2-irreducible word, because R\u2a7d2(u\u2032) starts with ab and by de\ufb01nition, w(abc)i ends with c but not cac. Therefore, w(abc)iR\u2a7d2(u\u2032) is the \u2a7d2-root of x\u2032, or, R\u2a7d2(x\u2032) = w(abc)iR\u2a7d2(u\u2032) and likewise R\u2a7d2(y\u2032) = w(abc)hR\u2a7d2(v\u2032) for some h \u2a7ej. Since the \ufb01rst symbol of R\u2a7d2(x\u2032 \\p\u2032) and R\u2a7d2(y\u2032 \\q\u2032) cannot be c, neither R\u2a7d2(u\u2032) nor R\u2a7d2(v\u2032) start with abc. Therefore, since R\u2a7d2(x\u2032) = R\u2a7d2(y\u2032), we can deduce that h = i and R\u2a7d2(u\u2032) = R\u2a7d2(v\u2032). Since h = i, we have R\u2a7d2(p\u2032) = R\u2a7d2(q\u2032). Together with p \u2217 = \u21d2 3d p\u2032, q \u2217 = \u21d2 3d q\u2032, by Theorem 8, we can have p and q are confusable. Since u \u2217 = \u21d2 3d u\u2032, v \u2217 = \u21d2 3d v\u2032, and R\u2a7d2(u\u2032) = R\u2a7d2(v\u2032), we can also conclude by Theorem 8 that u and v are confusable. Suppose that p and q are confusable, and u and v are confusable. Hence pu and qv are confusable. Note that pu is a descendant of x and qv is a descendant of y, hence x and y are also confusable. \u25a0 We can ef\ufb01ciently determine whether the pre\ufb01xes p and q are confusable. When the substring main(r) appears the same number of times in R\u2a7d2(p) as in R\u2a7d2(q), we may assume that R\u2a7d2(p) = w(abc)mab = R\u2a7d2(q). Hence, p and q are confusable by Proposition 2. In particular, we have Corollary 11. Consider z = abc, where a, b, c are three distinct symbols. Let Count(z, x) denote the number of times z appears as a substring in x, and Count(rot(z), x) denote the number of times abc, bca, or cab appears as a substring in x. Corollary 11. Suppose that x and y are two words such that r = R\u2a7d3(x) = R\u2a7d3(y) and r contains at least three distinct symbols. Set p = Ext(Reg(r), x) and q = Ext(Reg(r), y). Suppose further that Count(main(r), R\u2a7d2(p)) = Count(main(r), R\u2a7d2(q)). Then x and y are confusable if and only if x \\ \u2217Pref(r, x) and y \\ \u2217Pref(r, y) are confusable. On the other hand, when the number of appearances is not equal, we may modify the proof to obtain the following corollary. A detailed proof of the corollary is deferred to Appendix C. 7 \fAlgorithm 1 CONFUSE(x, y) Input: x, y \u2208\u03a3\u2217 q Output: true if x and y are confusable, false otherwise. 1: r \u2190R\u2a7d3(x) 2: r\u2032 \u2190R\u2a7d3(y) 3: if r \u0338= r\u2032 then 4: return false 5: else 6: return CONFUSE-RECURSION(x, y, r) 7: procedure CONFUSE-RECURSION(x,y,r) 8: if r has at most two distinct symbols then 9: return true 10: else 11: Compute main(r) and Reg(r) 12: p \u2190Ext(Reg(r), x), q \u2190Ext(Reg(r), y) 13: u \u2190x \\ \u2217Pref(r, x), v \u2190y \\ \u2217Pref(r, y) 14: Cp \u2190Count(main(r), R\u2a7d2(, p)), Cq \u2190Count(main(r), R\u2a7d2(q)) 15: r\u2217\u2190r \\ w(abc)\u2113, where Reg(r) = w(abc)\u2113ab 16: if Cp = Cq then 17: return CONFUSE-RECURSION(u, v, r\u2217) 18: else if Cp < Cq and Count(rot(main(r)), p) > 0 then 19: return CONFUSE-RECURSION(u, v, r\u2217) 20: else if Cp > Cq and Count(rot(main(r)), q) > 0 then 21: return CONFUSE-RECURSION(u, v, r\u2217) 22: else 23: return false Corollary 12. Suppose that x and y are two words such that r = R\u2a7d3(x) = R\u2a7d3(y) and r contains at least three distinct symbols. Set p = Ext(Reg(r), x) and q = Ext(Reg(r), y). Suppose further that Count(main(r), R\u2a7d2(p)) < Count(main(r), R\u2a7d2(q)). Then x and y are confusable if and only if Count(rot(main(r)), p) > 0 and x \\ \u2217Pref(r, x) and y \\ \u2217Pref(r, y) are confusable. B. Confusability Algorithm Based on the ideas in the previous subsection, we formally describe our algorithm CONFUSE in Algorithm 1 and demonstrate that the running time is linear. In the following analysis, we assume the de\ufb01nitions in Algorithm 1. \u2022 main(r) and Reg(r) can be computed in constant time. This is clear from de\ufb01nition. \u2022 The pre\ufb01xes p and q can be computed in time linear in |p| and |q|, respectively. From Proposition 3, we have that D\u2217 \u2a7d3(Reg(r)) is a regular language. Since the length of Reg(r) is at most six, the \ufb01nite automaton generating the language has a constant number of vertices. Hence, the longest pre\ufb01xes of x and y that are accepted by the \ufb01nite automaton are precisely p and q, respectively. In Appendix D, we show that the sum of the lengths of all pre\ufb01xes that are computed in all recursive calls is at most 3(|x| + |y|). Therefore, the computation of all pre\ufb01xes takes linear time. \u2022 In the recursive calls, the root r\u2217can be computed in constant time. As mentioned earlier, the \ufb01rst step is to compute the roots of x and y in linear time. When the roots are equal to r and certain conditions are met, we make the recursive call CONFUSE-RECURSION. If the roots are recomputed for the shorter words u and v, then the overall running time for CONFUSE is quadratic. To avoid this, we observe that the roots for the shorter words are always equal and in fact, the common root r\u2217of u and v may be easily inferred from r. More precisely, if Reg(r) = w(abc)\u2113ab, then r\u2217= r \\ w(abc)\u2113. \u2022 In the recursive calls, determining r has at most two symbols can be done in constant time. If r has length at least four, then r necessarily has at least three distinct symbols. Otherwise, it contradicts the fact that r is irreducible. Therefore, it suf\ufb01ces to check if the \ufb01rst four symbols r contain three distinct symbols. In summary, from the above observations, we conclude that the running time is O(max{|x|, |y|}). 8 \fIV. \u2a7d3-TANDEM-DUPLICATION CODES We use insights gained from the previous section to construct (n, \u2a7d3; q)-TD codes. Motivated by the concept of roots, we consider a \u2a7d3-irreducible word r and we say that a (n, \u2a7d3; q)-TD code C is an (n, \u2a7d3; r)-TD code if all words in C belong to D\u2217 \u2a7d3(r). For r \u2208Irr\u2a7d3(i, q) with i \u2a7dn, suppose that C(n, r) is an (n, \u2a7d3; r)-TD code. Then Proposition 2 implies that S r\u2208Irr\u2a7d3(i,q), i\u2a7dn C(n, r) is an (n, \u2a7d3; q)-TD code. Trivially, {\u03ben\u2212i(r)} is an (n, \u2a7d3; r)-TD code for all x \u2208 Irr\u2a7d3(i, q). Taking the union of these codes, we recover Construction 1. Therefore, in the rest of this section, our objective is to provide estimates on the size of an optimal (n, \u2a7d3; r)-TD code for \ufb01xed \u2a7d3-irreducible word r. To simplify our discussion, we focus on the case where q = 3 and let T(n) and T(n, r) to denote the sizes of an optimal (n, \u2a7d3; 3)-TD code and an optimal (n, \u2a7d3; r)-TD code, respectively. Pick x \u2208D\u2217 \u2a7d3(r). Using concepts from Section III, we de\ufb01ne the following combinatorial characterisation of x. Set x1 = x and r1 = r. For i \u2a7e2, set xi = xi\u22121 \\ \u2217Pref(ri\u22121, xi\u22121) and ri = R\u2a7d3(xi). We terminate this recursion when rm+1 has less than three distinct symbols. If rm is the last root to have three distinct symbols, we say that r has m regions and for 1 \u2a7di \u2a7dm, de\ufb01ne pi = Ext(Reg(ri), xi), ti = main(ri), ci = Count(ti, R\u2a7d2(pi)), and \u03b4i = ( +, if Count(rot(ti), pi) > 0, \u2212, otherwise. For the word x, we de\ufb01ne its label by Label(x) = (r, (c1, \u03b41), (c2, \u03b42), . . . , (cm, \u03b4m)). Example 7. Consider r = 01210. Then r has two regions with t1 = 012 and t2 = 210. Consider also the words 01210210, 01201210, and 01111210 that belongs to D\u2217 \u2a7d3(01210). Then their labels are as follow. Label(01210210) = (01210, (1, +), (2, +)), Label(01201210) = (01210, (2, +), (1, +)), Label(01112110) = (01210, (1, \u2212), (1, \u2212)). Then it follows from Theorem 10, Corollaries 11 and 12 that 01210210 and 01210210 are confusable, while 01210210 and 01112110 are not confusable. More generally, the following proposition is immediate from Theorem 10, Corollaries 11 and 12. Proposition 13. Let x and y belong to D\u2217 \u2a7d3(r). Suppose that Label(x) \u225c \u0010 r, \u0010 c(x) 1 , \u03b4(x) 1 \u0011 , \u0010 c(x) 2 , \u03b4(x) 2 \u0011 , . . . , \u0010 c(x) m , \u03b4(x) m \u0011\u0011 , Label(y) \u225c \u0010 r, \u0010 c(y) 1 , \u03b4(y) 1 \u0011 , \u0010 c(y) 2 , \u03b4(y) 2 \u0011 , . . . , \u0010 c(y) m , \u03b4(y) m \u0011\u0011 . Then x and y are not \u2a7d3-confusable if and only if for some 1 \u2a7di \u2a7dm, \u0010 c(x) i < c(y) i and \u03b4(x) i = \u2212 \u0011 or \u0010 c(x) i > c(y) i and \u03b4(y) i = \u2212 \u0011 . Immediate from Proposition 13 are certain suf\ufb01cient conditions for two words to be confusable. Corollary 14. Let x, y \u2208D\u2217 \u2a7d3(r) and their labels Label(x) and Label(y) be as de\ufb01ned in Proposition 13. (i) If c(x) i = c(y) i for all i, then x and y are \u2a7d3-confusable. (ii) If \u03b4(x) i = \u03b4(y) i = + for all i, then x and y are \u2a7d3-confusable. From Corollary 14, we use the number of integer solutions to certain equations as an upper bound for T(n, r). As this combinatorial argument is fairly technical, we state the upper bound and defer the proof to Appendix E. Proposition 15. Let i \u2a7dn. Suppose that r \u2208Irr\u2a7d3(i, 3) has m regions. Then T(n, r) \u2a7dU(n, i, m) \u225c (\u0000(n\u2212i)/3+m m \u0001 \u2212 \u0000(n\u2212i)/3+m\u22121 m\u22121 \u0001 + 1, if 3 divides n \u2212i, \u0000\u230a(n\u2212i)/3\u230b+m m \u0001 , otherwise. Immediate from Proposition 15 is that T(|r|, r) = T(|r| + 1, r) = T(|r| + 2, r) = 1. Next, we provide lower bounds for T(n, r) by constructing (n, \u2a7d3; r)-TD codes. We \ufb01rst show that there exists an (n, \u2a7d3; r)-TD code of size two. 9 \fConstruction 2. Let r \u2208Irr\u2a7d3(i, 3) for some i \u2a7e4. Set C2 \u225c \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 {r1r2r3r1r2r3r4 \u00b7 \u00b7 \u00b7 ri, r1r2r2r3r3r4r4 \u00b7 \u00b7 \u00b7 ri}, if r1 \u0338= r3, {r1r2r1r4r2r1r4r5 \u00b7 \u00b7 \u00b7 ri, r1r2r1r1r4r4r5r5 \u00b7 \u00b7 \u00b7 ri}, if r1 = r3 and i \u2a7e5 {r1r2r1r4r2r1r4, r1r2r1r1r4r4r4}, if r1 = r3 and i = 4 Then C2 is an (i + 3, \u2a7d3; r)-TD code of size two. By Proposition 15, C2 is optimal. Proof. Let C2 = {x, y} and the labels of the words be as given in Proposition 13. Observe that in all cases, \u0010 c(x) 1 , \u03b4(x) 1 \u0011 = (2, +) and \u0010 c(y) 1 , \u03b4(y) 1 \u0011 = (1, \u2212). Proposition 13 then implies that x and y are not confusable. \u25a0 Following the above construction, we have a improvement to Construction 1. Corollary 16. For n \u2a7e6, we have that T(n) \u2a7e X 1\u2a7di\u2a7d3, or n\u22122\u2a7di\u2a7dn Irr\u2a7d3(i, 3) + 2 n\u22123 X i=4 Irr\u2a7d3(i, 3). Proposition 13 suggests that we examine code constructions according to the number of regions of r. When |r| \u2a7d2, the number of regions is zero and hence T(n, r) = 1. We consider the smallest nontrivial case where r has exactly one region. Without loss of generality, assume that r \u2208R \u225c{012, 0120, 01201, 1012, 10120, 101201, 0121, 01202, 012010, 10121, 101202, 1012010}, and so, t1 = 012. In this case, we have the following construction whose proof is deferred to Appendix F. Construction 3. For r \u2208R and \u2113\u2a7e1, de\ufb01ne the words x(r, \u2113) and z(r, \u2113) with the following rule. r x(r, \u2113) z(r, \u2113) r x(r, \u2113) z(r, \u2113) 012 0(112200)\u2113\u22121112 (012)\u2113 1012 10(112200)\u2113\u22121112 1(012)\u2113 0120 0(112200)\u2113\u2212111220 (012)\u21130 10120 10(112200)\u2113\u2212111220 1(012)\u21130 01201 0(112200)\u2113\u221211122001 (012)\u211301 101201 10(112200)\u2113\u221211122001 1(012)\u211301 0121 0(112200)\u2113\u221211121 (012)\u21131 10121 10(112200)\u2113\u221211121 1(012)\u21131 01202 0(112200)\u2113\u22121112202 (012)\u211302 101202 10(112200)\u2113\u22121112202 1(012)\u211302 012010 0(112200)\u2113\u2212111220010 (012)\u2113010 1012010 10(112200)\u2113\u2212111220010 1(012)\u2113010 For r \u2208R and n \u2a7e|x(r, 2)|, set \u2113z = \u230a(n \u2212|r|)/3\u230b+ 1 and Cr(n) \u225c \b \u03ben\u2212|x(r,\u2113)|(x(r, \u2113)) : |x(r, \u2113)| \u2a7dn \t \u222a \b \u03ben\u2212|z(r,\u2113z)|(z(r, \u2113z)) \t . Then Cr(n) is an (n, \u2a7d3; r)-TD code. Furthermore, Cr(n) in optimal. Therefore, if we set n2 = |x(r, 2)|, we have T(r, n) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 \u0004 n\u2212n2 6 \u0005 + 3 if n \u2a7en2, 2, if |r| + 3 \u2a7dn < n2, 1, otherwise. Hence, the value of T(n, r) is completely determined whenever r has at most one region. Combining this result with Proposition 15, we have the following theorem. Theorem 17. Let I(i, m) denote the number of irreducible words in Irr\u2a7d3(i, 3) with exactly m regions, and U(n, i, m) be as de\ufb01ned in Proposition 15. Then T(n) \u2a7d X r\u2208R T(n, r) + n X i=5 i X m=2 I(i, m)U(n, i, m). (1) In addition to the above constructions, we construct tandem-duplication codes for small lengths by searching for them exhaustively. Speci\ufb01cally, \ufb01x r \u2208Irr\u2a7d3(i, 3) and n \u2a7ei, we construct the graph G(n, r), whose vertices correspond to the set of all labels of descendants of r of length n, or {L : L = Label(x) and x \u2208\u03a3n 3 \u2229D\u2217 \u2a7d3(r)}. Two vertices or labels L1 and L2 are connected in G(n, r) if and only if the words whose labels are L1 and L2 are not \u2a7d3-confusable. Hence, a clique of size M in the graph G(n, r) correspond to an (n, \u2a7d3; r)-TD code of size M and T(n, r) is the maximum size of a clique in G(n, r). 10 \fTABLE I: Estimates and Exact Values for T(n) Length Constr. 1 This paper Eq (1) Prop. 4 Length Constr. 1 This paper Eq (1) Prop. 4 1 3 3 3 3 16 5985 10641 12267 15495 2 9 9 9 9 17 8781 16287 19479 25077 3 21 21 21 21 18 12879 25005 30957 40581 4 39 39 39 39 19 18885 38223 49245 65667 5 69 69 69 69 20 27687 57957 78417 106257 6 111 117 117 117 21 40587 83619 125001 171933 7 171 195 195 195 22 59493 116145 199467 278199 8 261 315 315 321 23 87201 166761 318621 450141 9 393 495 495 525 24 127809 249159 509457 728349 10 585 777 777 855 25 187323 375129 815361 1178499 11 867 1221 1227 1389 26 274545 558573 1306107 1906857 12 1281 1887 1941 2253 27 402375 813771 2093967 3085365 13 1887 2913 3075 3651 28 589719 1164309 3359685 4992231 14 2775 4527 4875 5913 29 864285 1675935 5394369 8077605 15 4077 6969 7731 9573 30 1266681 2464419 8667075 13069845 Optimal values of T(n) are highlighted in bold. We use the exact algorithm MaxCliqueDyn [15] to determine the maximum size of the clique in these graph G(n, r) for n \u2a7d20. Since T(n) = P r\u2208Irr\u2a7d3(i,3), i\u2a7dn T(n, r), we tabulate the results T(n) in Table I. Remark 1. \u2022 Such a method to compute T(n, r) is only possible because we have developed the algorithm to determine confusability in Section III. Prior to this work, the necessary conditions for \u2a7d3-confusability was not known and hence, methods to compute T(n, r) were not available. \u2022 For a \ufb01xed length and root, even though the set of descendants is huge, the set of all labels are signi\ufb01cantly smaller. Hence, the task of computing maximum cliques remains feasible. More concretely, when n = 20, the order of the largest graph G(20, r) is 366, despite the fact that the average size1 of \u03a320 3 \u2229D\u2217 \u2a7d3(r) is 320/27687 \u2248125935. Finally, we develop recursive constructions in the following proposition, whose proof is deferred to Appendix F. Proposition 18. Let r = r1r2 \u00b7 \u00b7 \u00b7 ri \u2208Irr\u2a7d3(i, 3). Then the following holds. T(n, r) \u2a7e \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 T(n \u22121, r \\ r1), if r1 = r3, max{2T(n \u22124, r \\ r1), 3T(n \u22128, r \\ r1)}, if r1 \u0338= r3, r1 \u0338= r4, max{2T(n \u22125, r \\ r1r2), 3T(n \u221210, r \\ r1r2)}, if r1 \u0338= r3, r1 = r4, r2 \u0338= r5, max{2T(n \u22126, r \\ r1r2r3), 3T(n \u221212, r \\ r1r2r3)}, if r1 \u0338= r3, r1 = r4, r2 = r5. Furthermore, T(n, r) \u2a7eT(n \u22121, r) and T(n, r) = T(n, rR), where zR denotes the reverse of word z. Using Proposition 18 with Constructions 2 and 3 and the values computed by MaxCliqueDyn, we derive lower bounds for T(n) for 21 \u2a7dn \u2a7d30. The results are summarized in Table I. In addition to the lower bounds for the code size T(n), we also compare the upper bounds in Proposition 4 and (1). Observe that (1) is tight up to lengths at most ten and the constructions in this paper improve the rates2 for Construction 1 by as much as 6.74%. V. DISCUSSION We studied the problem of determining the \u2a7dk-confusability of two words. Combining the results of this paper, we have linear-time algorithms to solve the confusability problem for k \u2208{1, 2, 3}. It remains open whether there exist ef\ufb01cient algorithms to determine \u2a7d4-confusability. One key obstacle is the fact that Proposition 2(iii) does not hold for k = 4. In particular, there exists x and y such that R\u2a7d4(x) \u0338= R\u2a7d4(y), but x and y are \u2a7d4-confusable. An example is provided by Jain et al. [13], where x = 012 and y = 0121012 belongs to Irr\u2a7d4(3), but 012101212 is a common descendant of x and y. Another approach is to consider the intersection of the descendant cones D\u2217 \u2a7d4(x)\u2229D\u2a7d4(y) as formal languages. Unfortunately, Leupold et al. [6] demonstrated that while the language D\u2217 \u2a7d4(x) is context free for all x, the language is not regular in general. However, given two context free languages L1 and L2, determining whether L1 \u2229L2 is empty is an undecidable problem (see, for example, Sipser [16, Exercise 5.32]). While this does not imply that the confusability problem is undecidable, we nevertheless conjecture that it is undecidable for k \u2a7e4. 1The number of \u2a7d3-irreducible ternary words of length at most twenty is 27687. 2The rate of a code C of length n is given by (log3 |C|)/n. 11 \fACKNOWLEDGEMENT We are grateful for an anonymous reviewer who proposed the de\ufb01nition of label given in Section IV. This de\ufb01nition has led to a more succint presentation and a spectrum of new results, including Proposition 15 and the computation of the size of optimal codes of lengths up to twenty."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1701.07872v2",
+                    "title": "Cooling Codes: Thermal-Management Coding for High-Performance Interconnects",
+                    "abstract": "High temperatures have dramatic negative effects on interconnect performance\nand, hence, numerous techniques have been proposed to reduce the power\nconsumption of on-chip buses. However, existing methods fall short of fully\naddressing the thermal challenges posed by high-performance interconnects. In\nthis paper, we introduce new efficient coding schemes that make it possible to\ndirectly control the peak temperature of a bus by effectively cooling its\nhottest wires. This is achieved by avoiding state transitions on the hottest\nwires for as long as necessary until their temperature drops off. We also\nreduce the average power consumption by making sure that the total number of\nstate transitions on all the wires is below a prescribed threshold. We show how\neach of these two features can be coded for separately or, alternatively, how\nboth can be achieved at the same time. In addition, error-correction for the\ntransmitted information can be provided while controlling the peak temperature\nand/or the average power consumption.\n  In general, our cooling codes use $n > k$ wires to encode a given $k$-bit\nbus. One of our goals herein is to determine the minimum possible number of\nwires $n$ needed to encode $k$ bits while satisfying any combination of the\nthree desired properties. We provide full theoretical analysis in each case. In\nparticular, we show that $n = k+t+1$ suffices to cool the $t$ hottest wires,\nand this is the best possible. Moreover, although the proposed coding schemes\nmake use of sophisticated tools from combinatorics, discrete geometry, linear\nalgebra, and coding theory, the resulting encoders and decoders are fully\npractical. They do not require significant computational overhead and can be\nimplemented without sacrificing a large circuit area.",
+                    "authors": "Yeow Meng Chee, Tuvi Etzion, Han Mao Kiah, Alexander Vardy",
+                    "published": "2017-01-26",
+                    "updated": "2017-10-23",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION Power and heat dissipation limits have emerged as a \ufb01rst-order design constraint for chips, whether targeted for battery-powered devices or for high-end systems. With the migration to process geometries of 65 nm and below, power dissipation has become as important an issue as timing and signal integrity. Aggressive technology scaling results in smaller feature size, greater packing density, increasing microarchitectural complexity, and higher clock frequencies. This is pushing chip level power consumption to the edge. It is not uncommon for on-chip hot spots to have temperatures exceeding 100\u00b0C, while inter-chip temperature differentials often exceed 20\u00b0C. Power-aware design alone is not suf\ufb01cient to address this thermal challenge, since it does not directly target the spatial and temporal behavior of the operating environment. For this reason, thermally-aware approaches have emerged as one of the most important domains of research in chip design today. High temperatures have dramatic negative effects on circuit behavior, with interconnects being among the most impacted circuit components. This is due, in part, to the ever decreasing interconnect pitch and the introduction of low-k dielectric insulation which has low thermal conductivity. For example, as shown in [3], the Elmore delay [15] of an interconnect increases 5% to 6% for every 10\u00b0C increase in temperature, whereas the leakage current grows exponentially with temperature. Therefore, minimizing the temperature of interconnects is of paramount importance for thermally-aware design. Figure 1. Block diagram of the proposed bus architecture A. Related Work Numerous encoding techniques have been proposed in the literature [4,9,10,31,38,43,44,46,48,49,54,55] in order to reduce the overall power dissipation consumption of both on-chip and off-chip buses. It is known to be of importance from practical point of view for over thirty years and optimization of the related integrated circuits were considered by the electric companies [20]. It is well established [11,34,45,50,54,55] that bus power is directly proportional to the product of line capacitance and the average number of signal state transitions on the bus wires. Thus the general idea is to encode the data transmitted over the bus so as to reduce the average number of transitions. For example, the \u201cbus-invert\u201d code of [49] potentially complements the data on all the wires, according to the Hamming distance between consecutive transmissions, thereby ensuring that the total number of state transitions on n bus wires never exceeds n/2. Unfortunately, encoding techniques designed to minimize power consumption, do not directly address peak temperature minimization. In order to reduce the temperature of a wire, it is not suf\ufb01cient to minimize its average switching activity. Rather, it is necessary to control the temporal distribution of the \f3 state transitions on the wire. To reduce the peak temperature of an interconnect, it is necessary to exercise such control for all of its constituent wires. In [54,55], the authors propose a thermal spreading approach. They present an ef\ufb01cient encoding scheme that evenly spreads the switching activity among all the bus wires, using a simple architecture consisting of a shiftregister and a crossbar logic. This is designed to avoid the situation where a few wires get hot while the majority are at a lower temperature. This spreading approach is further extended in [9,43] using on-line monitoring of the switching activity on all the wires. Thermal spreading can be regarded as an attempt to control peak temperature indirectly, by equalizing the distribution of signal transitions over all the wires. Finally, analysis from information theory point of view, which is highly related to our work, including solutions with data compression are given in several papers, e.g. [30,40,47]. B. Our Contributions As technology continues to scale, existing methods may fall short of fully addressing the thermal challenges posed by high-performance interconnects in deep submicron (DSM) circuits. In this paper, we introduce new ef\ufb01cient coding schemes that simultaneously control both the peak temperature and the average power consumption of interconnects. The proposed coding schemes are distinguished from existing state-of-the-art by having some or all of the following features: A. We directly control the peak temperature of a bus by effectively cooling its hottest wires. This is achieved by avoiding state transitions on the hottest wires for as long as necessary until their temperature decreases. B. We reduce the overall power dissipation by guaranteeing that the total number of transitions on the bus wires is below a speci\ufb01ed threshold in every transmission. C. We combine properties A and/or B with coding for improved reliability (e.g., for low-swing signaling), using existing error-correcting codes. To achieve these desirable features, we propose to insert at the interface of the bus specialized circuits implementing the encoding and decoding functions, denoted herein by E and D, respectively. This is illustrated in Figure 1. The various coding schemes introduced in this paper employ tools from various \ufb01elds such as combinatorics, graph theory, block designs, discrete geometry, linear algebra, and the theory of error-correcting codes. Nonetheless, in each case the resulting encoders/decoders E and D are ef\ufb01cient: they do not require signi\ufb01cant computational overhead and can be implemented without sacri\ufb01cing a large circuit area. This is especially true for Property A, where the complexity of encoding and decoding scales linearly with the number of wires. We consider both adaptive and nonadaptive (memoryless) coding schemes. The advantage of nonadaptive schemes is that they are easier to implement and do not require memory. The disadvantage is that it is not possible to implement Property A with nonadaptive encoding. For this reason, most of the coding schemes developed in this paper will be adaptive, based on the idea of differential encoding. Notably, however, all of our schemes require the encoder and decoder circuits to keep track of only one (the most recent) previous transmission. Unlike the thermal spreading methods of [43,54,55] that lead to irredundant coding schemes, the solutions we propose do introduce redundancy: we require n > k wires to encode a given k-bit bus. A key consideration in this \f4 situation is the area overhead due to the additional n \u2212k wires. Therefore, it is important to determine the theoretically minimum possible number of wires n needed to encode k bits while satisfying the desired properties. We provide full theoretical analysis in each case. We moreover show that the number of additional wires required to satisfy Property A becomes negligible when k is large. C. Thermal Model Chiang, Banerjee, and Saraswat [11] came up with an analytic model that characterizes thermal effects due to Joule heating in high-performance Cu/low-k interconnects, under both steady-state and transient stress conditions. Shortly thereafter, Sotiriadis and Chandrakasan [45] gave a power dissipation model for DSM buses. These two models, accounting for thermal and power effects separately, were later uni\ufb01ed and re\ufb01ned by Sundaresan and Mahapatra [50]. Finally, building upon this work, Wang, Xie, Vijaykrishnan, and Irwin [55] proposed a more accurate thermal-andpower model for DSM buses. In all these papers, an n-bit bus (illustrated in Figure 2) is modeled in terms wire (1) heat flow to substrate layer \u2026 wire (2) wire (n) Rspread Rrect Rinter tILD Figure 2. Geometry used for calculating Rspread, Rrect, and Rinter substrate at ambient temperature Rinter \u2026 C1 C2 Cn R1 R2 Rn P1 P2 Pn Figure 3. Equivalent thermal RC-network for a k-bit bus of the equivalent thermal-RC network in Figure 3. Sundaresan and Mahapatra [50] show that this thermal-RC network is governed by the following differential equations: P1 = C1 \u2202\u03b81 \u2202t + \u03b81 \u2212\u03b80 R1 + \u03b81 \u2212\u03b82 Rinter , (1) Pn = Ck \u2202\u03b8k \u2202t + \u03b8k \u2212\u03b80 Rk + \u03b8k \u2212\u03b8k\u22121 Rinter , and (2) Pi = Ci \u2202\u03b8i \u2202t + \u03b8i \u2212\u03b80 Ri + 2\u03b8i \u2212\u03b8i\u22121 \u2212\u03b8i+1 Rinter , (3) for i = 2, 3, . . . , n\u22121, where Pi is the instantaneous power dissipated by wire i, Ci is the thermal capacitance per unit length of wire i, Ri = Rspread+Rrect is the thermal resistance per unit length of wire i along the heat transfer path downwards, Rinter is the lateral thermal resistance used to account for the parallel thermal coupling effect between the wires, \u03b8i is the temperature of wire i, and \u03b80 is the substrate ambient temperature. \f5 In any bus model for which (1) \u2013 (3) hold, the temperature of a wire will increase whenever the wire undergoes a state transition; conversely, in the absence of state transitions, the temperature will gradually decrease. We let \u03c3i denote the switching activity of wire i, which is the number of times the wire changes state. Then the power dissipated by a bus is determined by its total switching activity \u03c31 + \u03c32 + \u00b7 \u00b7 \u00b7 + \u03c3n. In order to directly control the peak temperature of a bus by avoiding transitions on its hottest wires, we need to know which wires are the hottest at every transmission. There are two general ways to obtain this information. We can use an analytical model [50], such as (1) \u2013 (3), to estimate the current temperatures of the wires. For each wire, such an estimate can be implemented with a counter that is incremented on transition and decremented on non-transition, where the precise magnitude of the increments/decrements is determined by the model. Alternatively, we can have actual temperature sensors for each wire. For DSM buses, accurate temperature sensing can be implemented using, for example, ring oscillators [13]. As shown in [13], sensors based on ring oscillators provide a resolution of 1\u00b0C while consuming an active power of only 65\u2013112\u00b5W. D. Organization The rest of this paper is organized as follows. We begin in the next section with a precise formulation of the coding problems that result from the thermal-management features we propose to implement \u2014 namely, Properties A, B, and C. In Section III, we present a nonadaptive coding scheme that combines Property B (reducing the average power dissipation) with the thermal spreading approach of [55]. Our constructions in Section III are based on the notions of anticodes and quorum systems, and use key results from the theory of combinatorial designs. Section IV is devoted to Property A: we show how state transitions on the t hottest wires can be avoided by using only t + 1 additional bus lines. This optimal construction is based on combining differential coding with the notion of spreads and partial spreads in projective geometry. The optimal construction can be applied when t + 1 \u2a7d(n + 1)/2. When t + 1 > (n + 1)/2 we use another technique from the theory of error-correcting codes to construct ef\ufb01cient codes. The designed codes can be viewed as sun\ufb02owers, while the partial spreads, are also sun\ufb02owers, and hence can be also viewed as a special case of these codes. The technique used is generalized with the notion of generalized Hamming weights. In Section V, we show how Properties A and B can be all achieved at the same time. That is, we design coding schemes that simultaneously control peak temperature and average power consumption in every transmission. For this purpose, we present three types of constructions. The \ufb01rst construction is based upon the Baranyai theorem on complete hypergraph decomposition into pairwise disjoint perfect matchings. The second construction is based on concatenation of low weights codes based on appropriate non-binary dual codes or nonbinary partial spreads. The third construction is the previous sun\ufb02ower construction, which also satisfy Property B. Section VI is devoted to codes which satisfy Property C simultaneously with either Property A or Property B or both, i.e. we add also correction for possible transmission errors on the bus wires. The constructions in this section will also be of two types. The \ufb01rst type of constructions is based on resolutions in block design. The second type of constructions will be to employ the previously given constructions, where our set of transitions is restricted to the set of codewords in a given error-correcting code. In all these sections our bounds and constructions are applied for in\ufb01nite sets of parameters, but there is no asymptotic analysis in any of these cases. The asymptotic analysis is postponed to Section VII, where the asymptotic behavior of our codes is analyzed. In particular we analyze area \f6 overhead of our constructions and prove that when k is large enough, the additional number of wires required to satisfy the desired properties is negligible. Finally, Section VIII summarizes our comprehensive work and presents a brief discussion of possible directions for future research. We would like to remark that this work is mainly of theoretical nature. A follow up work will present more constructions, especially for practical parameters. Examples and numerical experiments will be given and also comparison between the various constructions with emphasis on practical parameters. It will be illustrated and discussed how practical our methods and constructions are. II. PROBLEM FORMULATION AND PRELIMINARIES Let us now elaborate upon Properties A, B, C introduced in the previous section. For each of these properties, we will characterize the performance of the corresponding coding scheme by a single integer parameter. All of our coding schemes will use n > k wires to encode a k-bit bus. We assume that communication across the bus is synchronous, occurring in clocked cycles called transmissions. This leads to the following de\ufb01nition. De\ufb01nition 1. Consider a coding scheme for communication over a bus consisting of n wires. Let t, w, e be positive integers less than n. We say that the coding scheme has Property A(t): if every transmission does not cause state transitions on the t hottest wires; Property B(w): if the total number of state transitions on all the wires is at most w, in every transmission; Property C(e): if up to e transmission errors (0 received as 1, or 1 received as 0) on the n wires can be corrected. We presume that, at the time of transmission, it is known which t wires are the hottest; Property A(t) is required to hold assuming that any t wires can be designated as the hottest. The values of t, w, e are design parameters, to be determined by the speci\ufb01c thermal requirements of speci\ufb01c interconnects. The proposed coding schemes will work for various values of t, w, and e. Nevertheless, it might be helpful to think of t as a small constant, since signi\ufb01cant reductions in the peak temperature can be achieved by cooling only a few of the hottest wires. Thus, the most important values of t are small ones, say, less than half of the bus wires. But, our constructions in the following sections will consider also solutions for large values of t, speci\ufb01cally, any value of t. Similarly, w is also usually small since large w means a large number of state transitions on all the wires, which might result in too many hot wires. Finally, we also expect e to be small, especially as it must be smaller than w/2 as otherwise we won\u2019t be able to correct the errors. Codes which satisfy Properties A(t),B(w), and C(e) simultaneously in every transmission, will be called (n, t, w, e)low-power error-correcting cooling codes or (n, t, w, e)-LPECC codes for short. When a nonempty meaningful subset of the three properties (property C(e) is not interesting alone in our context) will be satis\ufb01ed, only the parameters and description related to this subset of properties will remain in the name. For example, (n, t, e)-LPEC codes stands for (n, t, e)-low-power error-correcting codes. Six such nonempty subsets exist and for each one we suggest coding schemes and related codes. It the conclusion of Section VIII a pointer will be given, where each one of these six subsets was considered. We view the collective state of the n wires during each transmission as a binary vector x = (x1, x2, . . . , xn). The set of all such binary vectors is the Hamming n-space H(n) = {0, 1}n. We will identify H(n) with the vector \f7 space Fn 2 . Given any x, y \u2208Fn 2 , the Hamming distance d(x, y) is the number of positions where x and y differ. The Hamming weight of a vector x \u2208Fn 2 , denoted wt(x), is the number of nonzero positions in x. Conventionally, a binary code C of length n is simply a subset of Fn 2 . The elements of C are called codewords. Given a code C, its minimum distance d(C) and diameter diam(C) are de\ufb01ned as follows: d(C) def = min x,y\u2208C d(x, y) and diam(C) def = max x,y\u2208C d(x, y) . Later, in Sections IV and V, we will need to modify and generalize this conventional de\ufb01nition of binary codes in an important way. This modi\ufb01cation will be needed for codes which satisfy Property A(t). III. NONADAPTIVE LOW-POWER CODES The encoding schemes considered in this section belong to the nonadaptive kind, in that the choice which codeword to transmit across the bus in the current transmission does not depend on codewords that have been transmitted earlier. Such coding schemes are also known as memoryless. The advantage of nonadaptive schemes is that they are simpler to implement: they do not need a continuously changing data model, and they do not require memory to track the history of previous transmissions. In the nonadaptive case, an n-bit coding scheme for a source S \u2286Fk 2 is a triple E = \u27e8C, E, D\u27e9, where 1) C is a binary code of length n, 2) E: S \u2192C is a bijective map called an encoding function, 3) D: C \u2192S is a bijective map called a decoding function, such that D \u0000E(u) \u0001 = u for all u \u2208S. Encoding and decoding circuits that implement E and D are inserted at the interface of the bus (see Figure 1). Suppose u, v \u2208S are two words that are to be communicated across the bus during consecutive transmissions. In the absence of a coding scheme, the total switching activity of the bus is then given by |{i : ui \u0338= vi}|. This is precisely the Hamming distance d(u, v), which could be as high as k. If an n-bit coding scheme is used, then x = E(u) and y = E(v) are transmitted instead. The resulting total switching activity of the bus is therefore d(x, y), which is upper bounded by diam(C). It follows that the coding scheme satis\ufb01es Property B(w) if and only if diam(C) \u2a7dw. As the power consumption of a bus is directly related to its total switching activity, we call such a code C an (n, w)-low-power code ((n, w)-LP code for short). In this section, we are interested in (n, w)-LP codes that also achieve low peak temperatures by spreading the switching activities among the bus wires as uniformly as possible. In doing so, we are following the analysis of [10, 54,55] and the resulting thermal spreading approach [10,43,54]. In order to quantify the thermal spreading achieved by a given coding scheme E = \u27e8C, E, D\u27e9, let us treat the source S as a random variable taking on values in Fk 2 , and assume that S is uniformly distributed. This is a common assumption in bus analysis \u2014 see, for example, [45]. With this assumption, the expected switching activity of wire i is given by \u00b5i = 1 |S|2 \u2211 u,v\u2208S \f \fE(u)i \u2212E(v)i \f \f = 2ri(|C| \u2212ri) |S|2 (4) \f8 where ri is the number of codewords (x1, x2, . . . , xn) \u2208C such that xi = 1. If \u00b51, \u00b52, . . . , \u00b5n are all equal, we say that the code C is thermal-optimal, since the expected switching activities of the bus wires are then uniformly distributed. This leads to the following problem: Given n and w, determine the maximum size of a thermal-optimal (n, w)-low-power code (5) The size of C is important because we wish to minimize the area overhead introduced by our coding scheme. This overhead is largely determined by the number n \u2212k of additional wires that we need to encode a given source S \u2286Fk 2 . Clearly, to encode such a source, we need a code C with |C| \u2a7e2k. It is easy to see from (4) that \u00b51, \u00b52, . . . , \u00b5n are all equal if and only if r1, r2, . . . , rn are all equal. Hence in a thermal-optimal code C, the number of codewords (x1, x2, . . . , xn)\u2208C having xi = 1 is the same for all i. Such codes are said to be equireplicate in the combinatorics literature. To construct such codes, we will need tools from the theory of set systems as was suggested by Chee, Colbourn, and Ling [10]. A. Set Systems Given a positive integer n, the set {1, 2, . . . , n} is abbreviated as [n]. For a \ufb01nite set X and k \u2a7d|X|, we de\ufb01ne 2X def = \b A : A \u2286X \t and \u0012X k \u0013 def = \b A \u22082X : |A| = k \t A set system of order n is a pair (X, A), where X is a \ufb01nite set of n points and A \u22862X. The elements of A are called blocks. A set system (X, 2X) is a complete set system. The replication number of x \u2208X is the number of blocks containing x. A set system is equireplicate if its replication numbers are all equal. There is a natural one-to-one correspondence between the Hamming n-space Fn 2 and the complete set system ([n], 2[n]) of order n. For a vector x = (x1, x2, . . . , xn) \u2208Fn 2 , the support of x is de\ufb01ned as supp(x) def = \b i \u2208[n] : xi \u0338= 0 \t With this, the positions of vectors in Fn 2 correspond to points in [n], each vector x \u2208Fn 2 corresponds to the block supp(x), and d(x, y) = | supp(x) \u25b3supp(y)|, where \u25b3stands for the symmetric difference. It follows from the above that there is a 1-1 correspondence between the set of all codes of length n and the set of all set systems of order n. Thus we may speak of the set system of a code or the code of a set system. B. Thermal-Optimal Low-Power Codes The set system ([n], A) of a thermal-optimal (n, w)-low-power code is de\ufb01ned by the following properties: 1) |A1 \u25b3A2| \u2a7dw for all A1, A2 \u2208A, and 2) ([n], A) is equireplicate. It follows that our problem in (5) can be recast as an equivalent problem in extremal set systems, as follows: Given n and w, determine T(n, w), the maximum size of an equireplicate set system (X, A) of order n such that |A1 \u25b3A2| \u2a7dw for all A1, A2 \u2208A (6) If the equireplication condition is removed, the resulting set system is known as an anticode of length n and diameter w. Hence, thermal-optimal low-power codes are equivalent to equireplicate anticodes. Anticodes in \f9 general, and the size of anticodes of maximum size have been a subject of intensive research in coding theory, see [1,2,8,16,35,42] and references therein. The determination of equireplicate anticodes of maximum size appears to be a new problem, also to the combinatorics and coding theory communities. However, the maximum size of an anticode has been completely determined by Kleitman [29], and even earlier by Katona [27], in a different but equivalent setting. Thus the following theorem is from [27] and [29]. Theorem 1. Let T(n, w) be the maximum number of blocks in a set system ([n], A) with |A1 \u25b3A2| \u2a7dw for all A1, A2 \u2208A. Then T(n, w) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 w/2 \u2211 i=0 \u0012n i \u0013 if w \u22610 (mod 2) \u0012n\u22121 w\u22121 2 \u0013 + w\u22121 2 \u2211 i=0 \u0012n i \u0013 if w \u22611 (mod 2) . For all even w, an extremal set system ([n], A) with T(n, w) blocks is given by: A = w/2 [ i=0 \u0012[n] i \u0013 . (7) If w is odd, let x be any \ufb01xed element of [n]. Then an extremal set system ([n], A) is given by: A = w\u22121 2 [ i=0 \u0012[n] i \u0013 [ ( A \u222a{x} : A \u2208 \u0012[n]\\{x} w\u22121 2 \u0013) . (8) We observe here that when w is even, the extremal set system in Theorem 1 is equireplicate. It consists of all the vectors of length n and weight at most w/2. Hence, we have the following result, which solves (5) and (6) for even w. Corollary 2. T(n, w) = w/2 \u2211 i=0 \u0012n i \u0013 when w \u22610 (mod 2) . The situation when w is odd is much more dif\ufb01cult. The set system in (8) is not equireplicate. In particular, we do not know if there exists an equireplicate anticode of order n and diameter w having size T(n, w). Hence, from Theorem 1 we can derive only that for all odd w, we have: w\u22121 2 \u2211 i=0 \u0012n i \u0013 \u2a7dT(n, w) \u2a7d \u0012n\u22121 w\u22121 2 \u0013 + w\u22121 2 \u2211 i=0 \u0012n i \u0013 . (9) The left hand side of the equation is obtained from a code which consists of all the vectors of length n and weight at most (w \u22121)/2. The right hand side is obtained from the upper bound on T(n, w) given in Theorem 1. The next three propositions establishes some exact values of T(n, w) for odd w. Corollary 3. T(n, 1) = 1 for n \u2a7e2 . Proof: Since the distance between two different vectors of length n and weight one is two, it follows that T(n, 1) = 1 when n \u2a7e2. A code with maximum size consists of the unique all-zero vector of length n. \f10 Proposition 4. T(n, n\u22121) = 2n\u22121 for n \u2a7e3 . Proof: When the distance between two codeword is at most n \u22121, the code cannot contain two complement codewords and hence its size is at most 2n\u22121. For odd n, an equireplicate set system of size 2n\u22121 is obtained from all vectors of length n and even weight. For even n, we give the following construction (which also works for any odd n \u2a7e5 if induction is applied). Let ([n \u22121], A) be a set system which attains T(n \u22121, n \u22122) = 2n\u22122 and each element is contained in 2n\u22123 blocks. We de\ufb01ne the following set system ([n], B). B def = {X \u222a{n} : X \u2208A} [ {X : X \u2208A} . We claim that B is an equireplicate set system which attains T(n, n \u22121) = 2n\u22121 and each element is contained in 2n\u22122 blocks of B. Clearly, |B| = 2|A| = 2n\u22121 and the fact that A does not contain complement blocks immediately implies that also B does not contain complement blocks. Finally, A is equireplicate set system of size 2n\u22122 and each element of [n \u22121] is contained in 2n\u22123 blocks of A. Therefore, each i \u2208[n] is contained in 2n\u22122 blocks in B. Hence, B is an equireplicate set system which attains T(n, n \u22121) = 2n\u22121. Proposition 5. T(n, 3) = n + 1 for n \u2a7e5 . Proof: First note, that by Proposition 4 we have T(4, 3) = 8. Also, T(n, 2) = \u22111 i=0 (n i) = n + 1 for all n \u2a7e2. Finally, it is easy to verify that T(5, 3) = 6, These facts will be used in the current proof that T(n, 3) = n + 1 if n > 4. Let C be the largest possible equireplicate anticode of length n > 5 and diameter 3. Let x and z be two codewords of C such that d(x, z) = 3. W.l.o.g. x and z differ in the last three coordinates and the \ufb01rst n \u22123 coordinates in both have x1, x2, ..., xn\u22123. Let \u03b1\u03b2\u03b3 and \u00af \u03b1 \u00af \u03b2 \u00af \u03b3 be the values of the last three coordinates in x and z, respectively. There is no other codeword in C which ends with either \u03b1\u03b2\u03b3 or \u00af \u03b1 \u00af \u03b2 \u00af \u03b3 since such a codeword should also start with x1, x2, ..., xn\u22123, to avoid distance greater than 3 from either x or z, and hence such a codeword will be equal to either x or z. Since each one of the last three columns has at least one zero and at least one one and the anticode is equireplicate, it follows that the weight of a column is at least 1 and at most |C| \u22121. The Hamming distance of any two of 100, 010, 001, 111 is 2 and hence the pre\ufb01xes of length n \u22123 related to the codewords ending with these suf\ufb01ces differ in at most one coordinate. Since anticode with diameter one has at most two codewords, it follows that all codewords with these suf\ufb01ces (if differ) have different values in exactly the same coordinate (in the pre\ufb01x of length n \u22123). The same argument holds also for the codewords ending with 011, 101, 110, and 000 (they have the same values in n \u22124 out of the \ufb01rst n \u22123 coordinates). Note that since the suf\ufb01x of either x or z is in {100, 010, 001, 111} and the other suf\ufb01ex is in {011, 101, 110, 000}, it follows that all the other n \u22125 coordinates (which don\u2019t have different values) of x and z have the same values for all the codewords. Since n > 5, it follows that each one of the n \u22125 coordinates (which exist) forms either a column of zeroes or a column of ones. If the column consists of zeroes we have a contradiction since the weight of the column is 0 which is smaller than 1. If the column consists of ones we have a contradiction since the weight of the column is |C| which is greater than |C| \u22121. Thus, for n > 5 there are no two codewords for which the Hamming distance is three. and hence for n > 6, T(n, 3) = T(n, 2) = n + 1, which completes the proof. \f11 For other odd values of w, we start with the extremal anticode A of diameter w \u22121 in (7) and add blocks to A while maintaining the equireplication requirement. Such blocks must contain exactly (w+1)/2 points to make sure that their distance with the blocks of A with (w \u22121)/2 points, or less, will not exceed w + 1. Any two blocks with (w+1)/2 points must intersect in at least one point as otherwise their distance will be w + 1. Interestingly, these properties precisely de\ufb01ne a regular uniform quorum system of rank (w+1)/2. Quorum systems have been studied extensively in the literature on fault-tolerant and distributed computing \u2014 see [53] for a recent survey. There are many types of such systems. For example, if (w + 1)/2 = q + 1, n = q2 + q + 1, and q is a prime power, then an optimal such system consists of q2 + q + 1 blocks which form a projective plane of order q [52, p. 224]. In other words Proposition 6. If w = 2q + 1, q is a prime power, then T(q2 + q + 1, 2q + 1) = q \u2211 i=0 \u0012q2 + q + 1 i \u0013 + q2 + q + 1 . For a proof of this proposition and other similar constructions, we refer the reader to the work in [12], where such systems are constructed from combinatorial designs. IV. COOLING CODES Unfortunately, it is not possible to satisfy Property A(t) with nonadaptive coding schemes, even for t = 1. Indeed, suppose we wish to avoid state transitions on just the one hottest wire, say wire i. If the encoder does not know the current state of wire i, the only way to guarantee that there is no state transition is to have xi = yi for any two codewords (x1, x2, . . . , xn) and (y1, y2, . . . , yn). Since any of the n wires could be the hottest, we must have (x1, x2, . . . , xn) = (y1, y2, . . . , yn). Thus all codewords are the same, and no communication is possible. In this section, we shall see that if the encoder and decoder keep track of just one previous transmission then Property A(t) can be satis\ufb01ed for any t with only t + 1 additional wires if 2(t + 1) \u2a7dn, by using spreads and partial spreads, notions from projective geometry. If t + 1 > n/2 we propose a construction based on a sun\ufb02ower for which the construction for 2(t + 1) \u2a7dn can be viewed as a special case. Finally, we provide a road map for our best lower bounds on the size of (n, t)-cooling codes in general, and in particular when 1 \u2a7dt < n \u2a7d100. A. Differential Encoding and Decoding The main idea of our differential encoding method is to encode the data to be communicated across the bus in the difference between the current transmission and the previous one. Similar ideas have been used in digital communications and in magnetic recording, among other applications. Why is differential coding useful? The most useful feature in our context is this. When we use the differential encoding method to transmit a codeword x = (x1, x2, . . . , xn), there is a state transition on wire i if and only if xi = 1, and the total number of transitions is precisely wt(x), the Hamming weight of x. This makes it possible to reduce the area overhead signi\ufb01cantly beyond the best overhead achievable with nonadaptive schemes. For example, under differential encoding, a code C satis\ufb01es Property B(w) \u2014 and so is an (n, w)-LP code \u2014 if and only if wt(x) \u2a7dw for all x \u2208C. It follows that the thermal-optimal (n, w)-LP code of maximum size is given by J+(n, w) def = \b x \u2208{0, 1}n : wt(x) \u2a7dw \t (10) \f12 This set, distinguished from the Johnson space de\ufb01ned by J(n, w)def ={x \u2208{0, 1}n : wt(x) = w}, is clearly equireplicate and its size is much larger than the size of the largest anticode of diameter w (cf. Theorem 1), for both odd and even w. B. De\ufb01nition of Cooling Codes Even under differential encoding, it is still not possible to satisfy Property A(t) with conventional binary codes. To see this, again suppose that we wish to avoid transitions on just the one hottest wire, say wire i. With differential encoding, in order to guarantee no state transitions on wire i while transmitting a codeword x = (x1, x2, . . . , xn), we must have xi = 0. But, once again, any of the n wires could be the hottest, which implies that x = 0. Since this must hold for any codeword, it follows that C = {0} and no communication is possible. Consequently, we henceforth modify our notion of a code C as follows. The elements of C will be sets of binary vectors of length n, say C1, C2, . . . , CM. We will refer to C1, C2, . . . , CM as codesets. We do not require these codesets to be of the same size, but we do require them to be disjoint: Ci \u2229Cj = \u2205for all i \u0338= j. The elements of each codeset Ci will be called codewords. The goal is to guarantee that no matter which codeset Ci is chosen, for each possible designation of t wires as the hottest, there is at least one codeword in Ci with zeros on the corresponding t positions. This leads to the following de\ufb01nition. De\ufb01nition 2. For positive integers n and t < n, an (n,t)-cooling code C of size M is de\ufb01ned as a set {C1, C2, . . . , CM}, where C1, C2, . . . , CM are disjoint subsets of Fn 2 satisfying the following property: for any set S \u2282[n] of size |S| = t and for all i \u2208[M], there exists a codeword x \u2208Ci with supp(x) \u2229S = \u2205. Given the foregoing de\ufb01nition of cooling codes, we also need to modify our notions of an encoding function and a decoding function, introduced in Section III. As before, we assume that the data to be communicated across the bus is represented by a source S taking on some M \u2a7d2k values in Fk 2 . In contrast to Section III, we no longer need to assume that S is uniformly distributed \u2014 in fact, no probabilistic model for S is required. On the other hand, the input to the encoding function E now comprises, in addition to a word u \u2208S, also a set S \u2282[n] of size t representing the positions of the t hottest wires. We let C def = C1 \u222aC2 \u222a\u00b7 \u00b7 \u00b7 \u222aCM . Then the output of the encoding function E is a vector x \u2208C such that supp(x) \u2229S = \u2205. For every possible S, the function E(\u00b7, S) is a bijective map from S to C. Since the codesets C1, C2, . . . , CM are disjoint, this allows the decoding function D to recover u \u2208S from the encoder output x \u2208C. We summarize the foregoing discussion in the next de\ufb01nition. De\ufb01nition 3. For integers n and t < n, an (n,t)-cooling coding scheme for a source S \u2286Fk 2 is a triple E = \u27e8C, E, D\u27e9, where 1) The code C is an (n, t)-cooling code; 2) The encoding function E: S \u00d7 \u0000[n] t \u0001 \u2192C is such that for all S \u2282[n] of size t and all u \u2208S, we have supp \u0000E(u, S) \u0001 \u2229S = \u2205; \f13 3) The decoding function D: C \u2192S is such that for all u \u2208S, we have D \u0000E(u, S) \u0001 = u regardless of the value of S. It follows immediately from Definition 3 that, under differential encoding, an (n, t)-cooling coding scheme satis\ufb01es Property A(t) by avoiding state transitions on the t hottest wires, which are represented by the subset S. Example. Consider n = 6 and t = 2. An (6, 3)-cooling code is given by the following eight codesets, C000 = {100000, 101011, 111010, 001011, 010001, 110001, 011010}, C001 = {010000, 100011, 011101, 110011, 111110, 101110, 001101}, C010 = {001000, 100111, 111000, 101111, 011111, 010111, 110000}, C011 = {000100, 100101, 011100, 100001, 111001, 111101, 011000}, C100 = {000010, 100100, 001110, 100110, 101010, 101000, 001100}, C101 = {000001, 010010, 000111, 010011, 010101, 010100, 000110}, C110 = {110110, 001001, 110101, 111111, 111100, 001010, 000011}, C111 = {011011, 110010, 101100, 101001, 011110, 000101, 110111}. We index the codesets by all three-bit messages. To verify Property A(2), we explicitly describe the encoding function E : S \u00d7 \u0000[6] 2 \u0001 \u2192C via the following lookup table. S 000 001 010 011 100 101 110 111 {1, 2} 001011 001101 001000 000100 000010 000001 001001 000101 {1, 3} 010001 010000 010111 000100 000010 000001 000011 000101 {1, 4} 001011 010000 001000 011000 000010 000001 001001 011011 {1, 5} 010001 010000 001000 000100 001100 000001 001001 000101 {1, 6} 011010 010000 001000 000100 000010 010010 001010 011110 {2, 3} 100000 100011 100111 000100 000010 000001 000011 000101 {2, 4} 100000 100011 001000 100001 000010 000001 001001 101001 {2, 5} 100000 001101 001000 000100 100100 000001 001001 101100 {2, 6} 100000 101110 001000 000100 000010 000110 001010 101100 {3, 4} 100000 010000 110000 100001 000010 000001 000011 110010 {3, 5} 100000 010000 110000 000100 100100 000001 110101 000101 {3, 6} 100000 010000 110000 000100 000010 010010 110110 110010 {4, 5} 100000 010000 001000 100001 101000 000001 001001 101001 {4, 6} 100000 010000 001000 011000 000010 010010 001010 110010 {5, 6} 100000 010000 001000 000100 100100 010100 111100 101100 C. Bounds on the Size of Cooling Codes In this subsection, we show that realizing an (n, t)-cooling coding scheme requires at least t + 1 additional wires. That is, the number of bits that can be communicated over an n-wire bus while satisfying Property A(t) is at most \f14 k \u2a7dn \u2212t \u22121. In the next subsection, we will present a construction that achieves this bound. Herein, let us begin with the following lemma. Lemma 7. Let C be an (n, t)-cooling code of size |C| = M. Then M \u2a7dt!(n\u2212t)! n! n\u2212t \u2211 w=0 \u0012n w \u0013\u0012n\u2212w t \u0013 = 2n\u2212t . (11) Proof: For convenience, we will refer to sets S \u2282[n] of size |S| = t as t-subsets. Given a t-subset S and a vector x \u2208Fn 2 , we shall say that x covers S if supp(x) \u2229S = \u2205. Observe that a vector of weight w covers exactly (n\u2212w t ) different t-subsets. Therefore, the total number of t-subsets (counted with multiplicity) covered by all the vectors in Fn 2 is given by N(n, t) def = n\u2212t \u2211 w=0 \u0012n w \u0013\u0012n\u2212w t \u0013 . Now consider a codeset C in C. By de\ufb01nition, for any t-subset S, there is at least one codeword x \u2208C that covers S. Hence, the total number of t-subsets (possibly counted with multiplicity) covered by all the codewords of C is at least (n t). Since this holds for each of the M codesets in C, the total number of t-subsets (again, counted with multiplicity) covered by all the vectors in C = C1 \u222aC2 \u222a\u00b7 \u00b7 \u00b7 \u222aCM is at least M(n t). But, since the codesets C1, C2, . . . , CM are disjoint and C \u2286Fn 2 , this number cannot exceed N(n, t), i.e. M(n t) \u2a7dN(n, t), and the lemma follows. The proof of Lemma 7 can be shorten by considering any given t-subset of coordinates that must be zeroes in at least one codeword of each codeset. There are 2n\u2212t such codewords of length n and hence the maximum number of codesets is 2n\u2212t. Nevertheless, there is one advantage in presenting the longer proof. It follows from the proof of Lemma 7 that an (n, t)-cooling code C of size |C| = 2n\u2212t would be perfect. In such a code, the codesets C1, C2, . . . , CM form a partition of Fn 2 and each of these codesets is a perfect covering of ([n] t ), i.e. all words with exactly t zeroes. Using these observations, we can prove that such cooling codes do not exist, unless t = 1 or t \u2a7en \u22121. Proposition 8. Perfect (n, t)-cooling codes do not exist, unless t = 1 or t \u2a7en \u22121. Proof: W.l.o.g. assume that the codeset C1 contains the codeword x which starts with a one followed by n \u22121 zeroes. The only words which are not covered by this codeword are all those words which start with a zero and have exactly t \u22121 zeroes in the other n \u22121 coordinates. Let z be one of these (n\u22121 t\u22121) words. A codeword y which covers z must starts with a zero. If y has at least t + 1 zeroes then it covers words with t zeroes which are also covered by x, and therefore C1 won\u2019t be a perfect covering. Hence, C1 contains x and the (n\u22121 t\u22121) codewords, which start with a zero, and have exactly t zeroes. Now, assume w.l.o.g. that C2 contains the codeowrd which starts with n \u22121 zeroes and ends with a one. With similar analysis as for C1 this codeset contains the (n\u22121 t\u22121) codewords, which end with a zero, and have exactly t zeroes. Now, by this analysis, both C1 and C2 must contain the (n\u22122 t\u22122) words which start and end with a zero and have exactly t zeroes. Therefore, if (n\u22122 t\u22122) > 0 a perfect code cannot exist. (n\u22122 t\u22122) > 0, unless t = 1 or t \u2a7en \u22121. \u2022 If t = 1, then there exists a perfect (n, 1)-cooling code, with 2n\u22121 codesets, each one contains a pair of complement codewords. \f15 \u2022 If t = n, then there exists a perfect (n, n)-cooling code, with one codeset which contains all codewords of length n. \u2022 If t = n \u22121 then there exists a perfect (n, n \u22121)-cooling code, with two codesets, one contains the all-zero codeword and the second contains all codewords of weight one. All other words can be partitioned arbitrarily between these two codesets. Lemma 7 and Proposition 8 imply the following result. Corollary 9. If 1 < t < n \u22121, then the size of any (n, t)-cooling code C is bounded by |C| \u2a7d2n\u2212t \u22121. Consequently, such a code cannot support the transmission of n \u2212t or more bits over an n-wire bus. Denote the maximum size of an (n, t)-cooling code by C(n, t). In the following subsection we will obtain lower bounds on C(n, t). We start in this subsection with a few simple bounds and values of C(n, t). Corollary 10. 1) C(n, 1) = 2n\u22121; 2) C(n, n \u22121) = 2; 3) If 2 \u2a7dt \u2a7dn \u22122, then C(n, t) \u2a7d2n\u2212t \u22121. Proposition 11. If 2 \u2a7dt \u2a7dn \u22122, then C(n, t) \u2a7en \u2212t + 1. Proof: For i \u2208[n \u2212t + 1], let Ci contain the set of all binary words of length n and weight n \u2212t \u2212i + 1, i.e. each codeword in Ci has t + i \u22121 \u2a7et zeroes. Clearly, the code C with these codesets is an (n, t)-cooling code. By Corollary 10 and Proposition 11 we have the following Corollary 12. C(n, n \u22122) = 3. D. Construction of Optimal Cooling Codes In this subsection, we construct (n, t)-cooling codes that support the transmission of up to n \u2212t \u22121 bits over an n-wire bus. By Corollary 9, such cooling codes are optimal. Our construction is based on the notion of spreads and partial spreads in projective geometry. We will give the related equivalent de\ufb01nition for vector spaces. In this section we need only spread and partial spread over F2, but we will de\ufb01ne and discuss them over Fq, as those will be required later in our exposition. Loosely speaking, a partial \u03c4-spread of the vector space Fn q is a collection of disjoint \u03c4-dimensional subspaces of Fn q . Formally, a collection V1, V2, . . . , VM of \u03c4-dimensional subspaces of Fn q is said to be a partial \u03c4-spread of Fn q if Vi \u2229Vj = {0} for all i \u0338= j , (12) Fn q \u2287V1 \u222aV2 \u222a\u00b7 \u00b7 \u00b7 \u222aVM . (13) If the \u03c4-dimensional subspaces form a partition of Fn q then the partial \u03c4-spread is called a \u03c4-spread. It is well known that such \u03c4-spreads exist if and only if \u03c4 divides n, in which case M = (qn\u22121)/(q\u03c4\u22121) > qn\u2212\u03c4. For the case where \u03c4 does not divide n, partial \u03c4-spreads with M \u2a7eqn\u2212\u03c4 have been constructed in [18, Theorem 11]. Let \f16 Mq(n, \u03c4) be the maximum size of a partial \u03c4-spread. The value of Mq(n, \u03c4) has been considered for many years in projective geometry, and a survey with the known results is given in [17]. Recently, there has been lot of activity and the question has been almost completely solved [32,33,37]. Theorem 13. Let V1,V2, . . . , VM be a partial (t+1)-spread of Fn 2 , and de\ufb01ne the code C = {V\u2217 1 , V\u2217 2 , . . . , V\u2217 M}, where V\u2217 i = Vi \\{0} for all i. Then C is an (n, t)-cooling code of size M \u2a7e2n\u2212t\u22121. Proof: It is obvious from (12) that the M codesets of C are disjoint subsets of Fn 2 . It remains to verify that for any set S \u2282[n] of size t, each of V\u2217 1 ,V\u2217 2 , . . . , V\u2217 M contains at least one vector whose support is disjoint from S. To this end, consider an arbitrary (t+1)-dimensional subspace V of Fn 2 , and suppose {v1, v2, . . . , vt+1} is a basis for V. Let v\u2032 1, v\u2032 2, . . . , v\u2032 t+1 denote the projections of the basis vectors on the t positions in S. These t + 1 vectors lie in a t-dimensional vector space \u2014 the projection of Fn 2 on S. Hence, these vectors must be linearly dependent, and there exist binary coef\ufb01cients a1, a2, . . . , at+1, not all zero, with a1v\u2032 1 + a2v\u2032 2 + \u00b7 \u00b7 \u00b7 + at+1v\u2032 t+1 = 0. But then x = a1v1 + a2v2 + \u00b7 \u00b7 \u00b7 + at+1vt+1 is a nonzero vector in V whose support does not include any of the positions in S. As this holds for an arbitrary (t+1)-dimensional subspace, it must hold for each of the subspaces V1,V2, . . . , VM in the partial spread. Whether we start with a spread or a partial spread, the size of the code C constructed in Theorem 13 will usually be strictly larger than 2n\u2212t\u22121. We omit also the possibility for adding another codset which contains the all-zero codeword since the encoding of k-bit data generated by the source S, requires only 2k codesets. In order to use such a code to communicate k = n \u2212t \u22121 bits, one can choose a subset of C, with 2k codesets, arbitrarily. We illustrate this point in the next subsection. Nevertheless, sometimes we will be interested in the exact number of codesets to \ufb01nd bounds on C(n, t), which will be discussed later in this section. E. Ef\ufb01cient Encoding and Decoding of Cooling Codes Several ef\ufb01cient algorithms for coding into spreads are known. In this subsection, we describe a particularly simple and powerful method that was originally developed by Dumer [14] in the context of coding for memories with defects. This method involves computations in the \ufb01nite \ufb01eld F2t+1). Since t is a small constant, such computations are very ef\ufb01cient. In fact, for most applications of cooling codes, t \u2a7d7 will be more than suf\ufb01cient to cool the bus wires. Thus the proposed encoder E and decoder D work with bytes, or even nibbles, of data. As in the previous subsection, we set \u03c4 = t + 1. For simplicity and w.l.o.g. we assume that \u03c4 divides n and, hence, also k = n \u2212\u03c4. Presented with a k-bit data word u generated by the source S, the encoder E \ufb01rst partitions u into m = k/\u03c4 blocks u1, u2, . . . , um, each consisting of \u03c4 bits. We will refer to these blocks as \u03c4-nibbles and think of them as elements in the \ufb01nite \ufb01eld F2\u03c4. The output of the encoder is the n-bit vector: E(u, S) = (\u03b2u1, \u03b2u2, . . . , \u03b2um, \u03b2) \u2208Fn 2 (14) for a carefully chosen nonzero element \u03b2 of F2\u03c4 that depends on both u and S. Note that, given \u03b2, computing E(u, S) amounts to k/\u03c4 multiplications in F2\u03c4. The operation of the decoder D is equally simple. Given its input \f17 x = E(u, S), the decoder \ufb01rst partitions x into \u03c4-nibbles x1, x2, . . . , xm+1 and reads off \u03b2 = xm+1. The decoder then computes \u03b2\u22121 from \u03b2 in F2\u03c4, and recovers the original data word u as follows: D(x) = (\u03b2\u22121x1, \u03b2\u22121x2, . . . , \u03b2\u22121xm) = (u1, u2, . . . , um) . Thus decoding amounts to one inversion and k/\u03c4 multiplications in F2\u03c4. It remains to explain how \u03b2 is computed. Computing \u03b2 from u and S is equivalent to solving a system of t linear equations in \u03c4 unknowns over F2. We illustrate this with the following example for the case \u03c4 = 3. Example. In this example, we will work with the \ufb01nite \ufb01eld F23 = {0, 1, \u03b1, \u03b12, \u03b13, \u03b14, \u03b15, \u03b16}, de\ufb01ned by \u03b13 = 1 + \u03b1. Notice that 1, \u03b1, \u03b12 is a basis for F23 over F2. Suppose that the t = 2 hottest wires, presented to the encoder via the set S, fall in the 3-nibbles xi = \u03b2ui and xj = \u03b2uj (if both fall in the same 3-nibble, the situation is similar). Let us write ui = u0 + u1\u03b1 + u2\u03b12 and uj = u\u2032 0 + u\u2032 1\u03b1 + u\u2032 2\u03b12 . Let us also write \u03b2 = b0 + b1\u03b1 + b2\u03b12; our goal is to determine the \u03c4 = 3 unknowns b0, b1, b2 from the t = 2 constraints. Computing (x0, x1, x2) = \u03b2ui and (x\u2032 0, x\u2032 1, x\u2032 2) = \u03b2uj, we have: x0 = u0b0 + u2b1 + u1b2 (15) x1 = u1b0 + (u0+u2)b1 + (u1+u2)b2 (16) x2 = u2b0 + u1b1 + (u0+u2)b2 (17) x\u2032 0 = u\u2032 0b0 + u\u2032 2b1 + u\u2032 1b2 (18) x\u2032 1 = u\u2032 1b0 + (u\u2032 0+u\u2032 2)b1 + (u\u2032 1+u\u2032 2)b2 (19) x\u2032 2 = u\u2032 2b0 + u\u2032 1b1 + (u\u2032 0+u\u2032 2)b2 . (20) Equating any two of x0, x1, x2, x\u2032 0, x\u2032 1, x\u2032 2 in (15) \u2013 (20) to zero generates a system of t = 2 linear equations in the unknowns b0, b1, b2 with coef\ufb01cients determined by u0, u1, u2, u\u2032 0, u\u2032 1, u\u2032 2. Such a system can be easily solved by Gaussian elimination or, if necessary, more ef\ufb01cient methods. \u25a1 In general, the complexity of computing \u03b2 from u and S is O(t3), which is very small for constant t. The overall complexity of encoding/decoding is linear in the number of wires n. F. Cooling Codes for Large t The cooling codes constructed based on spreads or partial spreads can be used only for small t, or more precisely when t + 1 \u2a7dn/2 since subspaces of dimension greater that t + 1 have a nontrivial intersection if t + 1 > n/2. Fortunately, this is probably what is usually required for the design parameters of a thermal system. Nevertheless, we want to \ufb01nd ef\ufb01cient cooling codes also for larger values of t. Therefore, when t + 1 > n/2 we have to use another construction which generates cooling codes of a large size. In this subsection we present a construction which forms a sun\ufb02ower whose heart of seeds (kernel) is a linear code C and his \ufb02owers are codes obtained by the sum of C and elements of a spread. The construction of subsection IV-D, which is based on a partial spreads can be viewed also as such a sun\ufb02ower, where the kernel is a trivial linear code (only the all-zero codeword), and the \ufb02owers are the elements of the partial spread without the all-zero codeword. \f18 For the new construction and for some constructions which follow we will use some basic and more sophisticated elements in coding theory which will be de\ufb01ned. First, an [n, \u03ba, d] code C over Fq is a \u03ba-dimensional subspace of Fn q , with minimum Hamming distance d. This is the only concept we need for the basic construction. The more sophisticated elements will be needed for a generalization of the construction which will be given later. Theorem 14. Let n, t, s, r, d be integers, such that r + t \u2a7d(n + s)/2. If there exists a binary [n, s, d] code and a binary [n \u2212t, r, d] code does not exist, then there exists an (n, t)-cooling code of size M > 2n\u2212t\u2212r. Proof: Let n, t, s, r, d be integers, such that r + t \u2a7d(n + s)/2, K a binary [n, s, d] code, and assume that a binary [n \u2212t, r, d] code does not exist. Let B be an (n \u2212s)-dimensional subspace of Fn 2 such that K + B = {a + b : a \u2208K, b \u2208B} = Fn 2 . The fact that r + t \u2a7d(n + s)/2 implies that r + t \u2212s \u2a7d(n \u2212s)/2 and hence there exists a partial (r + t \u2212s)spread of B whose size M is M2(n \u2212s, r + t \u2212s) \u2a7e2n\u2212r\u2212t. Let V1, V2, . . . , VM be the subspaces of a related partial spread and construct the following M sets: Ci = (Vi + K) \\ K for i \u2208[M] . Since Vi \u2229Vj is the null space for i \u0338= j, it follows that (Vi + K) \u2229(Vj + K) = K and hence Ci \u2229Cj = \u2205. Hence, we can take the Ci\u2019s as the codesets in a code C. To prove that C is an (n, t)-cooling code it remains to be shown that for any given subset S \u2208[n] of size t and a codeset Ci, 1 \u2208[M], there exists a codeword x \u2208Ci, such that supp(x) \u2229S = \u2205. The code K is an s-dimensional subspace, Vi in an (r + t \u2212s)-dimensional subspace, and K \u2229Vi = {0} (since Vi \u2282B and K \u2229B = {0}). Hence, Vi + K is an (r + t)-dimensional subspace. Let {v1, v2, . . . , vr+t} is a basis for Vi + K. Let v\u2032 1, v\u2032 2, . . . , v\u2032 r+t denote the projections of the basis vectors on the t positions in S. These r + t vectors lie in a t-dimensional vector space \u2014 the projection of Fn 2 on S. Hence, there exists an r-dimensional subspace Ui spanned by these r + t basis vectors (which span Vi + K), such that for each z \u2208Ui, we have supp(z) \u2229S = \u2205. We can remove the t coordinates which only have zero elements in Ui, from all the vectors of Ui, to obtain an [n \u2212t, r, \u03b4] code U\u2032 i. Since an [n \u2212t, r, d] code does not exist it follows that \u03b4 < d. Hence, Ui contains a vector x which is not contained in K. Since x \u2208Ui and x / \u2208K, it follows that x \u2208Ci, and hence C is an (n, t)-cooling code and the proof has been completed. Theorem 14 can be applied in various ways. For example, we derive the following result. Corollary 15. If n = 2m, m > 1, and n/2 < t \u2a7d(n + m \u22121)/2, then there exists an (n, t)-cooling code of size M > 2n\u2212t\u22121. Proof: Apply Theorem 14 with the [n, m + 1, n/2] \ufb01rst order Reed-Muller code as the kernel code K, i.e. s = m + 1. If r = 1 then r + t \u2a7d1 + (n + m \u22121)/2 \u2a7d(n + m + 1)/2 = (r + s)/2. Clearly, an [n \u2212t, 1, n/2] code does not exist since n \u2212t < n/2. Hence, by Theorem 14 we obtain an (n, t)-cooling code whose size is greater than 2n\u2212t\u22121. This code is optimal by Corollary 9. Theorem 14 can be generalized by using the concept of generalized Hamming weights which was de\ufb01ned by Wei [56]. For this de\ufb01nition we generalize the notion of support from a word to a subcode. The support of a \f19 subcode C\u2032 of C, supp(C\u2032) is de\ufb01ned as the set of coordinates on which C\u2032 contains codewords with nonzero coordinates, i.e. supp(C\u2032) def = \b i \u2208[n] : \u2203(x1, x2, . . . , xn) \u2208C\u2032, xi \u0338= 0 \t . Now, the rth generalized Hamming weight, dr(C), of C is de\ufb01ned as the minimum number of coordinates in supp(C\u2032), for an r-dimensional subcode C\u2032 of C, i.e. dr(C) def = min \b| supp(C\u2032)| : C\u2032 is an r-dimensional subcode of C \t . Theorem 16. If n, t, s, r, d are integers, such that r + t \u2a7d(n + s)/2, and the following two requirements are satis\ufb01ed: (R1) There exists a binary [n, s, d] code K. (R2) The rth generalized Hamming weight of K is larger than n \u2212t. Then there exists an (n, t)-cooling code of size M > 2n\u2212t\u2212r. Proof: The proof is along the same lines as the proof of Theorem 14, up to the point in which an [n \u2212t, r, \u03b4] code U\u2032 i is obtained from Ui. Since the rth generalized Hamming weight of K is larger than n \u2212t, it follows that the r-dimensional subspace Ui contains a vector x which is not contained in K. Since x \u2208Ui and x / \u2208K, it follows that x \u2208Ci, and hence C is an (n, t)-cooling code and the proof has been completed. Theorem 16 is stronger than and generalizes Theorem 14. Indeed, consider the [n, s, d] code K in Theorem 14. If there does not exist an [n \u2212t, r, d] code, then the rth generalized Hamming weight of K must be larger than n \u2212t. But, if for a given [n, s, d] code K the rth generalized Hamming weight of K is larger than n \u2212t, then an [n \u2212t, r, d] code might still exists. An independent question is whether Theorem 16 can improve on the results implied by Theorem 14, or all the results that can be obtained from Theorem 16 can also be obtained from Theorem 14. The answer is that Theorem 16 improves on some of the results obtained by Theorem 14. This is demonstrated with the following examples. The \ufb01rst interesting example is considered in the range 1 \u2a7dt < n \u2a7d100. Consider the following 8 \u00d7 20 parity-check matrix H over F2, H = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 1 0 0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 . One can verify that every six columns have rank at least 5. Therefore, there exists an [n, n \u22128, d] linear code with d2 \u2a7e7 for n \u2208{18, 19, 20}. Thus, by Theorem 16, there exists an (n, n \u22126) cooling code of size greater than 24 = 16 for n \u2208{18, 19, 20}. On other hand, suppose that Theorem 14 is used to obtain an (n, n \u22126)-cooling code of size greater than 24 for n \u2208{18, 19, 20}. It is required by Theorem 14 that t + r \u2a7d(n + s)/2, where t = n \u22126 and r = 2, and hence \f20 s \u2a7en \u22128. Using the online codetables.de [22], the best code with dimension n \u22128 for length n \u2208{18, 19, 20} has minimum Hamming distance 4. But, there exists a [6, 2, 4] code, and hence Theorem 14 cannot be applied. Another example is derived from Feng et al. [19] who computed the following bounds for the generalized Hamming weights for cyclic codes of length 255. (i) The primitive double-error-correcting BCH [255, 239, 5] code has d9 \u2a7e18. If t = 238, then by Theorem 16, there exists a (255, 238)-cooling code of size greater than 28. On other hand, suppose that Theorem 14 is used to obtain a (255, 238)-cooling code of size greater than 28. It is required by Theorem 14 that t + r \u2a7d(n + s)/2, where n = 255, t = 238 and r = 9, and hence s \u2a7e239. Using the online codetables.de [22], the best code with dimension 239 for length 255 has minimum Hamming distance 5. But, there exists a [17, 9, 5] code, and hence Theorem 14 cannot apply. (ii) The reversible cyclic double-error-correcting BCH [255, 238, 5] code has d9 \u2a7e19 [19]. If t = 237, then by Theorem 16, there exists a (255, 237)-cooling code with size greater than 29. On other hand, suppose that Theorem 14 is used to obtain a (255, 237)-cooling code of size greater than 29. It is required by Theorem 14 that 237 + 9 \u2a7d(255 + s)/2 and hence s \u2a7e237. Using the online codetables.de [22], the best code with dimension 237 has minimum Hamming distance 6. But, there exists an [18, 9, 6] code, and hence Theorem 14 cannot be applied. G. Best Lower Bounds on C(n, t) Now, we will summarize our constructions by providing our best lower bounds on C(n, t) in general and when 1 \u2a7dt < n \u2a7d100 in particular. C(n, t) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 = 2n\u2212t, if t = 1 or t = n \u22121, > 2n\u2212t\u22121, if t + 1 \u2a7dn/2, > 2n\u2212t\u22122, if (n, t) \u2208{(18, 12), (19, 13), (20, 14)}, > M2(n \u2212s, r + t \u2212s), if n, t, s, and r appears in TABLE I in the Appendix, and (n, t) / \u2208{(18, 12), (19, 13), (20, 14)}, \u2a7en \u2212t + 1, otherwise. V. LOW-POWER COOLING CODES In this section, we present coding schemes that satisfy Properties A(t) and B(w) simultaneously in every transmission. The corresponding codes are called (n, t, w)-low-power cooling codes (or (n, t, w)-LPC codes for short). We suggest two types of constructions. The \ufb01rst type is based on decomposition of the complete hypergraph into disjoint perfect matchings. The second type is based on cooling codes over GF(q), dual codes of [n, \u03ba, t + 1] codes, MDS codes, spreads, J+(r, \u03c9), and concatenation codes. Finally, we will show that also the sun\ufb02ower construction can be used to obtain (n, t, w)-LPC codes. As before, we assume that the coding schemes constructed in what follows are augmented by differential encoding. Since the codes are also (n,t)-cooling codes, they conform to Definition 2. Thus a code C is a collection of \f21 codesets C1, C2, . . . , CM, which are disjoint subsets of Fn 2 . In order to satisfy Property B(w), the codesets of the code C must satisfy that C1, C2, . . . , CM \u2282J+(n, w) . (21) As shown in Section IV-A, this guarantees that the total number of state transitions on the n bus wires is at most w. A. Decomposition of the Complete Hypergraph The \ufb01rst construction applies the well known Baranyai\u2019s theorem [52, p. 536]. The theorem can be stated in terms of set systems, but it is more known in the context of the decomposition of complete hypergraph on n vertices. The hyperedges of the complete hypergraph consist of all subsets of w vertices. In other words, the set of vertices is [n] and the set of edges consists of all w-subsets of [n]. If w divides n then the Baranyai\u2019s theorem asserts that the hyperedges of this complete hypergraph can be decomposed into pairwise disjoint perfect matchings, where each matching consists of disjoint hyperedges (two hyperedges do not contain the same vertex) and each vertex is contained in exactly one hyperedge of the matching. Clearly, each perfect matching contains n/w hyperedges, and such a decomposition contains w n (n w) = (n\u22121 w\u22121) disjoint matchings. How can such a decomposition can be used to construct (n, t, w)-LPC codes? The answer lies on a right choice of n, e.g., if n = w(t + 1 + \u03f5), where \u03f5 is a nonnegative integer, we can use each matching in such a decomposition as a codeset. To prove that such a code is an (n, t, w)-LPC code, we have to prove that it satis\ufb01es Property A(t). Given a set S \u2282[n] of size t with the numbers of the t hottest wires, and a perfect matching C, the elements of S are contained in at most t hyperedges. But, since a matching contains t + 1 + \u03f5 \u2a7et + 1 hyperedges, it follows that there is at least one hyperedge in the matching which does not contain any element from S. Such a hyperedge is the codeword x \u2208C for which supp(x) \u2229S = \u2205. The most effective sets of parameters on which this construction can be applied are when \u03f5 = 0, i.e. (w(t + 1), t, w)-LPC code. Clearly, we can add codesets with codewords whose weight is less than w. When n is large, the contribution of such codesets is minor. When n is small such a contribution can be important. For example, if n = 12, w = 3, and t = 3, then there are 55 codesets if we restrict ourself only to codewords of weight 3. If we use all words of weight at most 3, then we can have a code with 81 codesets. The advantage is reduced for larger n \u2014 for example, if n = 21, w = 3 and t = 6. There are 190 codesets if only codewords of weight 3 are used. If there are codewords of weight at most 3, then we can have a code with 224 codesets, which is less dramatic improvements compared to the previous example. What about encoding and decoding of this code? Here lies the big disadvantage of this method. Unless the parameters are relatively small there is no known ef\ufb01cient encoding and decoding algorithms. Anyway, in practice, usually small parameters are used and for many of these sets of parameters this method (code) is probably the most effective one. B. Constructions Based on Cooling Codes over Fq The low-power codes and the cooling codes are all binary codes, as a codeword indicates which transitions should be made on the bus-wires during the transmission. Hence, there is no use for non-binary codes as thermal codes. Nevertheless, non-binary codes might be useful in constructions of binary codes as will be demonstrated in this subsection. \f22 De\ufb01nition 4. For positive integers n and t < n, and for a prime power q, an (n,t)q-cooling code C of size M is de\ufb01ned as a set {C1, C2, . . . , CM}, where C1, C2, . . . , CM are disjoint subsets of Fn q satisfying the following property: for any set S \u2282[n] of size |S| = t and for all i \u2208[M], there exists a codeword x \u2208Ci with supp(x) \u2229S = \u2205. Similarly to Theorem 13 we can prove that Lemma 17. Let V1,V2, . . . , VM be a partial (t+1)-spread of Fn q , and de\ufb01ne the code C = {V\u2217 1 , V\u2217 2 , . . . , V\u2217 M}, where V\u2217 i = Vi \\{0} for all i, i = 1, 2, . . . , M. Then C is an (n, t)q-cooling code of size M \u2a7eqn\u2212t\u22121. Theorem 17 can be applied when t + 1 \u2a7dn/2. When t + 1 > n/2 we can generalize Theorem 14 to obtain (n, t)q-cooling codes. The generalization is straightforward and hence it will be omitted. A third construction, which is not a generalization of previous constructions, for (n, t)q-cooling codes is based on the cosets of a dual code for a linear code over Fq, whose minimum Hamming distance is at least t + 1. We start with the related de\ufb01nitions and properties. For more information and proofs of the claims given, the reader can consult with [35]. Each [n, \u03ba, d] code C over Fq induces a partition of Fn q , where Cz, z \u2208Fn q , is a part in this partition if Cz = \b z + c : c \u2208C \t . Each such part is called a coset of C and this partition contains qn\u2212\u03ba pairwise disjoint cosets. Each [n, \u03ba, d] code C over Fq has a dual code C\u22a5, which is the dual subspace of C. C\u22a5is an [n, n \u2212\u03ba, d\u2032] code. A \u03bb \u00b7 qt \u00d7 n matrix A with elements from Fq is called an orthogonal array with strength t if each t-tuple over Fq appears exactly \u03bb times in each projection on any t columns of A. The dual code C\u22a5of an [n, \u03ba, d] code C, is an orthogonal array of strength d \u22121. Finally, the Singleton bound for an [n, \u03ba, d] code over Fq asserts that d \u2a7dn \u2212\u03ba + 1. A code which attains this bound, with equality, is called a maximum distance separable code (an MDS code in short). Lemma 18. If there exists an [n, \u03ba, t + 1] code over Fq, then there exists an (n, t)q-cooling code of size q\u03ba. Proof: Let C be an [n, \u03ba, t + 1] code over Fq and C\u22a5be its dual code. C\u22a5is an orthogonal array of strength t, i.e., each t-tuple over Fq appears the same number of times, in each projection of t columns of C\u22a5. Since the size of C\u22a5is qn\u2212\u03ba, it follows that each such t-tuple (including the all-zero n-tuple) appears qn\u2212\u03ba\u2212t times in each projection (note that qn\u2212\u03ba\u2212t > 0 by the Singleton bound). Since a coset of C\u22a5is formed by adding a \ufb01xed vector of length n to all the codewords of C\u22a5, it follows that each coset is also an orthogonal array of strength t. Therefore, the code C\u22a5and its q\u03ba \u22121 cosets, can be taken as q\u03ba codesets, to form an (n, t)q-cooling code of size q\u03ba. Lemma 18 has some interesting consequences. First, one might asks why the construction implied by Lemma 18 was not given in Section IV? The answer is very simple. The binary codes obtained by this construction are not good enough as the codes presented in the constructions of Section IV. A second and more important observation from Lemma 18 is a construction of a large (n, t)q-cooling code if we use an MDS code as the original code C. Recall that an [n, \u03ba, d] code is an MDS code if and only if d = n \u2212\u03ba + 1. Moreover, the dual code C\u22a5of an MDS code is also an MDS code. Finally, there is a well known conjecture about the range in which MDS codes can exist and there are MDS codes for all parameters in this range. \f23 Conjecture 19. If d \u2a7e3, then there exists an [n, \u03ba, d] MDS code over Fq if and only if n \u2a7dq + 1 for all q and 2 \u2a7d\u03ba \u2a7dq \u22121, except when q is even and \u03ba \u2208{3, q \u22121}, in which case n \u2a7dq + 2. Theorem 20. If d \u2a7e3, then there exists an [n, \u03ba, d] MDS code over Fq if n \u2a7dq + 1 for all q and 2 \u2a7d\u03ba \u2a7dq \u22121, except when q is even and \u03ba \u2208{3, q \u22121}, in which case n \u2a7dq + 2.. Corollary 21. If n \u2a7dq + 1, then there exist an (n, t)q-cooling code of size qn\u2212t. We continue to present a construction which transform an (n, t)q-cooling code into an (ns, t, nw)-LPC code. This construction will be called a concatenation construction since it perform concatenations of the elements in J+(s, w) implied by an (n, t)q-cooling code. Theorem 22. If q \u2a7d\u2211w i=0 (s i) and there exists an (m, t)q-cooling code of size M, then there exists an (ms, t, mw)-LPC code of size M. Proof: Let \u03c8 be an injection from Fq to J+(s, w) such that \u03c8(0) = 0. Let \u03a8 be an injection from Fm q to Fqms de\ufb01ned by \u03a8(x1, x2, . . . , xm) = (\u03c8(x1), \u03c8(x2), . . . , \u03c8(xm)). Let C = {C1, C2, . . . , CM} be an (m, t)q-cooling code of size M. Let D = {D1, D2, . . . , DM} the image of C under \u03a8, i.e. Di = {\u03a8(x) : x \u2208Ci}, for 1 \u2a7di \u2a7dM. We claim that D is an (ms, t, mw)-LPC code. The length of codewords in the codesets of D is clearly ms and since \u03c8(xi) \u2a7dw for each xi \u2208Fq it follows that the weight of a codeword, in a codeset, is at most mw. It remains to show that for any given set S \u2282[ms] of size t, and a codeset Di, there exists a codeword y in Di such that supp(y) \u2229S = \u2205. Since C is an (m, t)q-cooling code, it follows that for any set S\u2032 \u2282[m] of size t, the codeset Ci contains a codeword x = (x1, x2, . . . , xm) such that supp(x) \u2229S\u2032 = \u2205. If we partition the set of ms coordinates of the codewords in D into m consecutive sets L1, L2, . . . , Lm, where L1 contains the \ufb01rst s coordinates, L2 the next s coordinates, and so on, then the elements in S are contained in t\u2032 \u2a7dt of these sets, say, Ij1, Ij2, . . . , Ijt\u2032 . If we de\ufb01ne S\u2032\u2032 = {j1, j2, . . . , jt\u2032}, then S\u2032\u2032 \u2282[m] is a set of size t\u2032 \u2a7dt. Hence, since C is an (m, t)q-cooling code, it follows that there exists a codeword z \u2208Ci such that supp(z) \u2229S\u2032\u2032 = \u2205. Since \u03c8(0) = 0, it follows from the de\ufb01nition of \u03a8 that \u03a8(z) \u2208Di and supp(\u03a8(z)) \u2229S = \u2205, i.e. we can take y = \u03a8(z) to complete the proof. Theorem 22 can be combined with Theorem 17 and Corollary 21 to obtain the following two results. Corollary 23. If q \u2a7d\u2211w\u2032 i=0 (s i) and t + 1 \u2a7dm/2, then there exists an (ms, t, mw\u2032)-LPC code of size M > qm\u2212t\u22121. Corollary 24. If q \u2a7d\u2211w\u2032 i=0 (s i) and m \u2a7dq + 1, then there exists an (ms, t, mw\u2032)-LPC code of size qm\u2212t. C. Constructions Based on Sun\ufb02owers We examine the code obtained by the sun\ufb02ower construction of Theorem 14. Note, that in the proof of Theorem 14 the codeword z of the related codeset used to show that supp(z) \u2229S = \u2205has weight less than d. Therefore, we can remove from the codesets all codewords of weight greater than d \u22121. Thus, the sun\ufb02ower construction yields an (n, t, d \u22121)-LPC code if the related constraints are satis\ufb01ed as follows. Corollary 25. Let n, t, s, r, d be integers, such that r + t \u2a7d(n + s)/2. If there exists an [n, s, d] code and an [n \u2212 t, r, d] linear code does not exist, then there exists an (n, t, d \u22121)-LPC code of size M > 2n\u2212t\u2212r. \f24 Note, that a similar theorem can be obtained by using the sun\ufb02ower of Theorem 16 which is based on the generalized Hamming weights. VI. ERROR-CORRECTING THERMAL CODES In this section, we construct thermal codes that satisfy Property C(e) and simultaneously Property A(t) or Property B(w). The idea will be to modify and generalize the constructions which were given in the previous sections, by adding error-correction into the constructions. We will present constructions in the same order in which they were presented so far in this work. First, we discuss (n, w, e)-LPEC codes, for both nonadaptive and adaptive schemes. We continue with (n, t, e)-ECC codes, and conclude with (n, t, w, e)-LPECC codes. A. Adaptive and Nonadaptive Low-Power Codes For codes which satisfy simultaneously Properties B(w) and C(e), we consider \ufb01rst nonadaptive codes along the lines discussed in Section III. The type of codes which were considered in Section III are anticodes and in particular equireplicate anticodes. For this purpose we will use again set systems and in particular a family of block design called Steiner systems. A Steiner system S(r, w, n) is a collection B of w-subsets (called blocks) from the n-set [n] such that each r-subset of [n] is contained in exactly one block of B. The blocks of such a set system can be translated into a binary code C of length n and constant weight w for the codewords. It is easy to verify that the code C has minimum Hamming distance 2(w \u2212r + 1), i.e. the code C can correct any w \u2212r errors and can detect any w \u2212r + 1 errors. The diameter of the related code is at most 2w, and it is less than 2w if an only if there is a nonempty intersection between any two codewords. If we are not restricted to equireplicated then we can use constant weight codes instead of Steiner systems. Constant weight codes have many applications an hence they were intensively investigated throughout the years, e.g. [7] and references therein. A constant weight code can be equireplicate, but it does not have to be such a code. In this subsection we consider only equireplicate codes. A Steiner system is equireplicate, so such systems will be the basis of our construction. Information on the known Steiner systems can be found in the main textbooks on block designs, e.g. [5]. There are well-known necessary conditions for the existence of a Steiner system S(r, w, n). For each i, 0 \u2a7di \u2a7dr the number (n\u2212i r\u2212i)/(w\u2212i r\u2212i) is an integer. It was recently proved in [24,28] that for any given 0 < r < w, these necessary conditions are also suf\ufb01cient, except for a \ufb01nite number of cases. As was discussed in Section III, to have at most w transitions on the bus wires in the nonadaptive scheme, the minimum distance of our code must be at most w, and each codeword must be of weight at most w/2 if w is even, and (w + 1)/2 if w is odd. For simplicity we will assume that w is even. Now, assume that we want our code to have Property C(e). For this purpose we need a Steiner system S(w/2 \u2212e, w/2, n). If 2e \u2a7dw/2 then we can add codewords of weight w/2 \u22122e. The number of possible such codewords is negligible compared to the number of codewords with weight w/2 and hence we omit these possible codewords in our discussion. Similar codes can be constructed for other parameters. Assume there exists a system set B which is a Steiner system S(w/2 \u2212e, w/2, n + 1) on the point set [n + 1]. Consider the set B\u2032def = \b X : X \u2208B, n + 1 / \u2208X \t [\b X \\ {n + 1} : X \u2208B, n + 1 \u2208X \t . \f25 It is easy to verify that B\u2032 is an (n, w, e)-LPEC code. Moreover, this code is also equireplicate and can be used for a nonadaptive scheme. In contrast to (n, w)-LP codes, where an optimal anticode C with codewords of weight w/2 implies that C has diameter w, the situation when we consider also Property C(e) can be slightly different. Clearly, the diameter of the code should be w, but the weight of a codeword can be larger than w/2. For example, if w/2 = q, n = q2 + q + 1, and q is a prime power, then an optimal such system consists of q2 + q + 1 blocks which form a projective plane of order q [52, p. 224]. Recall that such a structure was considered also in subsection III-B. The blocks of such a projective plane form a Steiner system S(2, q + 1, q2 + q + 1), where any two distinct blocks intersect in exactly one point and hence the diameter of the related code is 2q = w as required. The related code can correct any q \u22121 errors and can detect any q errors. Finally, for parameters where related Steiner systems do not exist or no ef\ufb01cient construction for such systems is known, we can use similar constant weight codes based on the rich literature of such codes. For adaptive (n, w)-LPEC codes we use as in most of our exposition the differential encoding method. The codes which are used are Steiner systems as in the nonadaptive case. The only difference is that the weight of the codewords will be at most w and not w/2 or (w + 2)/2 as in the nonadaptive case. The number of errors which are corrected is de\ufb01ned by the Steiner system as we discussed in this subsection. B. Error-Correcting Cooling Codes In this subsection we will adapt the construction based on spreads and the sun\ufb02ower construction, given in Section IV, to form codes which satisfy Property A(t) and Property C(e). The idea is start with a binary [n, \u03ba, 2e + 1] code C which corrects e errors. In C there exists at least one set S of \u03ba coordinates whose projection on C is F\u03ba 2 . This set of coordinates is called a systematic set of coordinates. On this set of S coordinates, we either apply the spread construction or the sun\ufb02ower construction to obtain a code which satis\ufb01es simultaneously Property A(t) and Property C(e). For the construction which is based on a partial (t + 1)-spread, we start with our favorite binary [n, \u03ba, 2e + 1] code C, where \u03ba \u2a7e2(t + 1). In addition, we take a partial (t + 1)-spread (or a (t + 1)-spread if t + 1 divides \u03ba) of F\u03ba 2 . Since C has dimension \u03ba, there exists at least one set of \u03ba coordinates whose projection on C spans F\u03ba 2 . The partial (t + 1)-spread can be formed on these coordinates. The codewords of C are partitioned into codesets related to this partial spread. Given any t coordinates, each codeset has at least one codeword with zeroes in these t coordinates since the partial spread has dimension t + 1. This is proved exactly as in the proof of Theorem 13. Moreover, the code can correct at least e errors since the codesets are disjoint and all the codewords in the codesets are contained in the code C. Thus, we have constructed an (n, t, e)-ECC code. We summary this construction with the following theorem. Theorem 26. If there exists a binary [n, \u03ba, 2e + 1] code and \u03ba \u2a7e2(t + 1), then there exists an (n, t, e)-ECC code of size M > 2\u03ba\u2212t\u22121. The sun\ufb02ower construction is adapted in a similar way to obtain an (n, t, e)-ECC code. We start with our favorite binary [n, \u03ba, 2e + 1] code C and apply the sun\ufb02ower construction on \u03ba systematic coordinates in C. Similarly to Theorem 14 we have the following theorem \f26 Theorem 27. Let n, t, s, r, \u03ba, d, be integers such that r + t \u2a7d(\u03ba + s)/2 and there exists a binary [n, \u03ba, 2e + 1] code. If there exists a [\u03ba, s, d] code and a binary [\u03ba \u2212t, r, d] code does not exist, then there exists an (n, t, e)-ECC code of size M > 2\u03ba\u2212t\u2212r. Similarly, to Theorem 27 we can adapt the construction of Theorem 16 to obtain an (n, t, e)-ECC code. C. Constructions of Low-Power Error-Correcting Cooling Codes In this subsection we consider codes which satisfy all Properties A(t), B(w), and C(e) simultaneously. The related codes are (n, t, w, e)-LPECC codes. We suggest three methods to construct such codes, which re\ufb02ect the three methods in Section V, where only Properties A(t) and B(w), without Property C(e), were considered. The \ufb01rst method, generalizes the decomposition of the complete hypergraph as described in subsection V-A, by considering resolvable Steiner systems. The second method is based on dual codes and concatenation modi\ufb01es the construction in subsection V-B. The last construction is based on the sun\ufb02ower construction similarly to subsection V-C. Our \ufb01rst method is a generalization for the decomposition of the complete hypergraph into pairwise disjoint perfect matchings. For this generalization we consider the hyperedges as blocks in a system set, or more precisely in a block design. The related concepts in block design are resolution and parallel classes. A block design (set system) is said to be resolvable if the blocks of the design can be partitioned into pairwise disjoint sets, called parallel classes, where each class forms a partition of the point sets into pairwise disjoint blocks. The whole process is called resolution. By this de\ufb01nition, the decomposition of the complete hypergraph on n vertices and hyperedges of size w is a resolution of all w-subsets of [n]. To have low-power error-correcting cooling codes we will use resolutions of Steiner system. Recall that a Steiner system S(r, w, n) is a binary code with minimum Hamming distance 2(w \u2212r + 1) and hence it can correct any w \u2212r errors and can detect any w \u2212r + 1 errors. Since we are interested in an (n, t, w, e)-LPECC code, we should start with a Steiner system S(w \u2212e, w, w(t + 1)) and partition it into pairwise disjoint Steiner systems S(1, w, n) (which are the parallel classes). Such partitions of Steiner systems are known in several cases given as follows: 1) If n \u22613(mod 6) then there exists a resolvable S(2, 3, n) [25]. 2) If n \u22614(mod 12) then there exists a resolvable S(2, 4, n) [25]. 3) If n \u2261w(mod w(w \u22121)) then for suf\ufb01ciently large n there exist a resolvable S(2, w, n) [41]. 4) If n \u22614 or 8(mod 12) then there exists a resolvable S(3, 4, n). This was proved in [23] for all n, except for 23 cases which were completed in [26]. 5) If q is a prime power then there exists a resolvable S(2, q, q2) derived from af\ufb01ne plane (which can be generated from a projective plane of order q which is equivalent to S(2, q + 1, q2 + q + 1)). These resolvable Steiner systems imply the existence of the following (n, t, w, e)-LPECC codes: Theorem 28. If n = 3(t + 1), where t is even, then there exists an (n, t, 3, 1)-LPECC code of size larger than n\u22121 2 . Proof: If t is even, then n = 3(t + 1) \u22613(mod 6), and hence there exists a resolvable Steiner triple system of order n whose size is n(n\u22121) 2\u00b73 , with n\u22121 2 parallel classes, each one of size n 3 . The addition of the all-zero vector of length n in a new codeset enlarge the size of the code. Similarly, we have \f27 Theorem 29. If n = 4(t + 1) \u22614 or 8(mod 12), then there exists an (n, t, 4, 1)-LPECC code of size larger than (n\u22121)(n\u22122) 6 . Theorem 30. If n = 4(t + 1) \u22614(mod 12), then there exists an (n, t, 4, 2)-LPECC code of size (n\u22121)(n\u22122) 12 . Theorem 31. If n = w(t + 1) \u2261w(mod w(w \u22121)), then for suf\ufb01ciently large n there exists an (n, t, w, w \u22122)LPECC code of size (n\u22121)(n\u22122) w(w\u22121) . Theorem 32. If q is a prime power, then there exists a (q2, q \u22121, q, q \u22122)-LPECC code of size q2. The constructions derived, in subsection V-B, from codes over Fq are based on the codes constructed in Corollaries 23 and 24. These corollaries are derived from the concatenation construction presented in Theorem 22. To adapt these results to form (n, t, w, e)-LPECC codes, we use constructions of two types. The \ufb01rst one starts with an error-correcting code C over Fq, partition of the codewords of C by using a partial spread, and apply the concatenation construction on the codesets derived from the partial spread. The second type of construction is based the (n, t, w)-LPC codes derived from dual codes of MDS codes (see Corollary 24). For the \ufb01rst construction we start with an [m, \u03ba, 2e + 1] code C over Fq, such that \u03ba \u2a7e2(t + 1). In addition, we take a partial (t + 1)-spread of F\u03ba q . Since C has dimension \u03ba, it follows that there exists at least one set of \u03ba systematic coordinates whose projection spans F\u03ba q . The partial (t + 1)-spread can be formed on these \u03ba systematic coordinates. The codewords of C can be partitioned into codesets related to this partial spread and form a new code C\u2032. Given any set S of size t, each codeset has at least one codeword with zeroes in the t coordinates of S, since the subspaces of the partial spread have dimension t + 1. The proof is exactly as the proof of Theorem 13. Moreover, the code C\u2032 can correct e errors, since the codesets of C\u2032 are disjoint and all the codewords in the codesets are contained in the code C which can correct any e errors. Finally, let q \u2a7d\u2211w\u2032 i=0 (s i), and we use the concatenation construction given in Theorem 22 and Corollary 23. Thus, we obtain an (ms, t, mw\u2032, e)-LPECC code. The size of the code C\u2032 depends on the largest dimension of an [m, \u03ba, 2e + 1] code C, subject to the requirement that \u03ba \u2a7e2(t + 1). The resulting code C\u2032 will have size q\u03ba\u2212t\u22121. The second construction is an immediate consequence of Corollary 24. The minimum Hamming distance of the low-power cooling code obtained in Corollary 24 is the same as the minimum Hamming distance of the related MDS code of length m, m \u2a7dq + 1. Therefore, the size of the code is qm\u2212t and its minimum Hamming distance t + 1. Corollary 33. If q \u2a7d\u2211w\u2032 i=0 (s i) and m \u2a7dq + 1, then there exists an (ms, t, mw\u2032, \u230at/2\u230b)-LPECC code of size qm\u2212t. Finally, we want to adapt the sun\ufb02ower construction to form an (n, t, w, e)-LPECC code. The idea is to use the arguments of Corollary 25, in the construction implied by Theorem 27, which yields the following result Corollary 34. Let n, t, s, r, \u03ba, d, be integers such that r + t \u2a7d(\u03ba + s)/2 and there exists a binary [n, \u03ba, 2e + 1] code. If there exists a [\u03ba, s, d] code and a binary [\u03ba \u2212t, r, d] code does not exist, then there exists an (n, t, d \u22121, e)-LPECC code of size M > 2\u03ba\u2212t\u2212r. Similarly, to Corollary 34 we can adapt the construction of Theorem 16 to obtain an (n, t, w, e)-LPECC code. \f28 VII. ASYMPTOTIC BEHAVIOR In this section we will analyze the asymptotic behavior of the thermal codes which were constructed in the previous sections. We start in subsection VII-A where we consider only cooling codes. As was proved in Section IV, when t + 1 \u2a7dn/2 our codes which use only t + 1 redundancy bits are optimal and the number of additional wires used is negligible when k or n are large enough. Hence, the interesting case is the asymptotic behavior when t + 1 > n/2 and the sun\ufb02ower construction is used. The case is even more complicated when we consider lowpower cooling codes. Our methods with ef\ufb01cient encoding and decoding algorithms are not optimal, but this does not exclude the possibility of being asymptotically optimal. The asymptotic behavior of these codes will be considered in subsection VII-B. In addition, to \ufb01nd asymptotically good codes, we will consider in this subsection a new Gilbert-Varshamov type method, called the expurgation method. A. Asymptotic Behavior of Cooling Codes Let B(n, d) be the largest dimension of a binary [n, \u03ba, d] code. For 0 < \u03b4, \u03c4 < 1, de\ufb01ne \u03b2(\u03b4) def = lim sup n\u2192\u221e B(n, \u230a\u03b4n\u230b) n , \u03f1(\u03c4) def = lim sup n\u2192\u221e log2 C(n, \u230a\u03c4n\u230b) n . By Lemma 7, we have that any (n, t)-cooling code of size M, satis\ufb01es M \u2a7d2n\u2212t. This immediately implies the following asymptotic upper bound on the rate of an (n, t)-cooling code. Corollary 35. If 0 < \u03c4 < 1, then \u03f1(\u03c4) \u2a7d1 \u2212\u03c4 . The following proposition for computing an asymptotic lower bound on the rate of an (n, t)-cooling code. Proposition 36. Assume 0 < \u03f5 < 1 satis\ufb01es the following conditions: \u03f5 > \u03b2(\u03b4) (22) \u03c4 + \u03f5(1 \u2212\u03c4) < 1 + \u03b2(\u03b4(1 \u2212\u03c4)) 2 . (23) Then \u03f1(\u03c4) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5). Proof: By the de\ufb01nition of \u03b2, there exists N1 such that B(M, \u230a\u03b4M\u230b) < \u03f5M, (24) for all M \u2a7eN1. If \u03b4\u2032 = \u03b4(1 \u2212\u03c4) and 0 < \u03f51 < 1, then for any given large enough integer N2, there exists an integer N \u2a7eN2, such that B(N, \u0004 \u03b4\u2032N \u0005) \u2a7e\u03b2(\u03b4\u2032)N \u2212\u03f51N . (25) Set s = \u230a\u03b2(\u03b4\u2032)N \u2212\u03f51N\u230b, d = \u230a\u03b4\u2032N\u230b, t = \u2308\u03c4N\u2309, M = N \u2212t, and r = \u2308\u03f5M\u2309. We claim that the conditions of Theorem 14 are met for these parameters. \f29 Observe that t + r = \u2308\u03c4N\u2309+ \u2308\u03f5M\u2309= \u2308\u03c4N\u2309+ \u2308\u03f5(N \u2212t)\u2309 = \u2308\u03c4N\u2309+ \u2308\u03f5(N \u2212\u2308\u03c4N\u2309)\u2309 \u2a7d(\u03c4N + 1) + (\u03f5(1 \u2212\u03c4)N + 1) = \u03c4N + \u03f5(1 \u2212\u03c4)N + 2. (26) On the other hand, N + s 2 = N + \u230a\u03b2(\u03b4\u2032)N \u2212\u03f51N\u230b 2 \u2a7eN + \u03b2(\u03b4\u2032)N \u2212\u03f51N \u22121 2 . (27) Now, let N2 \u2a7e 5/2 (1 + \u03b2(\u03b4\u2032) \u2212\u03f51)/2 \u2212(\u03c4 + \u03f5(1 \u2212\u03c4)). (28) Furthermore, since (23) holds, it follows that there exists 0 < \u03f52 < 1, such that for all 0 < \u03f51 < \u03f52 the denominator in the right hand side of (28) is strictly positive. Hence, since N > N2, we have that \u0000(1 + \u03b2(\u03b4\u2032) \u2212\u03f51)/2 \u2212(\u03c4 + \u03f5(1 \u2212\u03c4)) \u0001 N \u2a7e5/2, or, \u03c4N + \u03f5(1 \u2212\u03c4)N + 2 \u2a7dN + \u03b2(\u03b4\u2032)N \u2212\u03f51N \u22121 2 . (29) Combining (29) with Inequalities (26) and (27), we have that t + r \u2a7d(N + s)/2, and the \ufb01rst condition of Theorem 14 is satis\ufb01ed. Next, since s \u2a7d\u03b2(\u03b4\u2032)N \u2212\u03f51N, it follows from (25) that there exists an [N, s, d] code, and the second condition of Theorem 14 is satis\ufb01ed. Finally, observe that M = N \u2212t \u2a7eN \u2212(\u03c4N + 1) = (1 \u2212\u03c4)N \u22121. (30) If we choose N2 \u2a7eN1 + 1 1 \u2212\u03c4 , (31) then since N \u2a7eN2, it follows by (31) that N(1 \u2212\u03c4) \u2a7eN1 + 1 which implies by (30) that M \u2a7eN1. Therefore, from (24), we infer that B(M, \u230a\u03b4M\u230b) < \u03f5M \u2a7dr. Since \u230a\u03b4M\u230b\u2a7d\u230a\u03b4(1 \u2212\u03c4)N\u230b= d, it follows that an [N \u2212t, r, d] code does not exist, and the third condition of Theorem 14 is satis\ufb01ed. In summary, to satisfy the three conditions of Theorem 14, we need N2 to satisfy (28) and (31). Additionally, to guarantee that N \u2a7eN0 we require that N2 \u2a7eN0 and hence we have that N2 = max \u001a N0, N1 + 1 1 \u2212\u03bb , 5/2 (1 + \u03b2(\u03b4\u2032) \u2212\u03f51)/2 \u2212(\u03bb + \u03f5(1 \u2212\u03bb)) \u001b . \f30 Therefore, by Theorem 14 there exists an (N, t)-cooling code of size greater than 2N\u2212t\u2212r. Now, N \u2212t \u2212r = N \u2212\u2308\u03c4N\u2309\u2212\u2308\u03f5M\u2309 = N \u2212\u2308\u03c4N\u2309\u2212\u2308\u03f5(N \u2212\u2308\u03c4N\u2309)\u2309 \u2a7eN \u2212\u03c4N \u2212\u03f5N \u2212\u03f5\u03c4N \u22122 = N(1 \u2212\u03c4)(1 \u2212\u03f5) \u22122. This implies that there exists an (N, t)-cooling code of size greater than 2(1\u2212\u03bb)(1\u2212\u03f5)N\u22122. Since t = \u2308\u03bbN\u2309\u2a7e\u230a\u03bbN\u230b, it follows that there also exists an (N, \u230a\u03bbN\u230b)-cooling code of size greater than 2(1\u2212\u03bb)(1\u2212\u03f5)N\u22122. Thus, log2 C(N, \u230a\u03bbN\u230b) N \u2a7e(1 \u2212\u03bb)(1 \u2212\u03f5) \u22122 N , which implies that \u03f1(\u03c4) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5). To apply Proposition 36 we have to use the best known lower bound on \u03b2(\u00b7) in (22) and the best known upper bound on \u03b2(\u00b7) in (23). For lower bound we will use the Gilbert-Varshamov bound [21,51]. For upper bound we will use the McEliece-Rodemich-Rumsey-Welch (MRRW) bound [36]. Speci\ufb01cally, we have 1) For the lower bound, the Gilbert-Varshamov bound implies that \u03b2(\u03b4) \u2a7e1 \u2212H(\u03b4) + o(1), where H(\u00b7) is the binary entropy function, given by H(x) def = \u2212x log2x \u2212(1 \u2212x) log2(1 \u2212x) . 2) As for the upper bound, the MRRW bound (equation (1.4) in [36]) implies that \u03b2(\u03b4) \u2a7dMRRW(\u03b4) = min 0\u2a7du\u2a7d1\u22122\u03b4 1 + g \u0010 u2\u0011 \u2212g \u0010 u2 + 2\u03b4u + 2\u03b4 \u0011 , (32) where g(x) = H((1 \u2212\u221a 1 \u2212x)/2). Hence, we have the following corollary. Corollary 37. Suppose that 0 < \u03f5, \u03b4, \u03c4 < 1 satisfy the following conditions: \u03f5 > MRRW(\u03b4), (33) \u03c4 + \u03f5(1 \u2212\u03c4) < 2 \u2212H(\u03b4(1 \u2212\u03c4)) 2 . (34) Then \u03f1(\u03c4) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5). Proof: Since \u03f5 > MRRW(\u03b4) by (33) and \u03b2(\u03b4) \u2a7dMRRW(\u03b4) by (32), it follows that \u03f5 > \u03b2(\u03b4) and hence Inequality (22) is satis\ufb01ed. From the Gilbert-Varshamov bound, we have that \u03b2(\u03b4(1 \u2212\u03c4)) \u2a7e1 \u2212H(\u03b4(1 \u2212\u03c4)) + o(1). Hence, we have \u03c4 + \u03f5(1 \u2212\u03c4) < 2 \u2212H(\u03b4(1 \u2212\u03c4)) 2 = 1 + (1 \u2212H(\u03b4(1 \u2212\u03c4))) 2 \u2a7d1 + \u03b2(\u03b4(1 \u2212\u03c4)) 2 . Therefore, Inequality (23) is satis\ufb01ed. \f31 0.2 0.4 0.6 0.8 1 Relative Cooling Rate 0.2 0.4 0.6 0.8 1 Rate Upper Bound Sunflower Construction Sunflower Construction assuming GV is sharp Figure 1. Asymptotic Rates for Cooling Codes Since the conditions in Proposition 36 are satis\ufb01ed, it follows from the proposition that \u03f1(\u03c4) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5). Now, \ufb01x 0 < \u03c4 < 1 and in what follows, we compute a lower bound on \u03f1(\u03c4) implied by Corollary 37. De\ufb01ne the set E(\u03c4) def = {\u03f5 : there exists a \u03b4 such that \u03f5, \u03b4 satisfy (33) and (34)}. This de\ufb01nition implies that by Corollary 37, for all \u03f5 \u2208E(\u03c4), we can have that \u03f1(\u03c4) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5). Maximizing the right hand side, we obtain \u03f1(\u03c4) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5\u2217), (35) where \u03f5\u2217is the value of \u03f5 for which (1 \u2212\u03c4)(1 \u2212\u03f5) is maximized when \u03f5\u2217is substituted for \u03f5. For the next computation we de\ufb01ne E\u2217(\u03c4) def = \u03f5\u2217. Via numerical computations, we have that E\u2217(\u03c4) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \u22480, if 0 < \u03c4 \u2a7d0.687 between 0 and 1, if 0.687 \u2a7d\u03c4 \u2a7d0.737 \u22481 if 0.737 \u2a7d\u03c4 < 1. The plot of the lower bound (35) for \u03f1(\u03c4) is given in Figure 1. Without giving the formal analyze we also add a curve for the case that the Gilbert-Varshamov bound is tight (at least in the binary case). Finally, we obtain the following result which is consistent with the \ufb01gure, and proves that for \u03c4 \u2a7d0.687 our constructions are asymptotically optimal. Corollary 38. If \u03c4 \u2a7d0.687, then \u03f1(\u03c4) = 1 \u2212\u03c4. \f32 Proof: If \u03c4 \u2a7d0.687, then the value of 1 \u2212H((1 \u2212\u03c4)/2)/(1 \u2212\u03c4) is strictly positive. If we set \u03b4 = 1/2, then (33) and (34) are reduced to \u03f5 > 0 (36) \u03c4 + \u03f5(1 \u2212\u03c4) < 2 \u2212H((1 \u2212\u03c4)/2) 2 . (37) As a consequence, E(\u03c4) contains the interval (0, 1\u2212H((1\u2212\u03c4)/2)/(1\u2212\u03c4)). Hence, the expression (1 \u2212\u03c4)(1 \u2212\u03f5) is maximized for \u03f5 which tends to 0. Therefore, \u03f1(\u03c4) \u2a7e1 \u2212\u03c4 and since by Corollary 35, we have \u03f1(\u03c4) \u2a7d1 \u2212\u03c4, it follows that \u03f1(\u03c4) = 1 \u2212\u03c4. B. Asymptotic Behavior of low-power Cooling Codes The asymptotic analysis when the codes have Property B(w) in addition to Property A(t) is more complicated, needless to say that our methods are slightly less ef\ufb01cient compared to the methods used to construct codes when only A(t) is satis\ufb01ed. Let C(n, t, w) be the largest size of an (n, t, w)-LPC code and for a \ufb01xed 0 < \u03c4 < 1 and a \ufb01xed 0 < \u03c9 < 1, consider the asymptotic rate \u03f1(\u03c4, \u03c9) def = lim sup n\u2192\u221e log2 C(n, \u230a\u03c4n\u230b, \u230a\u03c9n\u230b) n . If t = o(n) and 0 < \u03c9 < 1, then we claim that Corollary 23 provides a family of (n, t, \u230a\u03c9n\u230b)-LPC codes whose rates approach at least H(\u03c9). Formally, we claim that if Cn is such an (n, t, \u230a\u03c9n\u230b)-LPC code, then lim n\u2192\u221e log2 |Cn| n \u2a7eH(\u03c9). (38) To prove (38), consider any given \u03f5 > 0, a prime power q and an integer s > 1 such that q \u2a7d \u230a\u03c9s\u230b \u2211 i=0 \u0012s i \u0013 and log2 q s \u2a7eH(\u03c9) \u2212\u03f5. For each n, set m = \u230an/s\u230b. Since t = o(n), it follows that t + 1 \u2a7dm/2 for suf\ufb01ciently large n. Applying Corollary 23, we obtain an (n, t, \u230a\u03c9n\u230b)-LPC code Cn of size at least qm\u2212t\u22121. Now, we have that lim n\u2192\u221e log |Cn| n \u2a7elim n\u2192\u221e log2 q s \u0012 1 \u2212t + 1 m \u0013 \u2a7e(H(\u03c9) \u2212\u03f5) \u00b7 lim n\u2192\u221e \u0012 1 \u2212t + 1 m \u0013 = H(\u03c9) \u2212\u03f5, and hence limn\u2192\u221e log |Cn| n \u2a7eH(\u03c9), where the last equality follows since t = o(n). The next theorem describes a very simple construction which will be called in the sequel the expurgation construction. Theorem 39. If M > \u2211n i=w+1 (n i) and an (n, t)q-cooling code C of size M exists, then there exists an (n, t, w)-LPC code of size at least M \u2212\u2211n i=w+1 (n i). Proof: The theorem follows as an immediate consequence implied by removing all codewords of weight larger than w from all the codesets of C. \f33 Corollary 40. If \u03c4 \u2a7d0.687, \u03c9 \u2a7e1/2, and H(\u03c9) < 1 \u2212\u03c4, then \u03f1(\u03c4, \u03c9) \u2a7e1 \u2212\u03c4 \u2212o(1). Proof: By the sun\ufb02ower construction (see Theorem 14) there exists a family of (n, t)-cooling codes whose size is at least 2n(1\u2212\u03c4\u2212\u03f5\u2032), where t = \u230a\u03c4n\u230b, r = \u230a\u03f5\u2032n\u230b, and \u03f5\u2032 = o(1) (see subsection VII-A). It is well known by using the Stirling\u2019s approximation [35, p. 310] that if w = \u230a\u03c9n\u230b, then \u2211n i=w+1 (n i) \u2a7d2nH(\u03c9). Now, applying the expurgation construction on a code from this family yields an (n, t, w)-LPC code whose size is at least 2n(1\u2212\u03c4\u2212\u03f5\u2032) \u0010 1 \u22122n(H(\u03c9)\u22121+\u03c4+\u03f5\u2032)\u0011 . In other words, the rate, \u03f1(\u03c4, \u03c9), of this family of (n, t)-cooling codes is at least (1 \u2212\u03c4 \u2212\u03f5\u2032) + log \u0010 1 \u22122n(H(\u03c9)\u22121+\u03c4+\u03f5\u2032)\u0011 n , which is at least 1 \u2212\u03c4 \u2212\u03f5\u2032. Clearly, \u03f1(\u03c4, \u03c9) \u2a7d\u03f1(\u03c4) and hence combining Corollaries 35 and 40 implies that Corollary 41. If \u03c4 \u2a7d0.687, \u03c9 \u2a7e1/2, and H(\u03c9) < 1 \u2212\u03c4, then \u03f1(\u03c4, \u03c9) = 1 \u2212\u03c4. Unfortunately, Theorem 39 is nonconstructive and the domain of (\u03c4, \u03c9), where Corollary 40 is applicable, is limited. In the sequel we will consider other parameters outside this domain. It should be no surprise that in most cases, we observed that codes obtained by the sun\ufb02ower construction have larger size than those obtained by the expurgation construction. Nevertheless, in certain instances, the LPC codes obtained from Theorem 39 has a larger size as compared to LPC codes resulting from the sun\ufb02ower construction. For such an example, consider t = 1, w = 2 3(n \u22121), where 3 divides n \u22121. By Proposition 8, there exists an (n, 1)-cooling code of size 2n\u22121. Theorem 39 yields an (n, 1, 2 3(n \u22121))-LPC code of size at least 2n\u22121 \u2212\u2211n i=w+1 (n i). Note that 2n\u22121 \u2212 n \u2211 i=w+1 \u0012n i \u0013 \u2a7e2n\u22121 \u22122nH(2/3) \u2a7e2n\u22121 \u221220.92n. (39) By the Griesmer bound [35, p. 546], there is no [n \u22121, 2, 2 3(n \u22121) + 1] code and hence Theorem 14 yields an (n, 1, 2 3(n \u22121))-LPC code of size M2(n, 3) + 1. Since 5 142n > 20.92n + 1 for n \u2a7e19, it follows that 2n\u22121 \u221220.92n > M2(n, 3) + 1, and hence the expurgation construction yields an (n, t, w)-LPC code of larger size, compared to the code obtained by the sun\ufb02ower construction, in this case. Recall, that Corollary 25 is used to apply the sun\ufb02ower construction and obtain (n, t, d \u22121)-LPC codes of size greater than M2(n \u2212s, r + t \u2212s) > 2n\u2212t\u2212r, where r is given in Corollary 25. Furthermore, for 0 < \u03c4, \u03c9 < 1, let \u03b4 = \u03c9/(1 \u2212\u03c4) and de\ufb01ne \u03f5 = \u03f5(\u03c4, \u03b4) as \u03f5(\u03c4, \u03b4) def = inf{\u03f5 : \u03f5 satisfy (33) and (34)}. With this setting, Corollary 37 implies that \u03f1(\u03c4, \u03c9) \u2a7e(1 \u2212\u03c4)(1 \u2212\u03f5), and by Corollary 38 and its proof we have Corollary 42. If \u03c4 \u2a7d0.687 and \u03c9 \u2a7e(1 \u2212\u03c4)/2, then \u03f1(\u03c4, \u03c9) = 1 \u2212\u03c4. We continue to examine the asymptotic behavior of (n, t, w)-LPC codes constructed from cooling codes over Fq with concatenation. \f34 Corollary 43. For given 0 < \u03c4, \u03c9 < 1, suppose that there exists a prime power q and an integer s > 1 such that q \u2a7d \u230a\u03c9s\u230b \u2211 i=0 \u0012s i \u0013 and \u03c4s \u2a7d1 2. (40) Then \u03f1(\u03c4, \u03c9) \u2a7e(1 \u2212\u03c4s) log2 q s . Proof: For a given n, set m = l n s + 1 \u03c4s m and t = \u230a\u03c4n\u230b. Since \u03c4s \u2a7d1/2, if follows that t + 1 \u2a7d\u03c4n + 1 = \u0012n s + 1 \u03c4s \u0013 \u03c4s \u2a7dm/2 . Hence, by Corollary 23, there exists an (ms, t, m \u230a\u03c9s\u230b)-LPC code of size at least qm\u2212t\u22121. Therefore, we have that \u03f1(\u03c4, \u03c9) \u2a7elim ms\u2192\u221e m \u2212t \u22121 ms log2 q = log2 q s lim ms\u2192\u221e sm \u2212st ms = log2 q s lim ms\u2192\u221e \u0012 1 \u2212t m \u0013 = log2 q s lim ms\u2192\u221e \u0012 1 \u2212 \u03c4n n/s + 1/(\u03c4s) \u0013 > log2 q s lim ms\u2192\u221e \u0012 1 \u2212\u03c4n n/s \u0013 = log2 q s (1 \u2212\u03c4s). Note, that to apply Corollary 43 we have to be careful in choosing q and s such that equation (40) is satis\ufb01ed. VIII."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1512.07719v2",
+                    "title": "Linear Size Constant-Composition Codes Meeting the Johnson Bound",
+                    "abstract": "The Johnson-type upper bound on the maximum size of a code of length $n$,\ndistance $d=2w-1$ and constant composition ${\\overline{w}}$ is\n$\\lfloor\\dfrac{n}{w_1}\\rfloor$, where $w$ is the total weight and $w_1$ is the\nlargest component of ${\\overline{w}}$. Recently, Chee et al. proved that this\nupper bound can be achieved for all constant-composition codes of sufficiently\nlarge lengths. Let $N_{ccc}({\\overline{w}})$ be the smallest such length. The\ndetermination of $N_{ccc}({\\overline{w}})$ is trivial for binary codes. This\npaper provides a lower bound on $N_{ccc}({\\overline{w}})$, which is shown to be\ntight for all ternary and quaternary codes by giving new combinatorial\nconstructions. Consequently, by refining method, we determine the values of\n$N_{ccc}({\\overline{w}})$ for all $q$-ary constant-composition codes provided\nthat $3w_1\\geq w$ with finite possible exceptions.",
+                    "authors": "Yeow Meng Chee, Xiande Zhang",
+                    "published": "2015-12-24",
+                    "updated": "2016-08-08",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "cs.IT",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION Constant-composition codes have attracted a lot attention [1]\u2013[21] in recent years due to their vast applications, such as in determining the zero error decision feedback capacity of discrete memoryless channels [22], [23], multiple-access communications [24], [25], spherical codes for modulation [26], DNA codes [27], [28], powerline communications [29], [30], and frequency hopping [4]. Although constant-composition codes have been used since the early 1980s to bound error and erasure probabilities in decision feedback channels [31], their systematic study only began in late 1990s with Svanstr\u00a8 om [32]. Nowadays, the problem of determining the maximum size of a constant-composition code constitutes a central problem in their study due to their close relations to combinatorial design theory [2]\u2013[5], [7], [9], [10], [14]\u2013[17], [19], [20]. For integers m \u2264n, the set of integers {m, m + 1, . . . , n} is denoted by [m, n]. When m = 1, the set [1, n] is further abbreviated to [n]. If m > n, then [m, n] is de\ufb01ned to be empty. The ring Z/nZ is denoted by Zn. For \ufb01nite sets R and X, RX denotes the set of vectors of length |X|, where each component of a vector u \u2208RX has value in R and is indexed by an element of X, that is, u = (ux)x\u2208X, and ux \u2208R for each x \u2208X. A q-ary code of length n is a set C \u2286ZX q , for some X of size n. The elements of C are called codewords. The support of a vector u \u2208ZX q is supp(u) = {x \u2208X : ux \u0338= 0}. The Hamming weight of a vector u \u2208ZX q is de\ufb01ned as \u2225u\u2225= |supp(u)|. The distance induced by this weight is the Hamming distance, denoted by dH(\u00b7, \u00b7), so that dH(u, v) = \u2225u \u2212v\u2225, for u, v \u2208ZX q . A code C is said to have distance d if dH(u, v) \u2265d for all distinct u, v \u2208C. The composition of a vector u \u2208ZX q is the tuple w = Jw1, . . . , wq\u22121K, where wi = |{x \u2208X : ux = i}|, i \u2208Zq \\ {0}. A code C is said to have constant weight w if every codeword in C has weight w, and have constant composition w if every codeword has composition w. Hence, every constant-composition code is a constant-weight code. In this paper, attention is restricted to constant-composition codes. For constant-weight codes, interested readers are referred to [33]. A q-ary code of length n, distance d, and constant composition w is denoted an (n, d, w)q-code. The maximum size of an (n, d, w)q-code is denoted Aq(n, d, w), and an (n, d, w)q-code attaining the maximum size is said to be optimal. In an (n, d, w)q-code, reordering the components of w or deleting zero components of w will not affect the distance and composition properties. Hence, through out this paper, when we talk about a composition w = Jw1, w2 . . . , wq\u22121K, we always assume that all components are positive and listed in non-increasing order, that is, w1 \u2265w2 \u2265\u00b7 \u00b7 \u00b7 \u2265wq\u22121 \u22651. For succinctness, de\ufb01ne the total weight w := Pq\u22121 i=1 wi. The Johnson-type bound of Svanstr\u00a8 om for ternary constant-composition codes [1] could be easily extended to the following (see also [4]). Proposition 1. (Johnson Bound): Aq(n, d, Jw1,w2 . . . , wq\u22121K) \u2264 \u0016 n w1 Aq(n \u22121, d, Jw1 \u22121, w2, . . . , wq\u22121K) \u0017 . Y. M. Chee (ymchee@ntu.edu.sg) is with Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371. X. Zhang (drzhangx@ustc.edu.cn) is with School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, China. Part of this work was done while X. Zhang was with the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371. The research of X. Zhang is supported by NSFC under grant 11301503. Copyright (c) 2012 IEEE. Personal use of this material is permitted. \fDe\ufb01nition 1. Let q > q\u2032 be two positive integers. A composition w = Jw1, . . . , wqK is a re\ufb01nement of v = Jv1, . . . , vq\u2032K if there exist pairwise disjoint sets S1, . . . , Sq\u2032 \u2282[q] satisfying \u222aj\u2208[q\u2032]Sj = [q], such that P i\u2208Sj wi = vj for each j \u2208[q\u2032]. Chu et al. [4] made the following observation. Lemma 1. If w is a re\ufb01nement of v, then Aq(n, d, w) \u2265Aq(n, d, v). In [2], Chee et al. showed that Aq(n, d, w) = O(n) if and only if d \u22652w \u22121. For d \u22652w, it is trivial to determine values of Aq(n, d, w). For d = 2w \u22121, Aq(n, 2w \u22121, w) \u2264 \u0016 n w1 \u0017 for all w by Proposition 1. When q = 2, we know that A2(n, 2w \u22121, Jw1K) = \u230an w1 \u230b, trivially. When q = 3, the values of A3(n, 2w \u22121, w) has been completely determined by Svanstr\u00a8 om et al. [3]. Besides this, the following asymptotic statement was proved in [2]. Theorem 1 (Chee et al. [2]). Let w = Jw1, w2 . . . , wq\u22121K. Then Aq(n, 2w \u22121, w) = \u230an w1 \u230bfor all suf\ufb01ciently large n. A. Problem Status and Contribution In Theorem 1, the hypothesis that n is suf\ufb01ciently large must be satis\ufb01ed. But how large must n be? More precisely, for a composition w = Jw1, w2 . . . , wq\u22121K, let Nccc(w) = min{n0 \u2208N : Aq(n, 2w \u22121, w) = \u0016 n w1 \u0017 for all n \u2265n0}, which was \ufb01rst de\ufb01ned in [2]. For binary codes, it is trivial that Nccc(Jw1K) = 1. Explicit bounds on Nccc(w) for general w were given in [2]. Proposition 2. For any composition w, we have w2 \u2212w1(w \u22121) \u2264Nccc(w) \u22644w1(w \u22121)2 + 1. The upper and lower bounds on Nccc(w) in Proposition 2 differ approximately by a factor of 4w1. Our interest in this paper is in determining the exact values of Nccc(w). In fact, a stronger lower bound of Nccc(w) is established in Section II, and proved to be tight for ternary constant-composition codes. In Section III, we provide a general combinatorial construction for optimal linear size constant composition codes. Based on this construction, Sections IV and V serve to prove that our new lower bound of Nccc(w) is also tight for quaternary constant-composition codes. Finally, by re\ufb01ning and lengthening techniques, we determine the values of Nccc(w) for all w provided that 3w1 \u2265w. Our main result is summarized as below. Main Theorem 1. Given a composition w with at least two components. Let \u03bb = \u2308w w1 \u2309and s = \u03bbw1 \u2212w. Then Nccc(w) \u2265\u03bb(\u03bb \u22121)w2 1 \u22122(\u03bb \u22121)sw1 + w1 \u2212\u230a2s \u03bb \u230b. In particular, equality holds for all w provided that 3w1 \u2265w, and w is not a re\ufb01nement of any composition in {J4, 4, 2K, J4, 3, 3K, J5, 5, 3K, J5, 4, 4K}. Previously, exact values of Nccc(w) were known only for binary codes or for compositions w with total weight at most six. II. LOWER BOUNDS In this section, we prove the lower bound of Nccc(w) in Main Theorem 1. Chee et al. [2] showed that the following two conditions are necessary and suf\ufb01cient for a q-ary code C of constant weight w to have distance 2w \u22121: (C1) for any distinct u, v \u2208C, |supp(u) \u2229supp(v)| \u22641, and (C2) for any distinct u, v \u2208C, if x \u2208supp(u) \u2229supp(v), then ux \u0338= vx. The idea of deducing our lower bound is based on the above two conditions, which have been used in [2] to obtain the lower bound in Proposition 2. Let C = {u(1), . . . , u(|C|)} be an (n, 2w \u22121, w)q-code. Then, C can be regarded as an |C| \u00d7 n matrix C, whose jth row is u(j), j \u2208[|C|]. Let Ni be the number of nonzero entries in column i of C, i \u2208[n]. Then, X i\u2208[n] Ni = |C|w. (1) In each column of C, we associate each pair of distinct nonzero entries with the pair of rows that contain these entries. There are \u0000Ni 2 \u0001 such pairs of nonzero entries in column i of C. Therefore, there are P i\u2208[n] \u0000Ni 2 \u0001 such pairs in all columns of \fC. Since there are no pairs of distinct codewords in C whose support intersect in two elements, the P i\u2208[n] \u0000Ni 2 \u0001 pairs of rows associated with the P i\u2208[n] \u0000Ni 2 \u0001 pairs of distinct nonzero entries are also all distinct. Hence, X i\u2208[n] \u0012Ni 2 \u0013 \u2264 \u0012|C| 2 \u0013 . (2) We will use Eq. (2) to obtain our lower bound on Nccc(w). Given a composition w, let \u03bb := \u2308w w1 \u2309and s := \u03bbw1 \u2212w. Since q \u22653, we have \u03bb \u22652 and 0 \u2264s < w1. We \ufb01rst deal with the case when w1|n. Let n = Mw1 and |C| = M. It is easy to show that the left hand side of Eq. (2), P i\u2208[n] \u0000Ni 2 \u0001 achieves the minimum value when all Ni have almost the same values, that is, Ni equals \u03bb or \u03bb \u22121, i \u2208[n] by Eq. (1). Assume that there are x columns such that Ni = \u03bb \u22121. Then by Eq. (1), Mw = \u03bb(Mw1 \u2212x) + (\u03bb \u22121)x. Hence x = \u03bbMw1 \u2212Mw = Ms. By Eq. (2), we have \u0012\u03bb 2 \u0013 (Mw1 \u2212Ms) + \u0012\u03bb \u22121 2 \u0013 Ms \u2264 \u0012M 2 \u0013 , which yields that M \u2265\u03bb(\u03bb \u22121)w1 \u22122(\u03bb \u22121)s + 1. Let \u00b5 := \u03bb(\u03bb \u22121)w1 \u22122(\u03bb \u22121)s. Then n \u2265(\u00b5 + 1)w1, that is, (\u00b5 + 1)w1 is the smallest possible length n which is a multiple of w1 such that Aq(n, 2w \u22121, w) = \u0016 n w1 \u0017 . Next, we deal with length n such that \u00b5w1 < n < (\u00b5 + 1)w1. Suppose that |C| = \u00b5 and n = \u00b5w1 + r, where 1 \u2264r < w1. We need to \ufb01nd the smallest integer r such that Eqs. (1) and (2) both hold. By doing the same arguments as the case when w1|n, we deduce that \u03bb(\u03bb \u22121)r \u2265\u00b5, that is, r \u2265\u2308 \u00b5 \u03bb(\u03bb \u22121)\u2309= w1 \u2212 \u00162s \u03bb \u0017 . Since r < w1, we need 2s \u2265\u03bb in this case. Now we have proved the following lower bound on Nccc(w). Proposition 3. Given a composition w with at least two components. Let \u03bb = \u2308w w1 \u2309and s = \u03bbw1 \u2212w. Then Nccc(w) \u2265\u00b5w1 + \u2308 \u00b5 \u03bb(\u03bb \u22121)\u2309= (\u00b5 + 1)w1 \u2212 \u00162s \u03bb \u0017 , where \u00b5 = \u03bb(\u03bb \u22121)w1 \u22122(\u03bb \u22121)s. As mentioned above, (\u00b5+1)w1 is the smallest possible integer n which is a multiple of w1 such that Aq(n, 2w\u22121, w) = \u230an w1 \u230b. Further, (\u00b5 + 1)w1 \u2212 \u00162s \u03bb \u0017 = w2 \u2212w1(w \u22121) + (w1s \u2212s2 \u2212\u230a2s \u03bb \u230b). Since w1s \u2212s2 \u2212\u230a2s \u03bb \u230b\u22650 when \u03bb \u22652, the lower bound in Proposition 3 is stronger than that in Proposition 2. Observe that in Proposition 3, the lower bound only depends on the total weight w and the biggest component w1. By Lemma 1, it is easy to prove the following fact. Lemma 2. Suppose that w is a re\ufb01nement of v such that w1 = v1. If Nccc(v) achieves the lower bound in Proposition 3, so does Nccc(w). Now we show that for ternary constant-composition codes, the lower bound in Proposition 3 is always achievable. Proposition 4. For all w1 \u2265w2 \u22651, we have Nccc(Jw1, w2K) = 2w1w2 + w2. Proof: Let w = Jw1, w2K. Then \u03bb = 2 and s = w1 \u2212w2. By Proposition 3, Nccc(w) \u22652w1w2 + w2. By [3], A3(n, 2w \u22121, w) = max{M : n \u2265M(w1 + max{w2 \u2212M \u22121 2 , 0})}. So we only need to check that for all n \u22652w1w2 + w2, A3(n, 2w \u22121, w) = \u230an w1 \u230b. Let F(n) := n \u2212M(w1 + max{w2 \u2212M \u22121 2 , 0}), \fwhere M = \u230an w1 \u230bis a function of n. Since A3(n, 2w \u22121, w) \u2264\u230an w1 \u230bfor all w, it suf\ufb01ces to check that F(n) \u22650 for all n \u22652w1w2 + w2. We prove it by induction on n. It is easy to show that F(2w1w2 + w2) = 0. Suppose that F(n) \u22650 for some n \u22652w1w2 + w2, we want to show that F(n + 1) \u22650 too. Let n = Mw1 + r, where 0 \u2264r < w1 and M \u22652w2. If r < w1 \u22121, then n + 1 = Mw1 + r + 1. Hence F(n + 1) = F(n) + 1 \u22651. If r = w1 \u22121, then n + 1 = (M + 1)w1. Hence F(n + 1) = 0. This completes the proof. III. A COMBINATORIAL CONSTRUCTION In this section, we provide a general combinatorial construction for optimal (n, 2w \u22121, w)q-codes of size \u230an w1 \u230b, when the length n \u2265(\u00b5 + 1)w1 \u2212 \u0004 2s \u03bb \u0005 , where \u00b5 = \u03bb(\u03bb \u22121)w1 \u22122(\u03bb \u22121)s, \u03bb = \u2308w w1 \u2309and s = \u03bbw1 \u2212w. From now on, we assume that n \u2265(\u00b5 + 1)w1 \u2212 \u0004 2s \u03bb \u0005 . By Lemma 2, we can also assume that the composition w is not a re\ufb01nement of any v such that w1 = v1. Note that Aq(n, 2w \u22121, w) \u2264M for all length n \u2208[Mw1, Mw1 + w1 \u22121]. If 2s < \u03bb, we only need to construct optimal codes for length n which is a multiple of w1, that is n \u2208{Mw1 : M \u2265\u00b5 + 1}. If 2s \u2265\u03bb, we also need to construct optimal codes for length n = (\u00b5 + 1)w1 \u2212 \u0004 2s \u03bb \u0005 = \u00b5w1 + \u2308 \u00b5 \u03bb(\u03bb\u22121)\u2309. In this case, \u00b5w1 < n < (\u00b5 + 1)w1, and the optimal codes have size upper bounded by \u00b5. For other length n, apply the lengthening method (adding zeros in the end of codewords) used in [2]. For convenience, let S(w) = {(M, Mw1) : M \u2265\u00b5 + 1} \u222a{(\u00b5, \u00b5w1 + \u2308 \u00b5 \u03bb(\u03bb \u22121)\u2309) : if 2s \u2265\u03bb}, which is the collection of pairs (M, n) that we need to construct an (n, 2w \u22121, w)q-code of size M. Before giving our construction, we introduce necessary terminology in combinatorial design theory. A set system is a pair (X, B) such that X is a \ufb01nite set of points and B is a set of subsets of X, called blocks. The order of the set system is |X|, the number of points. For a set of nonnegative integers K, a set system (X, B) is said to be K-uniform if |B| \u2208K for all B \u2208B. A (v, K)-packing is a K-uniform set system (X, B) of order v, such that each pair of X occurs in at most one block in B. The packing number D(v, K) is the maximum number of blocks in any (v, K)-packing. A (v, K)-packing (X, B) is said to be optimal if |B| = D(v, K). If K = {k}, then we write k instead of {k} for short. The values of D(v, k) have been determined for all v when k \u2208{3, 4} [35]. In particular, we have D(v, 3) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 \u0016v 3 \u0016v \u22121 2 \u0017\u0017 \u22121, if v \u22615 (mod 6); \u0016v 3 \u0016v \u22121 2 \u0017\u0017 , otherwise. (3) In fact, when v \u22611, 3 (mod 6), an optimal (v, 3)-packing is also called a Steiner triple system of order v, denoted by STS(v). In this case, each pair of points occurs exactly once. Suppose that C is an (n, 2w \u22121, w)q-code of size M, where (M, n) \u2208S(w). As in Section II, C can be regarded as an M \u00d7 n matrix C, whose rows are codewords of C. For each column c \u2208[n], we assume that Nc = \u03bb or \u03bb \u22121, although it is not necessarily the case. Further, each entry from [q \u22121] occurs at most once in each column. Let the rows of C be indexed by ZM. Then we can de\ufb01ne a (q \u22121)-tuple Ac = (a1, a2, . . . , aq\u22121) \u2208(ZM \u222a{\u2217})q\u22121 for each column c, with ai being the index of the row containing symbol i in column c, i \u2208[q \u22121]. If some symbol i does not occur in column c, then let ai = \u2217. Let A = {Ac : c \u2208[n]}, then A satis\ufb01es the following properties. (T1) For each c, all the elements in Ac excluding \u2217are distinct. Let Bc be the set containing all elements in Ac excluding \u2217and B = {Bc : c \u2208[n]}. Then (ZM, B) is an (M, {\u03bb, \u03bb \u22121})-packing of size n. (T2) For each position i \u2208[q \u22121] (referring positions in Ac), each element of ZM occurs in position i exactly wi times in A. Example 1. Let w = J2, 2, 1K. Then \u03bb = 3 and s = 1. The following is the matrix form of an optimal (18, 9, w)4-code of size 9 from [2]. Let the rows of C be indexed with elements in Z9. Then the corresponding triples of A are listed below each column of C. C = \uf8ee \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8ef \uf8f0 1 0 0 1 2 0 0 0 0 2 0 3 0 0 0 0 0 0 0 0 1 0 0 1 2 0 0 0 0 2 0 3 0 0 0 0 0 0 0 0 1 0 0 1 2 0 0 0 0 2 0 3 0 0 0 0 0 0 0 0 1 0 0 1 2 0 0 0 0 2 0 3 0 3 0 0 0 0 0 0 1 0 0 1 2 0 0 0 0 2 0 2 0 3 0 0 0 0 0 0 1 0 0 1 2 0 0 0 0 0 0 2 0 3 0 0 0 0 0 0 1 0 0 1 2 0 2 0 0 0 0 2 0 3 0 0 0 0 0 0 1 0 0 1 0 1 2 0 0 0 0 2 0 3 0 0 0 0 0 0 1 0 \uf8f9 \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fa \uf8fb \fA : \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 \u2193 0 8 1 0 2 1 3 2 4 3 5 4 6 5 7 6 8 7 7 5 8 6 0 7 1 8 2 0 3 1 4 2 5 3 6 4 \u2217 4 \u2217 5 \u2217 6 \u2217 7 \u2217 8 \u2217 0 \u2217 1 \u2217 2 \u2217 3 It is easy to check that A satis\ufb01es the properties (T1) and (T2). The converse is true. Given a pair (ZM, A), where A \u2282(ZM \u222a{\u2217})q\u22121. If A satis\ufb01es (T1) and (T2) for a composition w, then we can construct an M \u00d7 |A| matrix C in a natural way, where the rows of C form an (|A|, 2w \u22121, w)q-code of size M. In fact, (T1) guarantees the code has minimum distance 2w \u22121, while (T2) guarantees each codeword is of constant composition w. Such a pair (ZM, A) is called a w-balanced (M, q \u22121)-packing. Proposition 5. If there exists a w-balanced (M, q \u22121)-packing of size n, then there exists an (n, 2w \u22121, w)q-code of size M. We aim to construct optimal (n, 2w\u22121, w)q-codes of size M by establishing the existence of w-balanced (M, q\u22121)-packings of size n for (M, n) \u2208S(w). By the similar arguments as in Section II, we can compute the numbers of blocks of sizes \u03bb and \u03bb \u22121 in the (M, {\u03bb, \u03bb \u22121})-packing de\ufb01ned in (T1). The details of these numbers are listed in the following table. TABLE I: Distribution of block sizes \u03bb and \u03bb \u22121 n # blocks of size \u03bb # blocks of size \u03bb \u22121 M \u2265\u00b5 + 1 Mw1 Mw1 \u2212Ms Ms M = \u00b5 Mw1 + \u2308 M \u03bb(\u03bb\u22121)\u2309 Mw1 + (1 \u2212\u03bb)\u2308 M \u03bb(\u03bb\u22121) \u2309\u2212Ms \u03bb\u2308 M \u03bb(\u03bb\u22121) \u2309+ Ms The next two sections will study linear size quaternary constant-composition codes. Given a composition w = Jw1, w2, w3K, if w1 \u2265w2 + w3, then w is a re\ufb01nement of Jw1, w2 + w3K. By Lemma 2 and Proposition 4, the value of Nccc(w) can be determined for this case. Hence we assume that w1 < w2 + w3, that is, \u00b5 = 6w1 \u22124s, \u03bb = 3 and s = 2w1 \u2212w2 \u2212w3 in the remaining of this paper. Distribution of different block sizes in this case is listed below. TABLE II: Distribution of block sizes 3 and 2 n # blocks of size 3 # blocks of size 2 M \u2265\u00b5 + 1 Mw1 M(w1 \u2212s) Ms M = \u00b5 Mw1 + \u2308M 6 \u2309 Mw1 \u22122\u2308M 6 \u2309\u2212Ms 3\u2308M 6 \u2309+ Ms IV. CONSTRUCTIONS FOR n \u2265(\u00b5 + 1)w1 In this section, we show that A4(n, 2w \u22121, w) = \u230an w1 \u230bfor all n \u2265(\u00b5 + 1)w1 = 6w2 1 \u22124sw1 + w1 based on the existence of difference families. A. Difference Families Let B = {b1, . . . , bk} be a k-subset of Zn. The list of differences from B is the multiset \u2206B = \u27e8bi \u2212bj : i, j \u2208[k], i \u0338= j\u27e9. A collection {B1, . . . , Bt} of k-subsets of Zn forms an (n, k; t) difference packing, or t-DP(n, k), if every nonzero element of Zn occurs at most once in \u2206B1 \u222a\u00b7 \u00b7 \u00b7 \u222a\u2206Bt. The sets Bi are base blocks. If every nonzero element of Zn occurs exactly once in \u2206B1 \u222a\u00b7 \u00b7 \u00b7 \u222a\u2206Bt, it is known as an (n, k) difference family, or DF(n, k) [34]. The parameter t is omitted since it could be computed from n and k, that is, t = n\u22121 k(k\u22121). Since t must be an integer, for a DF(n, k) exists, we must have n \u22611 (mod k(k \u22121)). The sizes of base blocks are same in a difference packing. It is natural to generalize difference packings to a collection of subsets with the same property but with varying block sizes. If t = e1 + . . . + es, and if there are ei base blocks of size ki, then the generalized difference packing is of block type ke1 1 \u00b7 \u00b7 \u00b7 kes s , and denoted by GDP(n, ke1 1 \u00b7 \u00b7 \u00b7 kes s ). Without loss of generality, we assume that k1 \u2265\u00b7 \u00b7 \u00b7 \u2265ks \u22652. Given a triple A = (a1, a2, a3) \u2208(ZM \u222a{\u2217})3, de\ufb01ne OrbZM A = {(a1 + i, a2 + i, a3 + i) : i \u2208ZM}, where \u2217+ i = \u2217for any i \u2208ZM. Proposition 6. Suppose that there exists a GDP(M, 3e12e2). Let w1 = e1+e2, w2 and w3 be integers such that w1 \u2265w2 \u2265w3 and w2 + w3 = 2e1 + e2. Then there exists a w-balanced (M, 3)-packing of size n, where n = w1M and w = Jw1, w2, w3K. \fProof: Given a GDP(M, 3e12e2), partition the set B of base blocks into three parts B1, B2 and B3, where B1 consists of all e1 blocks of size three, B2 contains w2 \u2212e1 blocks of size two, and B3 contains the remaining w3 \u2212e1 blocks of size two. For any B = {a, b, c} \u2208B1, de\ufb01ne AB = (a, b, c); for any B = {a, b} \u2208B2, de\ufb01ne AB = (a, b, \u2217) and; for any B = {a, c} \u2208B3, de\ufb01ne AB = (a, \u2217, c). Let A = \u222aB\u2208BOrbZMAB, then (ZM, A) is a w-balanced (M, 3)-packing of size n, where n = w1M and w = Jw1, w2, w3K. In a DF(n, k), the t blocks Bi = {bi,1, . . . , bi,k}, i \u2208[t], form a perfect (n, k) difference family over Zn if the tk(k \u22121)/2 differences bi,h \u2212bi,g (i \u2208[t], 1 \u2264g < h \u2264k) cover the set {1, 2, . . ., (n\u22121)/2}. If instead, they cover the set {1, 2, . . ., (n\u2212 3)/2}\u222a{(n+1)/2}, then the difference family is quasi-perfect. We denote them by PDF(n, k) and quasi-PDF(n, k) respectively. The existences of PDF(n, k)s and quasi-PDF(n, k)s are known when k = 3. Theorem 2. [34] A PDF(n, 3) exists when n \u22611 or 7 (mod 24), and a quasi-PDF(n, 3) exists when n \u226113 or 19 (mod 24). Corollary 1. Let e1, e2 \u22650 be two integers. Then a GDP(M, 3e12e2) exists for all M \u22656e1+2e2+1 except when e1 \u22612 or 3 (mod 4) and (M, e2) = (6e1 + 2, 0). Proof: For each e1 \u22610 or 1 (mod 4), let m = 6e1 + 1. By Theorem 2, there exists a PDF(m, 3) over Zm. Let B be the collection of all e1 base blocks. Given any e2 \u22650, let Pi = {0, m\u22121 2 + i}, i \u2208[e2]. Then B \u222a{Pi : i \u2208[e2]} is a GDP(M, 3e12e2) for all M \u2265m + 2e2. For each e1 \u22612 or 3 (mod 4), let m = 6e1 + 1. By Theorem 2, there exists a quasi-PDF(m, 3) over Zm, which is also a GDP(M, 3e1) for all M \u2265m except when M = m + 1. Let B be the collection of all e1 base blocks. Given any e2 \u22651, let P1 = {0, m\u22121 2 } and Pi = {0, m\u22121 2 + i} for all i \u2208[2, e2]. Then B \u222a{Pi : i \u2208[e2]} is a GDP(M, 3e12e2) for all M \u2265m+ 2e2. By the relations among all parameters in Proposition 6, it is easy to show that 6e1 + 2e2 + 1 = 6w1 \u22124s + 1 = \u00b5 + 1. Combining Corollary 1, Propositions 5 and 6, it is immediate that the following result holds. Proposition 7. Let w = Jw1, w2, w3K. Then A4(n, 2w \u22121, w) = \u230an w1 \u230bfor all n = Mw1, where M \u2265\u00b5 + 1, except when w1 = w2 = w3 \u22612 or 3 (mod 4) and n = 6w2 1 + 2w1. B. Exceptions in Proposition 7 Now we settle the exceptional cases in Proposition 7. That is, we need to prove that A4(6w2 1 + 2w1, 2w \u22121, Jw1, w1, w1K) = 6w1 + 2 for all w1 \u22612 or 3 (mod 4). By Proposition 5, we need to construct a w-balanced (6w1 + 2, 3)-packing of size 6w2 1 + 2w1 for all w1 \u22612 or 3 (mod 4), where w = Jw1, w1, w1K. Actually, they exist for all positive integers w1. Before stating our general construction, we give a small example \ufb01rst. Note that in this case, s = 0, so there are only blocks of size three in the (6w1 + 2, {3, 2})-packing by Table II, which is further optimal by the packing number in Eq. (3). Example 2. Let w1 = 2 and G = Z5 \u2295Z3. Write xy for the pair (x, y) \u2208Z5 \u2295Z3. Let B0 = (00, 01, 02), B1 = (00, 20, 11) and B2 = (00, 40, 21). Let B\u2032 = \u222ai\u2208[0,2]OrbGBi. If we consider all triples in B\u2032 as unordered 3-subsets, then B\u2032 is the block set of an STS(15) over G due to Skolem [36]. Let B = B\u2032 \\ {B \u2208B\u2032 : 42 \u2208B} and X = G \\ {42}, then (X, B) is an optimal (14, 3)-packing of size 28 if again consider triples as unordered sets. We show the reordering procedures in Table III. First look at the set OrbGB1\u222aOrbGB2, in which each element from G occurs twice in each position. After deleting triples containing the element 42, elements 21, 31, 02, 22 occurs only once in the \ufb01rst position, elements 01, 11, 12, 32 occurs only once in the second position, and elements 00, 10, 20, 30 occurs only once in the third position. It is natural to think of reordering triples from OrbGB0 to increase the occurrences of these elements. After reordering the remaining triples of OrbGB0 as in Table III, element 12 occurs three times in the \ufb01rst position but only once in the second position, while element 22 occurs only once in the \ufb01rst position but three times in the second position. Finally, we exchange the \ufb01rst two elements in the triple (12, 32, 20) from OrbGB1 and (32, 22, 00) from OrbGB2 to balance the occurrences. Proposition 8. For all positive integers w1, there exists a w-balanced (6w1 + 2, 3)-packing of size 6w2 1 + 2w1, where w = Jw1, w1, w1K. Proof: Let u = 2w1 + 1. We start from an STS(3u) which is due to Skolem [36]. Let G = Zu \u2295Z3. Choose base blocks A0 = (00, 01, 02) and Ax = (00, (2x)0, x1), x \u2208[w1]. It is easy to see that |OrbGA0| = u and |OrbGAx| = 3u, x \u2208[w1]. Let A\u2032 = \u222ax\u2208[0,w1]OrbGAx, A = A\u2032 \\ {A \u2208A\u2032 : (2w1)2 \u2208A} and X = G \\ {(2w1)2}. If we consider triples as unordered sets, then (G, A\u2032) is an STS(3u), and (X, A) is an optimal (6w1 + 2, 3)-packing of size 6w2 1 + 2w1. Note that A doesn\u2019t satisfy (T2) at this moment. The \ufb01rst three rows of Table IV point out the sets of elements occurring w1\u22121 or w1+1 in each position, all others occur in the corresponding positions exactly w1 times in A. The w1\u22121 occurrences happen when deleting triples from OrbGAx, x \u2208[w1], while w1 + 1 happens because of triples from OrbGA0. \fTABLE III: Reordering Procedures in Example 2 OrbGB0 reordering (00, 01, 02) \u2192(02, 01, 00) (10, 11, 12) \u2192(12, 11, 10) (20, 21, 22) \u2192(21, 22, 20) (30, 31, 32) \u2192(31, 32, 30) (40, 41, 42) OrbGB1 reordering OrbGB2 reordering (00, 20, 11) (00, 40, 21) (10, 30, 21) (10, 00, 31) (20, 40, 31) (20, 10, 41) (30, 00, 41) (30, 20, 01) (40, 10, 01) (40, 30, 11) (01, 21, 12) (01, 41, 22) (11, 31, 22) (11, 01, 32) (21, 41, 32) (21, 11, 42) (31, 01, 42) (31, 21, 02) (41, 11, 02) (41, 31, 12) (02, 22, 10) (02, 42, 20) (12, 32, 20) \u2192(32, 12, 20) (12, 02, 30) (22, 42, 30) (22, 12, 40) (32, 02, 40) (32, 22, 00) \u2192(22, 32, 00) (42, 12, 00) (42, 32, 10) For convenience, denote \u03b9 = \u230aw1 2 \u230band \u03ba = \u2308w1 2 \u2309. We follow the steps below. (S1) In OrbGA0, change ((2i + 1)0, (2i + 1)1, (2i + 1)2) to ((2i + 1)1, (2i + 1)2, (2i + 1)0) for all i \u2208[\u03b9, w1 \u22121]. (S2) In OrbGA0, change ((2i)0, (2i)1, (2i)2) to ((2i)1, (2i)2, (2i)0) for all i \u2208[\u03ba, w1 \u22121]. (S3) In OrbGA0, change (i0, i1, i2) to (i2, i1, i0) for all i \u2208[0, w1 \u22121]. (S4) Finally, in OrbGA\u03ba, change ((2i+1)2, (2\u03ba+2i+1)2, (\u03ba+2i+1)0) to ((2\u03ba+2i+1)2, (2i+1)2, (\u03ba+2i+1)0) for all i \u2208[0, \u03b9 \u22121]. At the same time, in OrbGAw1, change ((2i+1)2, (2i)2, (w1+2i+1)0) to ((2i)2, (2i+1)2, (w1+2i+1)0) for all i \u2208[\u03ba, w1 \u22121]. Note that {(2\u03ba + 2i + 1)2 : i \u2208[0, \u03b9 \u22121]} = {(2i + 1)2 : i \u2208[\u03ba, w1 \u22121]}. After each step, all elements occur at least w1 \u22121 and at most w1 + 1 times in each position. We list the elements occurring w1 \u22121 or w1 + 1 in each position after each step in Table IV. It is routine to check that after (S4), all elements occur w1 times in each position. Thus triples in A can be reordered to satis\ufb01es (T2). TABLE IV: Abnormal occurrences in three positions in Proposition 8 Originally 1 2 3 w1 \u22121 {(2i)2 : i \u2208[0, w1 \u22121]} \u222a{(w1)1 : i \u2208[w1, 2w1 \u22121]} {(2i + 1)2 : i \u2208[0, w1 \u22121]} {i0 : i \u2208[0, 2w1 \u22121]} w1 + 1 {i0 : i \u2208[0, 2w1 \u22121]} {(w1)1 : i \u2208[w1, 2w1 \u22121]} {i2 : i \u2208[0, 2w1 \u22121]} After (S1) 1 2 3 w1 \u22121 {(2i)2 : i \u2208[0, w1 \u22121]} \u222a{(2i)1 : i \u2208[\u03ba, w1 \u22121]} {(2i + 1)2 : i \u2208[0, \u03b9 \u22121]} {i0 : i \u2208[0, w1 \u22121]} \u222a{(2i)0 : i \u2208[\u03ba, w1 \u22121]} w1 + 1 {i0 : i \u2208[0, w1 \u22121]} \u222a{(2i)0 : i \u2208[\u03ba, w1 \u22121]} {(2i)1 : i \u2208[\u03ba, w1 \u22121]} {i2 : i \u2208[0, w1 \u22121]} \u222a{(2i)2 : i \u2208[\u03ba, w1 \u22121]} After (S2) 1 2 3 w1 \u22121 {(2i)2 : i \u2208[0, w1 \u22121]} {(2i + 1)2 : i \u2208[0, \u03b9 \u22121]} {i0 : i \u2208[0, w1 \u22121]} w1 + 1 {i0 : i \u2208[0, w1 \u22121]} {(2i)2 : i \u2208[\u03ba, w1 \u22121]} {i2 : i \u2208[0, w1 \u22121]} After (S3) 1 2 3 w1 \u22121 {(2i)2 : i \u2208[\u03ba, w1 \u22121]} {(2i + 1)2 : i \u2208[0, \u03b9 \u22121]} \u2205 w1 + 1 {(2i + 1)2 : i \u2208[0, \u03b9 \u22121]} {(2i)2 : i \u2208[\u03ba, w1 \u22121]} \u2205 The following consequence is immediate. Proposition 9. For all positive integers w1 and n = 6w2 1 + 2w1, A4(n, 2w \u22121, Jw1, w1, w1K) = 6w1 + 2. Combining Propositions 7 and 9, we have shown that A4(n, 2w\u22121, Jw1, w2, w3K) = \u230an w1 \u230bfor all n = Mw1, where M \u2265\u00b5+ 1. By the lengthening method (adding zeros in the end of codewords) used in [2], we have A4(n, 2w\u22121, Jw1, w2, w3K) = \u230an w1 \u230b for all n \u2265(\u00b5 + 1)w1. V. CONSTRUCTIONS FOR n = \u00b5w1 + \u2308\u00b5 6 \u2309 To determine values of Nccc(w), we still need to prove that A4(n, 2w \u22121, Jw1, w2, w3K) = \u230an w1 \u230bfor n = \u00b5w1 + \u2308\u00b5 6 \u2309if 2s \u2265\u03bb = 3. From now on, we assume that s \u22652, that is 2w1 \u2265w2 + w3 + 2. \fBy Proposition 5, we need to construct a w-balanced (\u00b5, 3)-packing of size \u00b5w1 + \u2308\u00b5 6 \u2309for all w = Jw1, w2, w3K satisfying that s \u22652. Here we use different method from that in Proposition 8. We \ufb01rst \ufb01nd a candidate of A satisfying (T2), then try to modify it to satisfy (T1). We show this idea in the following example. Example 3. Let M = 10 and w = J3, 2, 2K. Let A1 = (0, 1, 6), A2 = (0, 2, \u2217) and A3 = (0, \u2217, 3). Then A = \u222ai\u2208[3]OrbZ10Ai is a candidate satisfying (T2) over Z10. Note that the difference 5 occurs twice in \u2206A1. We \ufb01rst do the following changes to triples in OrbZ10A1: (0, 1, 6) \u2192(0, \u2217, 6), (1, 2, 7) \u2192(1, 2, \u2217), (3, 4, 9) \u2192(3, \u2217, 9), (4, 5, 0) \u2192(4, 5, \u2217), (7, 8, 3) \u2192(7, 8, \u2217). Then add two more triples (\u2217, 1, 0), (\u2217, 4, 3). Finally, change the following triple in OrbZ10A2 (1, 3, \u2217) \u2192(1, 3, 7). Note that we do not change the positions of symbols from Z10 appearing in A. For example, in the \ufb01rst triple (0, 1, 6), the symbol 1 in the second position disappears, but appears later in (\u2217, 1, 0) in the same position. Further, the pairs newly occurring in the last two steps are pairs deleted in the \ufb01rst step. For example, the pair {0, 1} appears in the second step when adding (\u2217, 1, 0), but it was deleted before in the \ufb01rst step when changing (0, 1, 6) to (0, \u2217, 6). So the pair {0, 1} still occurs only once after these three steps. Thus it is easy to check that we have a set of 32 triples satisfying both (T1) and (T2), which yields that A4(32, 13, J3, 2, 2K) = 10 and Nccc(J3, 2, 2K) = 32. Proposition 10. Let e1 \u22650, e2 \u22653, 2|M and M \u22658 be integers. Suppose that there exists a GDP(M, 3e12e2) with three speci\ufb01ed base blocks {0, 1}, {0, 2} and {0, M/2\u22121}. Let w1 = e1+e2\u22121, w2 and w3 be any integers such that w1 \u2265w2 \u2265w3 and w2 + w3 = 2e1 + e2. Then there exists a w-balanced (M, 3)-packing of size n = Mw1 + \u2308M 6 \u2309, where w = Jw1, w2, w3K. Proof: Suppose that B\u2032 is the given set of base blocks of a GDP(M, 3e12e2) over ZM. Let B1 = {0, 1, M/2 + 1}, B2 = {0, 2} and B = (B\u2032 \\ {{0, 1}, {0, M/2 \u22121}}) \u222a{B1}. Note that B has e1+e2\u22121 = w1 blocks. Partition B into three parts B1, B2 and B3, where B1 consists of all e1+1 blocks of size three, B2 contains w2 \u2212e1 \u22121 blocks of size two including B2 speci\ufb01cally, and B3 contains the remaining w3 \u2212e1 \u22121 blocks of size two. For each block B \u2208B1, let AB be an ordered triple with elements from B. For each block B = {a, b} \u2208B2, let AB = (a, b, \u2217). For each block B = {a, c} \u2208B3, let AB = (a, \u2217, c). Speci\ufb01cally, let AB1 = (0, 1, M/2 + 1) and AB2 = (0, 2, \u2217). Let A = \u222aB\u2208BOrbZMAB, then A is a candidate of Mw1 triples over ZM \u222a{\u2217} satisfying (T2). Now we do modi\ufb01cations on triples in A to make it satisfy both (T1) and (T2). The main idea is as follows. The set A does not satisfy (T1) since the difference M/2 occurs twice in \u2206B1. Thus, we \ufb01rst choose M/2 triples from OrbZMAB1, then change one symbol from each repeated pair to \u2217. Besides the M/2 repeated pairs, there are M/2 other pairs also broken in this step. We let them appear somewhere else by adding \u2308M 6 \u2309triples of type (\u2217, \u00b7, \u00b7), and changing the symbol \u2217in M/2 \u22122\u2308M 6 \u2309 triples from OrbZMAB2 to some symbol of ZM. Details for different congruent classes of M are listed in Table V. By Proposition 10, we need to construct a GDP(\u00b5, 3e12e2) with three speci\ufb01ed base blocks {0, 1}, {0, 2} and {0, \u00b5/2 \u22121}, where e1 = w1 \u2212s \u22121 and e2 = s + 2, for all w1 > s \u22652. Proposition 11. Given integers w1 > s \u22652 and (w1, s) \u0338\u2208{(4, 2), (5, 2)}, let e1 = w1 \u2212s \u22121 and e2 = s + 2. Then there exists a GDP(\u00b5, 3e12e2) over Z\u00b5 with three speci\ufb01ed base blocks {0, 1}, {0, 2} and {0, \u00b5/2 \u22121}. Proof: We split it into four cases based on the values of w1 \u2212s. For all cases, the e2 base blocks of size two are of type {0, d}, where d covers the values of all differences that do not appear in the list of differences from base blocks of size three. To save space, we only list the e1 base blocks of size three in each case. Note that the differences 1, 2, \u00b5/2 and \u00b5/2 \u22121 do not appear in any base block of size three. Thus the GDP contains the three mentioned base blocks of size two. Note that in the \ufb01rst two cases, that is when w1 \u2212s \u22610 or 1 (mod 4), the set of base blocks of size three are obtained by modifying some blocks of a DF(\u00b5, 3) in [37]. \fTABLE V: Modi\ufb01cations made on triples of A in Proposition 10 M = 6k, k \u22652 OrbZM AB1 (3i, 3i + 1, 3i + 3k + 1) \u2192(3i, 3i + 1, \u2217), i \u2208[0, k \u22121] (3i + 3k \u22121, 3i + 3k, 3i) \u2192(3i + 3k \u22121, 3i + 3k, \u2217), i \u2208[0, k \u22121] (3i + 3k + 1, 3i + 3k + 2, 3i + 2) \u2192(3i + 3k + 1, \u2217, 3i + 2), i \u2208[0, k \u22121] Add triples (\u2217, 3i + 3k + 2, 3i + 3k + 1), i \u2208[0, k \u22121] OrbZM AB2 (3i + 3k \u22121, 3i + 3k + 1, \u2217) \u2192(3i + 3k \u22121, 3i + 3k + 1, 3i), i \u2208[0, k \u22121] M = 6k + 2, k \u22651 OrbZM AB1 (3i + 1, 3i + 2, 3i + 3k + 3) \u2192(3i + 1, 3i + 2, \u2217), i \u2208[0, k \u22121] (3i + 3k, 3i + 3k + 1, 3i) \u2192(3i + 3k, 3i + 3k + 1, \u2217), i \u2208[0, k \u22121] (3i, 3i + 1, 3i + 3k + 2) \u2192(3i, \u2217, 3i + 3k + 2), i \u2208[0, k \u22121] (6k, 6k + 1, 3k) \u2192(6k, \u2217, 3k) Add triples (\u2217, 3i + 1, 3i), i \u2208[0, k \u22121] (\u2217, 6k + 1, 6k) OrbZM AB2 (3i + 1, 3i + 3, \u2217) \u2192(3i + 1, 3i + 3, 3i + 3k + 3), i \u2208[0, k \u22122] M = 6k + 4, k \u22651 OrbZM AB1 (3i + 1, 3i + 2, 3i + 3k + 3) \u2192(3i + 1, 3i + 2, \u2217), i \u2208[0, k \u22121] (3i + 3k + 1, 3i + 3k + 2, 3i) \u2192(3i + 3k + 1, 3i + 3k + 2, \u2217), i \u2208[0, k] (3i, 3i + 1, 3i + 3k + 3) \u2192(3i, \u2217, 3i + 3k + 3), i \u2208[0, k] Add triples (\u2217, 3i + 1, 3i), i \u2208[0, k \u22121] OrbZM AB2 (3i + 1, 3i + 3, \u2217) \u2192(3i + 1, 3i + 3, 3i + 3k + 4), i \u2208[0, k \u22121] When w1 \u2212s = 4k, then \u00b5 = 24k + 2s and e1 = 4k \u22121. If k \u22652, then use e1 base blocks of size three as below. {0, 6k \u22121, 18k + 2s \u22121}, {0, 4k \u22121, 9k \u22121}, {0, 2k, 10k \u22121}, {0, 4k, 10k}, {0, 2k + 2r \u22121, 7k + r \u22121}, r \u2208[1, k \u22121], {0, 2k + 2r, 11k + r \u22121}, r \u2208[1, k \u22121], {0, 2r + 1, 10k + r}, r \u2208[1, k \u22121], {0, 2r, 6k + r}, r \u2208[2, k \u22121]. If k = 1, and s = 2, that is \u00b5 = 28, then three base blocks are {0, 3, 11}, {0, 4, 9}, {0, 6, 16}. If s \u22653, then three base blocks are {0, 3, 13}, {0, 4, 12}, {0, 5, 11}. When w1 \u2212s = 4k + 1, then \u00b5 = 24k + 2s + 6 and e1 = 4k. If k \u22652, then the e1 base blocks of size three are {0, 6k, 18k + 2s + 4}, {0, 4k \u22121, 9k}, {0, 4k, 10k + 1}, {0, 4k + 1, 12k + 1}, {0, 2k, 12k}, {0, 2k + 2r \u22121, 7k + r}, r \u2208[1, k \u22121], {0, 2k + 2r, 11k + r}, r \u2208[1, k \u22121], {0, 2r + 1, 10k + r + 1}, r \u2208[1, k \u22121], {0, 2r, 6k + r + 1}, r \u2208[2, k \u22121]. If k = 1, and s \u22652, then four base blocks are {0, 3, 14}, {0, 4, 12}, {0, 5, 15}, {0, 6, 13}. When w1 \u2212s = 4k + 2, then \u00b5 = 24k + 2s + 12 and e1 = 4k + 1. If k \u22651, then the e1 base blocks are {0, 4k + 1, 10k + 6}, {0, 4k + 3, 10k + 7}, {0, 2r, 6k + r + 5}, r \u2208[2, 2k + 1], {0, 2r + 1, 10k + r + 7}, r \u2208[1, k], {0, 2k + 2r + 1, 11k + r + 7}, r \u2208[1, k \u22121]. If k = 0, and s \u22653, then let {0, 3, 7} be the only base block of size three. When w1 \u2212s = 4k + 3, then \u00b5 = 24k + 2s + 18 and e1 = 4k + 2. If k \u22651, then the e1 base blocks are {0, 4k + 1, 10k + 9}, {0, 4k + 3, 10k + 10}, {0, 4k + 4, 8k + 9}, {0, 2r, 6k + r + 7}, r \u2208[2, 2k + 1], {0, 2r + 1, 10k + r + 10}, r \u2208[1, k \u22121], {0, 2k + 2r \u22121, 11k + r + 9}, r \u2208[1, k]. If k = 0, and s \u22653, then two base blocks are {0, 3, 8}, {0, 4, 10}. Combining Propositions 5, 10 and 11, we obtain the following result. \fProposition 12. Given a composition w = Jw1, w2, w3K such that 2 \u2264s < w1, where s = 3w1 \u2212w. We have A4(\u00b5w1 + \u2308\u00b5 6 \u2309, 2w \u22121, Jw1, w2, w3K) = \u00b5, for all w1 \u2265w2 \u2265w3 \u22651 such that (w1, s) \u0338\u2208{(4, 2), (5, 2)}. By Propositions 7, 9, 12 and the lengthening method, we have determined the value of Nccc(w) for almost all compositions with three components. We state it in the following proposition. Proposition 13. Given a composition w = Jw1, w2, w3K such that w1 < w2 + w3. We have Nccc(w) = 6w2 1 \u22124sw1 + w1 \u2212\u230a2s 3 \u230b, where s = 3w1 \u2212w, except possibly when w \u2208{J4, 4, 2K, J4, 3, 3K, J5, 5, 3K, J5, 4, 4K}. VI."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1411.2513v1",
+                    "title": "Constructions of Optimal and Near-Optimal Multiply Constant-Weight Codes",
+                    "abstract": "Multiply constant-weight codes (MCWCs) have been recently studied to improve\nthe reliability of certain physically unclonable function response. In this\npaper, we give combinatorial constructions for MCWCs which yield several new\ninfinite families of optimal MCWCs. Furthermore, we demonstrate that the\nJohnson type upper bounds of MCWCs are asymptotically tight for fixed weights\nand distances. Finally, we provide bounds and constructions of two dimensional\nMCWCs.",
+                    "authors": "Yeow Meng Chee, Han Mao Kiah, Hui Zhang, Xiande Zhang",
+                    "published": "2014-11-10",
+                    "updated": "2014-11-10",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.CO",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION A multiply constant-weight code (MCWC) is a binary code of length mn where each codeword is partitioned into m equal parts and has weight exactly w in each part [1], [2]. This de\ufb01nition therefore generalizes the class of constant-weight codes (where m = 1) and a subclass of doubly constant-weight codes (where m = 2), introduced by Johnson [3] and Levenshte\u02d8 \u0131n [4]. Multiply constant-weight codes have attracted recent attention due to an application in the design of certain physically unclonable functions (PUFs) [1], [2], [5]. Introduced by Pappu et al. [6], PUFs provide innovative low-cost authentication methods that are derived from complex physical characteristics of electronic devices and have recently become an attractive option to provide security in low cost devices such as RFIDs and smart cards [5]\u2013[8]. Reliability and implementation considerations on programmable circuits for the design of Loop PUFs [5] lead to the study of multiply constant-weight codes. If we arrange each codeword in an MCWC as an m \u00d7 n array, then an MCWC can be regarded as a code over binary matrices, where each matrix has constant row weight w. Furthermore, if certain weight constraints are also satis\ufb01ed by all columns, we obtain two-dimensional weight-constrained codes that are used in the design of optical storage in holographic memory [9], and have possible applications in next generation memory technologies based on crossbar arrays of resistive devices, such as memristors [10]. Recently, these codes were also studied by Chee et al. [11] in an application for power line communications. Y. M. Chee, H. Zhang and X. Zhang are with the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore (emails:{YMChee, huizhang, XiandeZhang}@ntu.edu.sg). Han Mao Kiah is with Coordinated Science Lab, University of Illinois, Urbana-Champaign, 1308 W. Main Street, Urbana, IL 61801, USA, (email:hmkiah@illinois.edu). \f2 The theory of MCWC is at a rudimentary stage. Chee et al. established certain preliminary upper and lower bounds for possible sizes of MCWCs [1]. In particular, they extended techniques of Johnson [3] to derive certain upper bounds and showed that these bounds are asymptotically tight to a constant factor. However, the only nontrivial in\ufb01nite class of optimal codes was constructed from Reed-Solomon codes under the conditions that n/w \u2265mw \u22121 is a prime power and d \u22652m(w \u22121) + 2. In this paper, we continue this study and provide constructions of optimal or near-optimal MCWCs based on combinatorial techniques. Our main contributions are as follows: (i) determining completely the maximum sizes of MCWCs with total weight four and distance four; (ii) constructing in\ufb01nite families of optimal MCWCs with distance four and weight two or three in each part; (iii) establishing that the Johnson type upper bounds are asymptotically tight for \ufb01xed weight and distances. Our paper is organized as follows. In Section II, we give necessary de\ufb01nitions and notation, where a more general concept of MCWCs with different lengths and weights in different parts is introduced. Section III links the general MCWCs to generalized packing designs, a class of combinatorial objects recently studied by Bailey and Burgess [12]. By establishing the existence of optimal generalized packing designs, we completely determine the maximum sizes of MCWCs with total weight four and distance four. Combining results in [12], the maximum sizes of MCWCs with total weight less than or equal to four are determined except for only one small case. Section IV gives a product construction of MCWCs with equal length and equal weight in each part. Based on the existence of large sets of optimal packings, this construction yields in\ufb01nite families of optimal MCWCs with distance four and weight two or three in each part. Section V exhibits that the Johnson type upper bounds of general MCWCs are asymptotically tight for \ufb01xed weights and distance by applying a theorem on fractional matchings due to Kahn [13]. This improves the previous result in [1] which states that the bounds are asymptotically tight to a constant factor and for a smaller class of MCWCs. Finally in Section VI, we formally de\ufb01ne the notion of two dimensional MCWCs and provide several bounds and constructions. II. PRELIMINARIES Given a positive integer n, the set {1, 2, . . . , n} is denoted by In, and the ring Z/nZ is denoted by Zn. Through out this paper, let R = {0, 1} and let X be a \ufb01nite set of size N. Then RX denotes the set of all binary vectors of length N, where each component of a vector u \u2208RX is indexed by an element of X, that is, u = (ux)x\u2208X, and ux \u2208R for each x \u2208X. A binary code of length N is a set C \u2286RX for some X of size N. The elements of C are called codewords. The Hamming norm of a vector u \u2208RX is de\ufb01ned as \u2225u\u2225= |{x \u2208X : ux = 1}|. The distance induced by this norm is called the Hamming distance, denoted dH, so that dH(u, v) = \u2225u \u2212v\u2225, for u, v \u2208RX. A code C is said to have (minimum Hamming) distance d if dH(u, v) \u2265d for all distinct u, v \u2208C, denoted by (N, d) code. The largest size of an (N, d) code is denoted by A(N, d). Suppose that X1, X2, . . . , Xm is a partition of X with |Xi| = ni, i \u2208Im. Clearly, N = n1 + \u00b7 \u00b7 \u00b7 + nm. De\ufb01ne X = (X1, X2, . . . , Xm) and n = (n1, n2, . . . , nm). For any vector u \u2208RX, de\ufb01ne the support of \f3 u associated with X as supp(u)X = (U1, U2, . . . , Um), where Ui = {x \u2208Xi : ux = 1}. If the subscript X is omitted, then supp(u) = \u222ai\u2208ImUi is the usual support set. Let x = (x1, . . . , xm) and y = (y1, . . . , ym) be two m-tuples of nonnegative integers. We write x \u2264y to mean that xi \u2264yi for all i \u2208Im. For any integer k, use the notation \u0000X k \u0001 to denote the set of all k-subsets of X. Suppose w = (w1, w2, . . . , wm) is an m-tuple of integers such that w \u2264n. Let W \u225cw1+\u00b7 \u00b7 \u00b7+wm. De\ufb01ne \u0012X w \u0013 = \u0012X1 w1 \u0013 \u00d7 \u0012X2 w2 \u0013 \u00d7 \u00b7 \u00b7 \u00b7 \u00d7 \u0012Xm wm \u0013 , i.e., a member of \u0000X w \u0001 is an m-tuple of sets, of sizes (w1, w2, . . . , wm). Let C \u2286RX be an (N, d) code. If for each u \u2208C, supp(u)X \u2208 \u0000X w \u0001 , then C is said to be of multiply constant-weight, denoted by MCWC(n, d, w). The number of codewords in an MCWC(n, d, w) is called the size of the code. The maximum size of an MCWC(n, d, w) is denoted T(n, d, w), and the MCWC(n, d, w) achieving this size is said to be optimal. Speci\ufb01cally, when m = 1, an MCWC(n, d, w) is a constant-weight code, denoted by CWC(n, d, w) with n = n1 and w = w1; when m = 2, an MCWC(n, d, w) is a doubly constant-weight code [14]. The largest size of a CWC(n, d, w) is denoted by A(n, d, w). For general m, when ni = n and wi = w for all i \u2208Im, C is denoted by MCWC(m, n, d, w). In this case, we use the notation M(m, n, d, w) to denote the maximum size of such a code. Observe that by de\ufb01nition, M(1, n, d, w) = A(n, d, w), M(2, n, d, w) = T((n, n), d, (w, w)). Moreover, the functions A(n, d, w) and T((n, n), d, (w, w)) have been well studied, see for example [3], [14]\u2013[17]. For the general case T(n, d, w), some lower and upper bounds were studied in [1]. The techniques of Johnson bound have been applied to get the following recursive bounds in [1], [2]. Proposition II.1. We have 1) For each i \u2208Im, T(n, d, w) \u2264\u230ani wiT(n\u2032, d, w\u2032)\u230bwhere n\u2032 = (n1, . . . , ni\u22121, ni \u22121, ni+1, . . . , nm) and w\u2032 = (w1, . . . , wi\u22121, wi \u22121, wi+1, . . . , wm). 2) M(m, n, d, w) \u2264 \u0004 nm wmM(m, n \u22121, d, w \u22121) \u0005 . Since all the codes we consider in this paper are binary, we can assume that the distance d is even and let d \u225c2\u03b4 for convenience. Trivial cases are as follows. Lemma II.1. If d > 2 P i\u2208Im wi, then T(n, d, w) = 1; if d = 2 Pm i=1 wi, then T(n, d, w) = mini\u2208Im\u230ani wi\u230b; if d = 2, then T(n, d, w) = Q i\u2208Im \u0000ni wi \u0001 . III. OPTIMAL MCWCS WITH SMALL WEIGHT We construct optimal multiply constant-weight codes with total weight four in this section. Our approach is based on combinatorial design theory. First, we introduce necessary concepts and establish connections to multiply constant-weight codes. \f4 A. Generalized packing designs A set system is a pair (X, B) such that X is a \ufb01nite set of points and B is a set of subsets of X, called blocks. The order of the set system is |X|, the number of points. For a set K of nonnegative integers, a set system (X, B) is said to be K-uniform if |B| \u2208K for all B \u2208B. When K = {k}, we simply write that the system is k-uniform. Let N \u2265t and \u03bb \u22651. A t-(N, K, \u03bb) packing is a K-uniform set system (X, B) of order N, such that every t-subset of X occurs in at most \u03bb blocks in B. If K = {k} and each pair occurs in exactly \u03bb blocks, then we call it a balanced incomplete block design, and denote it by BIBD(N, k, \u03bb). A BIBD(N, 3, 1) is known as a Steiner triple systems of order N, denoted by STS(N). In [18], Cameron introduced a new class of combinatorial designs, which simultaneously generalizes various well-known classes of designs, including t-designs, mutually orthogonal Latin squares, orthogonal arrays and 1-factorizations of complete graphs. In a recent paper [12], Bailey and Burgess considered the analogue of Cameron\u2019s generalization for packings, which are called generalized packing designs. To de\ufb01ne generalized packing designs, we require various pieces of notation and terminology. If A = (A1, . . . , Am) and B = (B1, . . . , Bm) are m-tuples of sets, we write A \u2286B to mean that Ai \u2286Bi for all i \u2208Im, and say A is contained in B. Again let X = (X1, X2, . . . , Xm) and n = (n1, n2, . . . , nm) as in Section II. Assume that k = (k1, . . . , km) is an m-tuple of positive integers with sum k such that k \u2264n. Let t = (t1, . . . , tm) be an m-tuple of non-negative integers. We say t is (k, t)-admissible if t \u2264k and P i\u2208Im ti = t. In a similar manner, if T = (T1, . . . , Tm) is an m-tuple of disjoint sets, we say that T is (n, k, t)-admissible if each Ti is a ti-subset of Xi, where (t1, . . . , tm) is (k, t)-admissible. Note that since ti is allowed to be zero, the corresponding set Ti is allowed to be empty. A t-(n, k, \u03bb) generalized packing design, or more succinctly a generalized packing, is a pair (X, P), such that P is a family of elements of \u0000X k \u0001 , called blocks, with the property that every T = (T1, . . . , Tm) which is (n, k, t)-admissible is contained in at most \u03bb blocks in P. As with ordinary packings, the generalized packing number D\u03bb(n, k, t) is the maximum possible number of blocks in a t-(n, k, \u03bb) generalized packing. When n and k have only one component, we simply write D\u03bb(N, k, t), which is the usual packing number. When \u03bb is omitted, we mean \u03bb = 1. The equivalence between multiply constant-weight codes and generalized packing designs is obvious. Proposition III.1. There exists an MCWC(n, d, w) of size b if and only if a t-(n, w, 1) generalized packing of size b exists, where t = W \u2212\u03b4 + 1. Corollary III.1. T(n, d, w) = D(n, w, t) where t = W \u2212\u03b4 + 1. A few general upper bounds for D(n, k, t) can be found in [12]. The cases for t = 2 and k = 3 or 4 were completely determined except when k = (2, 2), n1, n2 are both odd and n1 \u2264n2 \u22642n1 \u22121 [12]. When t = 3 and k = 3, the designs are trivial. In the following subsection, we determine D(n, k, 3) for k = 4 completely. Note that the upper bounds for all cases discussed later could be easily obtained by applying techniques of Johnson. \f5 B. Determination of D(n, k, 3) for k = 4 We split the problem into \ufb01ve cases. Case 1: k = (4). Let n = (n), then a 3-(n, k, 1) generalized packing design is indeed a 3-(n, 4, 1) packing, for which the determination of packing numbers D(n, 4, 3) has been completed by Bao and Ji in [19]. Hence, we have the following result. Proposition III.2. For any positive integer n, D((n), (4), 3) = \uf8f1 \uf8f2 \uf8f3 \u0004 n 4 \u0004 n\u22121 3 \u0004 n\u22122 2 \u0005\u0005\u0005 n \u0338\u22610 (mod 6), \u0004 n 4 \u0004 n\u22121 3 \u0004 n\u22122 2 \u0005\u0005 \u22121 \u0005 n \u22610 (mod 6). Case 2: k = (3, 1). Let n = (n1, n2), then we have D(n, k, 3) \u2264min{ \u0000n1 3 \u0001 , \u230an2D(n1, 3, 2)\u230b}. We prove that the upper bound is achievable by using disjoint partial triple systems. Two 2-(N, k, 1) packings (X, A) and (X, B) are disjoint if A \u2229B = \u2205. When n \u22611, 3 (mod 6), n \u0338= 7, there exists a large set of Steiner triple systems of order n, i.e., a set of n \u22122 pairwise disjoint optimal 2-(n, 3, 1) packings, [20]\u2013[22]. By collecting from each STS(n) the blocks not containing a \ufb01xed point, we obtain a set of n \u22122 disjoint optimal 2-(n, 3, 1) packings of order n \u22121. For n \u22614 (mod 6), it was proved by Etzion [23], [24] that there exists a partition of all triples into n \u22121 optimal 2-(n, 3, 1) packings and one 2-(n, 3, 1) packing of size (n \u22121)/3. For n \u22615 (mod 6), n \u226511, Ji [25] proved that there is a partition of triples into n \u22122 optimal 2-(n, 3, 1) packings and one packing of size 4(n\u22122)/3. Let M(n) denote the maximum number of disjoint optimal 2-(n, 3, 1) packings. Then for n \u22658, M(n) = n \u22122 when n is odd and M(n) = n \u22121 when n is even. Proposition III.3. D((n1, n2), (3, 1), 3) = min{ \u0000n1 3 \u0001 , \u230an2D(n1, 3, 2)\u230b} for all n1, n2 > 0 except for (n1, n2) \u2208{(6, 5), (7, 3), (7, 4), (7, 5)}, whose values are listed below. (n1, n2) (6, 5) (7, 3) (7, 4) (7, 5) D((n1, n2), (3, 1), 3) 18 20 26 31 Proof: Let X = (X1, X2) with |Xi| = ni, i = 1, 2. Assume that (X1, Bi), i \u2208IM(n1)+\u03b4 are disjoint 2-(n1, 3, 1) packings as above with the \ufb01rst M(n1) being optimal. Here \u03b4 = 1 when n \u22614, 5 (mod 6) and 0 otherwise. For each j \u2264min{M(n1) + \u03b4, n2}, de\ufb01ne C(j) = {(B, {x}) : B \u2208Bi and x is i-th element of X2, 1 \u2264i \u2264j}. For n1 \u22658, if n2 \u2264M(n1), then (X, C(n2)) is an optimal generalized packing of size \u230an2D(n1, 3, 2)\u230b; if n2 > M(n1), then (X, C(M(n1 + \u03b4))) is optimal with \u0000n1 3 \u0001 blocks. For n1 \u22644, the optimal packings are trivial. For n1 = 5, the triples in X1 can be partitioned into \ufb01ve disjoint optimal 2-(5, 3, 1) packings, i.e., M(5) = 5. For example, let B1 = {125, 345}, B2 = {135, 234}, B3 = {145, 123}, B4 = {235, 124} and B5 = {245, 134}, then (I5, Bi), i \u2208I5 are disjoint optimal 2(5, 3, 1) packings. Then by the same construction, we have D(n, k, 3) = 2n2 if n2 \u22645 and D(n, k, 3) = \f6 \u00005 3 \u0001 = 10 if n2 > 5. For n1 = 7, Cayley [26] showed that there are only two mutually disjoint STS(7)s, i.e., M(7) = 2. Now consider the following six disjoint 2-(7, 3, 1) packings (I7, Bi) with i \u2208I6: B1 = {123, 145, 167, 246, 257, 347, 356}, B2 = {124, 137, 156, 235, 267, 346, 457}, B3 = {146, 157, 247, 256, 345, 367}, B4 = {125, 136, 147, 234, 357, 456}, B5 = {126, 135, 237, 245, 567}, B6 = {127, 134, 236, 467}. So for n2 \u22642, (X, C(n2)) is an optimal generalized packing of size \u230an2D(n1, 3, 2)\u230b; for n2 = 3 or 4, (X, C(n2)) is an optimal generalized packing of size \u230an2D(n1, 3, 2)\u230b\u2212n2 + 2; for n2 = 5, (X, C(5)) is a generalized packing of size 31, which is also optimal by an exhaustive computer search; for all n2 \u22656, (X, C(6)) is an optimal generalized packing of size \u0000n1 3 \u0001 . For n1 = 6, delete all blocks containing the element 2 from the above six 2-(7, 3, 1) packings over I7, then we get six disjoint 2-(6, 3, 1) packings over I7 \\ {2} with blocks set B\u2032 i, where |B\u2032 i| = 4 for 1 \u2264i \u22644 and |B\u2032 i| = 2 for i = 5, 6. So for n2 \u22644, (X, C(n2)) is an optimal generalized packing of size \u230an2D(n1, 3, 2)\u230b; for n2 = 5, (X, C(5)) is optimal of size 18 by an exhaustive computer search; for all n2 \u22656, (X, C(6)) is an optimal generalized packing of size \u0000n1 3 \u0001 . Case 3: k = (2, 2). Let n = (n1, n2) and assume n1 \u2264n2 without loss of generality. Then the upper bound is D(n, k, 3) \u2264 n1(n1\u22121) 2 \u0004 n2 2 \u0005 . Denote Kn a complete graph with n vertices. Let F(n) = {F1, F2, . . . , F\u03b3(n)} be a 1-factorization of Kn if n \u22610 (mod 2), or a near 1-factorization of Kn if n \u22611 (mod 2). Then \u03b3(n) = n \u22121 if n \u22610 (mod 2), and \u03b3(n) = n if n \u22611 (mod 2). Further, each 1-factor Fi is of size \u0004 n 2 \u0005 , i \u2208I\u03b3(n). Proposition III.4. D((n1, n2), (2, 2), 3) = n1(n1\u22121) 2 \u0004 n2 2 \u0005 , where n1 \u2264n2. Proof: Let X = (X1, X2) with |Xi| = ni, i = 1, 2. Suppose that F(ni) is a (near) 1-factorization of Kni with vertex set Xi, i = 1, 2. The upper bound is achieved by taking all blocks in {(P, Q) : P \u2208Fi \u2208 F(n1), Q \u2208F \u2032 i \u2208F(n2), 1 \u2264i \u2264\u03b3(n1)}. Case 4: k = (2, 1, 1). Let n = (n1, n2, n3) and assume n2 \u2264n3 without loss of generality. Then the upper bound is D(n, k, 3) \u2264min{ n1(n1\u22121)n2 2 , \u0004n1 2 \u0005 n2n3}. An n2 \u00d7 n3 Latin rectangle (LR(n2, n3)) is an n2 \u00d7 n3 array over In3 such that each symbol occurs exactly once in each row, and at most once in each column. When n2 = n3, it is also called a Latin square of order n3. An LR(n2, n3) can always be obtained from a Latin square of order n3 by collecting any n2 rows. \f7 Proposition III.5. Let n2 \u2264n3. Then D((n1, n2, n3), (2, 1, 1), 3) = min{n1(n1 \u22121)n2 2 , jn1 2 k n2n3}. Proof: Let X = (In1, In2, In3). Suppose that M = (mij) is an LR(n2, n3) over In3 and F(n1) is a (near) 1-factorization of Kn1 with vertex set In1. For each s \u2264min{\u03b3(n1), n3}, de\ufb01ne C(s) = {(P, {i}, {j}) : P \u2208Fx \u2208F(n1) and i \u2208In2, j \u2208In3 with mij = x, 1 \u2264x \u2264s}. Then when \u03b3(n1) \u2265n3, (X, C(n3)) is an optimal generalized packing of size \u0004 n1 2 \u0005 n2n3; when \u03b3(n1) < n3, (X, C(\u03b3(n1))) is an optimal generalized packing of size n1(n1\u22121)n2 2 . Case 5: k = (1, 1, 1, 1). Proposition III.6. Let n = (n1, n2, n3, n4) and n1 \u2264n2 \u2264n3 \u2264n4. Then D(n, (1, 1, 1, 1), 3) = n1n2n3. Proof: It is clear that D(n, k, 3) \u2264n1n2n3. Consider Zn4 and let Xi = {si : 0 \u2264si \u2264ni \u22121} for i \u2208I4. Equality then follows from considering all blocks of the form ({s1}, {s2}, {s3}, {s4}), where si \u2208Xi and P i\u2208I4 si = 0. Here the addition is over Zn4. Combining results in [12], we have determined T(n, d, w) for total weight less than or equal to four completely except when w = (2, 2), d = 6 and n1 \u2264n2 \u22642n1 \u22121, both n1 and n2 are odd. IV. OPTIMAL MCWCS WITH MINIMUM DISTANCE FOUR In this section we handle the multiply constant-weight codes MCWC(m, n, d, w) with minimum distance four and small weight w. For convenience, we sometimes arrange a codeword as an m\u00d7n binary matrix. When w = 1, an MCWC(m, n, d, 1) is equivalent to an n-ary code of length m and distance \u03b4 by a bijection from row words to the set {0, 1, . . . , n \u22121}. Lemma IV.1 (Chee et al. [1]). M(m, n, d, 1) = An(m, \u03b4), where An(m, \u03b4) is the maximum size of an n-ary code of length m and distance \u03b4. When \u03b4 = 2, An(m, 2) = nm\u22121 which can be achieved by a code over Zn consisting of all m-tuples with a constant sum. Corollary IV.1. M(m, n, 4, 1) = nm\u22121. Applying Proposition II.1(i) iteratively w times for each i \u2208Im\u22121, we obtain the following consequence. Proposition IV.1. If d \u22642w, then M(m, n, d, w) \u2264 \u0000n w \u0001m\u22121A(n, d, w). By Corollary IV.1, the bound in Proposition IV.1 is tight when w = 1 and d = 4. Next, we show that it is also tight for some other small values w. The following two corollaries are immediate consequences of Proposition IV.1. Corollary IV.2. M(m, n, 4, 2) \u2264 \u0000n 2 \u0001m\u22121\u230an 2\u230b. Corollary IV.3. M(m, n, 4, 3) \u2264 \u0000n 3 \u0001m\u22121\u230an 3\u230an\u22121 2 \u230b\u230b. \f8 A. Product Construction We give a product construction as follows by generalizing the method used in [14, Section 6]. Let Y be a \ufb01nite set of size n. Suppose that Ci \u2286RY is a CWC(n, d1, w) for each i \u2208Is and \u222ai\u2208IsCi is a CWC(n, d2, w). Obviously, we have d2 \u2264d1. Further, let C0 \u2282RIm\u00d7Is be an MCWC(m, s, d3, 1). Let X = Im \u00d7 Y and X = ({1} \u00d7 Y, . . . , {m} \u00d7 Y ). For each codeword u \u2208C0, de\ufb01ne a code Du \u2282RX as Du = {vj1|vj2| \u00b7 \u00b7 \u00b7 |vjm : vji \u2208Cji, (i, ji) \u2208supp(u) and i \u2208Im}, where the | operator concatenates codewords as strings. Then for each c \u2208Du, supp(c)X \u2208 \u0000X w \u0001 , where w = (w, . . . , w). Let D = \u222au\u2208C0Du. It is easy to check that D is an MCWC(m, n, d, w) with minimum distance d \u2265min(d3d2/2, d1). In fact, for any two codewords in Du, the distance is at least d1. For two codewords c \u2208Du and c\u2032 \u2208Dv with u \u0338= v, since dH(u, v) \u2265d3, |supp(u) \\ supp(v)| \u2265d3/2. Hence dH(c, c\u2032) \u2265d3d2/2. Further, if all Ci have the same size M, then |D| = |C0|Mm. B. Optimal MCWCs Now we apply the product construction with special families of constant-weight codes to obtain optimal multiply constant-weight codes. Proposition IV.2. M(m, n, 4, 2) = \u0000n 2 \u0001m\u22121\u230an 2\u230bfor all positive integers n and m. Proof: When n is even, let s = n \u22121. For each i \u2208Is, let Ci \u2282RY be a CWC(n, 4, 2) such that Fi = {supp(u) : u \u2208Ci} is a 1-factor and {Fi : i \u2208Is} is a 1-factorization of Kn with vertex set Y . Then apply the product construction with d1 = d3 = 4, d2 = 2 and w = 2. When n is odd, let s = n. Then use the similar codes such that each Fi is a near 1-factor and {Fi : i \u2208Is} is a near 1-factorization. To get the following result, we again use the existence of partitions of triples into disjoint optimal packings as described in Section III. Proposition IV.3. M(m, n, 4, 3) = \u0000n 3 \u0001m\u22121\u230an 3\u230an\u22121 2 \u230b\u230bfor all positive integers m and n \u22610, 1, 2, 3 (mod 6) with n \u0338= 6, 7. Proof: For each n \u22611, 3 (mod 6) and n \u0338= 7, let s = n \u22122. For each i \u2208Is, let Ci \u2282RY be a CWC(n, 4, 3) such that Fi = {supp(u) : u \u2208Ci} is the block set of a Steiner triple system over Y and {Fi : i \u2208Is} forms a large set of Steiner triple systems. Then apply the product construction with d1 = d3 = 4, d2 = 2 and w = 3. For each n \u22610, 2 (mod 6) and n \u0338= 6, let s = n \u22121. For each i \u2208Is, let Ci \u2282RY be a CWC(n, 4, 3) such that Fi = {supp(u) : u \u2208Ci} is an optimal 2-(n, 3, 1) packing over Y of size n(n \u22122)/6 and {Fi : i \u2208Is} is a set of n \u22121 disjoint optimal 2-(n, 3, 1) packings. Then apply the product construction with same values of d1, d2, d3 and w. Note that when n \u22614, 5 (mod 6), there is no partition of all triples into optimal packings [23]\u2013[25], but only a partition with almost all packings except one being optimal. By the product construction, if \f9 not all Ci have the same size, then it is dif\ufb01cult to compute the exact size of D. So we can only get a lower bound of D by using disjoint optimal packings for these cases. Proposition IV.4. Let n \u226510. For all positive integers m, M(m, n, 4, 3) \u2265( n(n\u22122)\u22122 6 )m(n \u22121)m\u22121 when n \u22614 (mod 6) and M(m, n, 4, 3) \u2265( n(n\u22121)\u22128 6 )m(n \u22122)m\u22121 when n \u22615 (mod 6). V. ASYMPTOTIC SIZES OF T(n, d, w) The following result for M(m, n, d, w) was proved in [1]. Proposition V.1 (Chee et al. [1]). Given m, d, w, let s be the smallest integer such that m(w\u2212s)\u2212\u03b4+1 < m and r = m(w \u2212s) \u2212\u03b4 + 1. Then M(m, n, d, w) \u2264 \u0016 nm wm \u0016 (n \u22121)m (w \u22121)m \u00b7 \u00b7 \u00b7 \u0016 (n \u2212s + 1)m (w \u2212s + 1)m \u0016 (n \u2212s)r (w \u2212s)r \u0017\u0017 \u00b7 \u00b7 \u00b7 \u0017\u0017 . (1) Further, let t = mw \u2212\u03b4 + 1 and consider M(m, n, d, w) as a function of n, then 1 \u2264lim sup n\u2192\u221e M(m, n, d, w) nt/wt \u2264 wt (w \u2212s)t. In addition, when t \u2264m, n/w \u2265mw \u22121 and n/w is a prime power, M(m, n, d, w) = nt wt holds. The last statement of Proposition V.1 shows that under certain restrictions, the upper estimate (1) is asymptotically sharp. In this section, we prove that (1) is asymptotically sharp for all cases. In fact, we give a more general result for T(n, d, w) when all the components of n grow in any \ufb01xed proportion. A. Asymptotic theorem Since T(n, d, w) = D(n, w, t), where t = W \u2212\u03b4 + 1 by Corollary III.1, we study the upper bound of generalized packing numbers D(n, w, t) instead. Let X, n, w, t be de\ufb01ned as before. Suppose that ni = civ, i \u2208Im for some integer v. Given a (w, t)-admissible t, since every element of \u0000X t \u0001 occurs at most once in a block, we have D(n, w, t) \u2264 Y i\u2208Im \u0012civ ti \u0013 / \u0012wi ti \u0013 \u2264 vt Q i\u2208Im wi! Y i\u2208Im cti i (wi \u2212ti)!. (2) Let C \u225cmin{Q i\u2208Im cti i (wi \u2212ti)! : t is (w, t)-admissible}. Then our asymptotic result can be stated as follows. Theorem V.1. Let t = W \u2212\u03b4 + 1 and d = 2\u03b4. Then lim v\u2192\u221e T(n, d, w) vt = lim v\u2192\u221e D(n, w, t) vt = C Q i\u2208Im wi!. For MCWC(m, n, d, w), we can let ci = 1 for all i \u2208Im and v = n. Then C can be achieved when the m-tuple t is an almost constant tuple, that is, values of ti differ at most one. Applying Theorem V.1, we get the asymptotic sizes for M(m, n, d, w). \f10 Corollary V.1. Given m, d, w, let s be the smallest integer such that m(w \u2212s) \u2212\u03b4 + 1 < m and r = m(w \u2212s) \u2212\u03b4 + 1. Let t = mw \u2212\u03b4 + 1. Then lim n\u2192\u221e M(m, n, d, w) nt = 1 wm(w \u22121)m \u00b7 \u00b7 \u00b7 (w \u2212s + 1)m(w \u2212s)r . B. Proof of Theorem V.1 Given a hypergraph H, let E(H) be the edge set and V (H) be the vertex set. Denote \u03bd(H) the maximum size of a matching in H. A function \u03b8 : E(H) \u2212 \u2192R is a fractional matching of the hypergraph H if P e\u2208E(H);x\u2208e \u03b8(e) \u22641 holds for every vertex x \u2208V (H). Let \u03b8(H) = P e\u2208E(H) \u03b8(e). The fractional matching number, denoted \u03bd\u2217(H) is the maximum of \u03b8(H) over all fractional matchings. Clearly, \u03bd(H) \u2264\u03bd\u2217(H). Kahn [13] proved that under certain conditions, asymptotic equality holds. For a subset W \u2282V (H), de\ufb01ne \u00af \u03b8(W) = P e\u2208E(H);W \u2282e \u03b8(e) and \u03b1(\u03b8) = max{\u00af \u03b8({x, y}) : x, y \u2208V (H), x \u0338= y}. Here \u03b1(\u03b8) is a fractional generalization of the codegree. We say that H is l-bounded if each of its edges has size at most l. Theorem V.2 (Kahn [13]). For every l and every \u03b5 > 0 there is a \u03c3 such that whenever H is an l-bounded hypergraph and \u03b8 is a fractional matching with \u03b1(\u03b8) < \u03c3, then \u03bd(H) > (1 \u2212\u03b5)\u03b8(H). We apply Kahn\u2019s Theorem to prove Theorem V.1. First, we de\ufb01ne a hypergraph \u0393v with v de\ufb01ned in n = (c1v, . . . , cmv). Let T be the set of all (w, t)-admissible m-tuples. Given A \u2208 \u0000X w \u0001 , let E(A) = \u222at\u2208T \u0000A t \u0001 . Then construct a hypergraph \u0393v with vertex set \u222at\u2208T \u0000X t \u0001 and edge set {E(A) : A \u2208 \u0000X w \u0001 }. Note that for any two distinct blocks A1 and A2 in the generalized packing, we have E(A1)\u2229E(A2) = \u2205. Hence a t-(n, w, 1) generalized packing corresponds to a matching in \u0393v, i.e., \u03bd(\u0393v) = D(n, w, t). It suf\ufb01ces to verify the conditions of Theorem V.2 and to produce a fractional matching \u03b8 of the hypergraph \u0393v of the desired size. It is easy to know that \u0393v is l-bounded with l = P t\u2208T Q i\u2208Im \u0000wi ti \u0001 . Now consider the function \u03b8 : E(\u0393v) \u2212 \u2192R by \u03b8(e) = C vW \u2212t Q i\u2208Im cwi i , for every e \u2208E(\u0393v). We \ufb01rst check \u03b8 is a fractional matching. For any vertex x \u2208V (\u0393v), which is an (n, w, t)-admissible m-tuple of disjoint sets of sizes t, we have deg(x) = Y i\u2208Im \u0012civ \u2212ti wi \u2212ti \u0013 \u2264 Y i\u2208Im (civ)wi\u2212ti (wi \u2212ti)! =vW \u2212t Y i\u2208Im cwi i cti i (wi \u2212ti)! \u2264vW \u2212t Q i\u2208Im cwi i C . \f11 Hence, P e\u2208E(\u0393v);x\u2208e \u03b8(e) \u22641 and \u03b8 is indeed a fractional matching. Next, we compute \u03b1(\u03b8). For every x, y \u2208V (\u0393v), then x \u222ay is (n, w, t\u2032)-admissible with t\u2032 \u2265t + 1. Here, the union operation is componentwise. Then the codegree of x and y is deg(x, y) = O(vW \u2212t\u22121). Hence \u03b1(\u03b8) = deg(x, y) \u00b7 \u03b8(e) = o(1) when v \u2192\u221e. Finally, we apply Kahn\u2019s Theorem. lim v\u2192\u221e D(n, w, t) vt = lim v\u2192\u221e \u03bd(\u0393v) vt \u2265lim v\u2192\u221e \u03b8(\u0393v) vt = lim v\u2192\u221e |E(\u0393v)| \u00d7 \u03b8(e) vt = lim v\u2192\u221e Y i\u2208Im \u0012civ wi \u0013 \u00d7 C vW \u2212t Q i\u2208Im cwi i /vt = C Q i\u2208Im wi!. The other inequality comes from the upper bound (2). VI. TWO DIMENSIONAL MULTIPLY CONSTANT-WEIGHT CODES Recall that when the lengths of different parts of a codeword are constant, say n = (n, n, . . . , n), then each codeword could be considered as an m\u00d7n binary matrix. In this section, we impose additional weight constraints on all columns. We note that these codes have applications in optical storage in holographic memory [9], crossbar arrays of resistive devices [10], and power line communications [11]. Let n = (n, n, . . . , n) and w = (w1, . . . , wm). A two dimensional multiply constant-weight code 2DMCWC(m, n, d, w, l) is an MCWC(n, d, w) in a matrix form such that each column of codewords has constant weight l. Let M(m, n, d, w, l) denote the largest size of a 2DMCWC(m, n, d, w, l). If wi = w for all i \u2208Im, we simply write 2DMCWC(m, n, d, w, l) and M(m, n, d, w, l). A. Upper Bounds An \u03b1-parallel class of a set system is a subset of the blocks that each point appears exactly \u03b1 times. De\ufb01nition VI.1. Let (Y, B) be a 2-(M, K, \u03bb) packing. If the blocks can be arranged into an m\u00d7n array R such that (1) each entry in R is either empty or a block; (2) the blocks in the i-th row form a wi-parallel class; (3) the blocks in each column form an l-parallel class. Then, we call it a doubly resolvable packing, and denote it by DRP(M, \u03bb; w, l; m, n). Again we write DRP(M, \u03bb; w, l; m, n) if wi = w for all i \u2208Im. Proposition VI.1. A DRP(M, \u03bb; w, l; m, n) is equivalent to a 2DMCWC(m, n, d, w, l) of size M, where d = 2(W \u2212\u03bb). Proof: Suppose (Y, B) is a DRP(M, \u03bb; w, l; m, n) such that the blocks are arranged into an m \u00d7 n array R. Denote the (i, j)-th entry of R by Rij, where i \u2208Im and j \u2208In. \f12 For each x \u2208Y , we construct an m \u00d7 n binary matrix ux with the (i, j)-th entry de\ufb01ned as ux ij = \uf8f1 \uf8f2 \uf8f3 1, if x \u2208Rij; 0, otherwise. Then C = {ux | x \u2208Y } is a 2DMCWC(m, n, d, w, l) of size M, where d = 2(W \u2212\u03bb). The construction can be easily reversed to obtain the converse. Example VI.1. Here is an example of a DRP(3, 3; 2, 2; 3, 3) over Y = Z3, 01 12 02 02 01 12 12 02 01 from which we obtain a 2DMCWC(3, 3, 6, 2, 2) by taking the codewords u0 = \uf8eb \uf8ec \uf8ed 1 0 1 1 1 0 0 1 1 \uf8f6 \uf8f7 \uf8f8, u1 = \uf8eb \uf8ec \uf8ed 1 1 0 0 1 1 1 0 1 \uf8f6 \uf8f7 \uf8f8, u2 = \uf8eb \uf8ec \uf8ed 0 1 1 1 0 1 1 1 0 \uf8f6 \uf8f7 \uf8f8. Lemma VI.1 (Upper Bound). If P i\u2208Im w2 i \u2212n\u03bb > 0, and there exists a DRP(M, \u03bb; w, l; m, n), or equivalently a 2DMCWC(m, n, d, w, l) of size M, where d = 2(W \u2212\u03bb), then M \u2264 n(nl \u2212\u03bb) P i\u2208Im w2 i \u2212n\u03bb. Proof: Let R be the m \u00d7 n array of a DRP(M, \u03bb; w, l; m, n) and fij = |Rij|. Then X i\u2208Im X j\u2208In fij = Mnl = M X i\u2208Im wi. By the de\ufb01nition of a doubly resolvable packing, \u03bbM(M \u22121) \u2265 X i\u2208Im X j\u2208In fij(fij \u22121) = X i\u2208Im X j\u2208In f 2 ij \u2212Mnl. By Cauchy\u2013Schwartz inequality, for any i \u2208Im, X j\u2208In f 2 ij \u2265 (P j\u2208In fij)2 n = (Mwi)2 n . So \u03bbM(M \u22121) \u2265 X i\u2208Im (Mwi)2 n \u2212Mnl. Hence M \u2264 n(nl \u2212\u03bb) P i\u2208Im w2 i \u2212n\u03bb, \f13 provided that P i\u2208Im w2 i \u2212n\u03bb > 0. Note that the right hand side is always positive since n2l = n P i\u2208Im wi \u2265 P i\u2208Im w2 i . If the bound in Lemma VI.1 is achieved, then all the blocks in the i-th row are of the same size Mwi n , and each pair appears in exactly \u03bb blocks. It is easy to check that the 2DMCWC(3, 3, 6, 2, 2) constructed in Example VI.1 is optimal. Further, if we let f = \u0016Ml m \u0017 and r = Ml \u2212mf. Then we can improve the above bound by using X i\u2208Im f 2 ij \u2265(m \u2212r)f 2 + r(f + 1)2. Lemma VI.2 (Improved Upper Bound). If there exists a DRP(M, \u03bb; w, l; m, n), or equivalently a 2DMCWC( m, n, d, w, l) of size M, where d = 2(W \u2212\u03bb), then \u03bbM(M \u22121) \u2265 n[(m \u2212r)f 2 + r(f + 1)2] \u2212Mnl = n(mf 2 + 2rf + r) \u2212Mnl, where f = \u0016Ml m \u0017 and r = Ml \u2212mf. Note that the upper bound of M(m, n, d, w, l) is the biggest M satis\ufb01es the inequality in Lemma VI.2, which is usually achieved when equality holds. Example VI.2. Here are two examples of optimal two dimensional multiply constant-weight codes that achieve the bound in Lemma VI.2. We list the equivalent doubly resolvable packings instead. An optimal 2DMCWC(6, 6, 20, 2, 2) of size 4: 01 23 0 1 2 3 23 01 3 0 1 2 2 3 02 13 0 1 1 2 13 02 3 0 0 1 2 3 03 12 3 0 1 2 12 03 . An optimal 2DMCWC(9, 9, 32, 2, 2) of size 6: 01 45 12 2 3 0 3 5 4 25 13 04 3 0 1 4 2 5 34 02 35 0 1 2 5 4 1 4 5 5 03 14 03 2 1 2 3 4 2 15 05 24 1 3 0 5 1 4 24 23 15 0 0 3 0 2 3 5 4 3 01 45 12 1 3 0 4 2 5 25 13 04 2 0 1 1 5 4 34 02 35 . \f14 B. Constructions First we show that concatenating small optimal two dimensional MCWCs gives big optimal two dimensional MCWCs. Proposition VI.2. Let a be a positive integer. Suppose there exists an optimal 2DMCWC(m, n, d, w, l) of size M achieving the bound in Lemma VI.1 or Lemma VI.2. Then there exists an optimal 2DMCWC(am, n, ad, w\u2032, al) of size M, where w\u2032 is a vector of length am by copying w a times. Proof: First, we check that the two codes have the same upper bounds of sizes. Suppose that a 2DMCWC(m, n, d, w, l) corresponds to a DRP(M, \u03bb; w, l; m, n) and a 2DMCWC(am, n, ad, w\u2032, al) corresponds to a DRP(M1, \u03bb1; w\u2032, al; am, n), then \u03bb1 = a\u03bb. By Lemma VI.1, M and M1 satisfy the same inequality. Now we check for Lemma VI.2. Let f1 = \u0016M1al am \u0017 = \u0016M1l m \u0017 and r1 = M1al \u2212amf1 = a(M1l \u2212mf1). Then \u03bb1M1(M1 \u22121) \u2265n[(am \u2212r1)f 2 1 + r1(f1 + 1)2] \u2212aM1nl, that is, \u03bbM1(M1 \u22121) \u2265n[(m \u2212r1/a)f 2 1 + (r1/a)(f1 + 1)2] \u2212M1nl. Being considered as an inequality with indeterminate M1, it is exactly the same inequality as in Lemma VI.2. Hence, M and M1 have the same restrictions. So if there exists an optimal 2DMCWC(m, n, d, w, l) C, then we can obtain an optimal 2DMCWC(am, n, ad, w\u2032, al) by concatenating each codeword from C a times in the vertical direction. By applying the same technique but concatenating each codeword in C b times in the horizontal direction, we have that M(m, n, d, w, l) = M achieving the bound in Lemma VI.1 or Lemma VI.2 implies M(am, bn, abd, bw\u2032, al) = M. Hence, we only consider the codes with parameters m, l and d (or n, w and d) having no common divisors. Lemma VI.3. For any positive integer n, M(n, n, 2n, 1, 1) = n. Proof: A Latin square of order n is a DRP(n, 0; 1, 1; n, n). The construction of 2DMCWC(nl, n, d, 1, l)s has been investigated in many papers as equidistant frequency permutation arrays and constant-composition codes [27], [28]. Most of the constructions can be generalized to construct 2DMCWC(m, n, d, w, l). Construction VI.1. If there exists an \u03b1-resolvable BIBD(M, k, \u03bb) with r = (M\u22121)\u03bb (k\u22121)\u03b1 \u03b1-parallel classes (each has b = \u03b1M k blocks), then for any pair of positive integers (s, t) with st = r, there exists a DRP(M, b\u03bb; \u03b1t, \u03b1s; bs, bt), i.e., an optimal 2DMCWC(bs, bt, d, \u03b1t, \u03b1s) of distance d = 2b(\u03b1r \u2212\u03bb). Proof: We construct the bs \u00d7 bt array of a DRP(M, b\u03bb; \u03b1t, \u03b1s; bs, bt) as follows. For each 1 \u2264i \u2264r, let Ai be a b \u00d7 b array with the \ufb01rst column occupied by the blocks in the i-th \u03b1-parallel class, and other columns being cyclic shifts of the \ufb01rst one. Then the bs \u00d7 bt array R is formed by arranging all Ai\u2019s into an s \u00d7 t array. The optimality is easy to be checked by Lemma VI.1. \f15 By Construction VI.1, the existence of \u03b1-resolvable BIBD(M, k, \u03bb)\u2019s implies the existence of optimal two dimensional multiply constant-weight codes with certain parameters. Necessary conditions for the existence of an \u03b1-resolvable BIBD(M, k, \u03bb) are (1) \u03bb(M \u22121) \u22610 (mod \u03b1(k \u22121)), (2) \u03bbM(M \u22121) \u22610 (mod k(k \u22121)), and (3) \u03b1M \u22610 (mod k). For k \u2208{2, 3, 4}, the necessary conditions are suf\ufb01cient except for (M, k, \u03bb, \u03b1) \u2208{(6, 3, \u03bb, 1) : \u03bb \u22612 (mod 4)} \u222a{(10, 4, 2, 2)} [29], [30]. Hence we can obtain several families of optimal two dimensional multiply constant-weight codes by Construction VI.1. VII."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1401.3928v1",
+                    "title": "Multiply Constant-Weight Codes and the Reliability of Loop Physically Unclonable Functions",
+                    "abstract": "We introduce the class of multiply constant-weight codes to improve the\nreliability of certain physically unclonable function (PUF) response. We extend\nclassical coding methods to construct multiply constant-weight codes from known\n$q$-ary and constant-weight codes. Analogues of Johnson bounds are derived and\nare shown to be asymptotically tight to a constant factor under certain\nconditions. We also examine the rates of the multiply constant-weight codes and\ninterestingly, demonstrate that these rates are the same as those of\nconstant-weight codes of suitable parameters. Asymptotic analysis of our code\nconstructions is provided.",
+                    "authors": "Yeow Meng Chee, Zouha Cherif, Jean-Luc Danger, Sylvain Guilley, Han Mao Kiah, Jon-Lark Kim, Patrick Sol\u00e9, Xiande Zhang",
+                    "published": "2014-01-16",
+                    "updated": "2014-01-16",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "math.CO",
+                        "math.IT"
+                    ],
+                    "main_content": "INTRODUCTION Physically unclonable functions (PUFs) introduced by Pappu et al. [1] provide innovative lowcost authentication methods that are derived from complex physical characteristics of electronic devices. Recently, PUFs have become an attractive option to provide security in low cost devices such as RFIDs and smart cards [1]\u2013[4]. Reliability and implementation considerations on programmable circuits for the design of Loop PUFs [4] lead to the investigation of a new class of codes called multiply constant-weight codes (MCWC). In an MCWC, each codeword is a binary word of length mn which is partitioned into m equal parts and has weight exactly w in each part [5]. This de\ufb01nition therefore generalizes the class of constant-weight codes (where m = 1) and a subclass of doubly constant-weight codes, introduced by Johnson [6] and Levenshte\u02d8 \u0131n [7] (where m = 2). In this paper, we consider upper and lower bounds for the possible sizes of MCWCs. Our constructions make use of both classical concatenation techniques [8], [9] and a method due to Zinoviev (for constant-weight codes) [10], that was later independently given by Etzion (for doubly constant-weight codes) [11]. A construction technique using resolvable designs is also examined. For upper bounds, we extend the techniques of Johnson [6] and exhibit that these bounds are asymptotically tight to a constant factor, provided m, w and d are \ufb01xed. We also examine the rates of the MCWCs and interestingly, demonstrate that these rates are the same as those of constant-weight codes of length mn and weight mw. We remark that if the codewords in an MCWC are regarded as m by n arrays, then an MCWC can be regarded as a code over binary matrices, where each matrix has constant row weight w. These codes were studied by Chee et al. [12] in an application for power line communications. The relevance of MCWCs for the latter context is an area for future research. The rest of this article is structured as follows. Section II collects the necessary de\ufb01nitions and notation, and Section III examines an application of MCWCs in the \ufb01eld of PUFs. Section IV deals with constructions and attached lower bounds, while Section V contains the upper bounds. Section VI studies asymptotic versions of the bounds of Section IV and Section V. Some of our results were initially reported in [5] and the present paper contains many new results and generalizations. II. DEFINITIONS AND NOTATION Let X be a set of q symbols. A q-ary code C of length n over the alphabet X is a subset of X n. Elements of C are called codewords. Endow the space X n with the Hamming distance metric. A code C is said to have distance d if the (Hamming) distance between any two distinct codewords of C is at least d. A q-ary code of length n and distance d is called an (n, d)q code. When q = 2, we assume X = F2. An (n, d)2 code is simply called an (n, d) code. The (Hamming) weight of a codeword u \u2208X n is given by the number of nonzero coordinates in u. \f3 Fix m, n1, n2, . . . , nm to be positive integers and let N = n1 +n2 +\u00b7 \u00b7 \u00b7+nm. An (N, d)2 code is said to be of multiply constant-weight and denoted by MCWC(w1, n1; w2, n2; . . . ; wm, nm; d), if each codeword has weight w1 in the \ufb01rst n1 coordinates, weight w2 in the next n2 coordinates, and so on and so forth. When m = 1, an MCWC(n, w; d) is a constant-weight code, denoted by CWC(n, d, w); when m = 2, an MCWC(w1, n1; w2, n2; d) is a doubly constant-weight code. When w1 = w2 = \u00b7 \u00b7 \u00b7 = wm = w and n1 = n2 = \u00b7 \u00b7 \u00b7 = nm = n, we simply denote this multiply constant-weight code of length N = mn by MCWC(m, n, d, w). Unless speci\ufb01ed otherwise, a multiply constant code refers to an MCWC(m, n, d, w) in this paper. The largest size of an (n, d)q code is denoted by Aq(n, d). When q = 2, this size is simply denoted by A(n, d). The largest of size of an MCWC(w1, n1; w2, n2; . . . ; wm, nm; d) is given by T(w1, n1; w2, n2; . . . ; wm, nm; d); the largest of size of an MCWC(m, n, d, w) is given by M(m, n, d, w); and the largest of size of a CWC(n, d, w) is given by A(n, d, w). In this paper, we are mainly interested in determining M(m, n, d, w). Observe that by de\ufb01nition, M(1, n, d, w) = A(n, d, w), M(2, n, d, w) = T(w, n; w, n; d). Moreover, the functions A(n, d, w) and T(w, n; w, n; d) have been well studied (see for example, [6], [11], [13]\u2013[15]). Online tables of the lower bounds for A(n, d, w) can be found at [16] while upper bounds for A(n, d, w) and T(w, n; w, n; d) can be found at [17]. In this paper, we are mainly interested in building multiply constant-weight codes from known q-ary codes and constant-weight codes. One such class of codes is the class of binary linear codes. A binary linear code of length n, dimension k and distance d is called a linear [n, k, d] code and we denote the largest quantity 2k of a binary linear [n, k, d] code by B(n, d). Unfortunately, an MCWC cannot be linear and hence, we look at possible generalization of linearity. A possible generalization given by the notion of systematic codes. A code of size 2k is said to be systematic if there is a set I of k coordinates such that the code when restricted to the coordinate set I is exactly Fk 2. The largest sizes of a systematic (n, d) code and CWC(n, d, w) are denoted by S(n, d) = 2s(n,d) and S(n, d, w) = 2s(n,d,w) respectively. We remark that systematic constant-weight codes have been studied in [18], [19]. Finally, as mentioned in the introduction, a codeword in an MCWC(m, n, d, w) can be regarded as a binary m by n matrix with constant row weight w. Throughout the rest of this paper, we shall regard a codeword in an MCWC as either a word of length mn or an m by n matrix. \f4 III. APPLICATION TO LOOP PUFS The need of an MCWC arises from the generation of some type of PUFs in trusted electronic circuits. In this section, we demonstrate the relevance of MCWC in the implementation of Loop PUF on Field Programmable Gate Array (FPGA) and in enhancing the reliability of PUF response. First, we present the principle behind Loop PUF. A. Loop PUF Principle In general, the PUF provides a unique signature to a device without the need for the user to program an internal memory [1]. This signature allows the user to build lightweight authentication protocols or even protect a master key in cryptographic implementations. Such a key can be used for standard cryptographic protocols, or for internal cryptography (e.g., memory encryption). Essentially, the PUF takes advantage of technological process variations to differentiate between two devices. For instance, consider two delay lines with the same structure. In theory, the propagation time is the same for both two delay lines. However, actual measurements of the propagation time differ between the delay lines due to imbalances between the physical elements. Furthermore, as these measurements cannot be predicted accurately, they are well suited for cryptographic purposes. Here, we consider the Loop PUF [4] that is a set of n identical delay lines laid out on a programmable circuit. The delay lines form a loop that oscillates as a single ring oscillator when closed by an inverter (see Figure 1) and this setup enhances the accuracy of delay measurements. Furthermore, each delay line is a series of m delay elements and the delay of the ith element of the jth line is controlled by the (i, j)-th bit of some control word u of length mn. Hence, corresponding to a control word u, we have a delay measurement, denoted by D(u). For expository purposes, we illustrate how a general binary code can be used in conjunction with the Loop PUF [4] to generate a set of Challenge-Response pairs for authentication purposes. For other cryptographic applications, we refer the interested reader to [1]\u2013[4]. Given a binary code C of length mn, the set of Challenge-Response pairs is given by n\u0010 (u, v), sign(D(u) \u2212D(v)) \u0011 : u \u0338= v, u, v \u2208C o . In other words, each challenge is an ordered pair of distinct codewords (u, v) from the binary code and the corresponding response is the sign of the delay difference between the pair of codewords1. For the set of Challenge-Response pairs to be used for authentication, it is important that we are unable to infer the sign of the delay difference with only knowledge of u and v. To achieve 1Here, we consider a response that consists of one bit. A different control strategy can be used to extract a response with more bits and this is described in [4]. \f5 Delay Chains Controller Frequency or delay measurement u Fig. 1. Loop PUF structure this unpredictability of response, we show that C needs to be an MCWC in Section III-B. On the other hand, it is also important that the measured response (or the sign of delay difference) remains the same despite environmental noise. The reliability of response is then shown to be associated with the minimum distance of the code C in Section III-C. Therefore, MCWCs are needed to satisfy both requirements of unpredictability and reliability. B. MCWC to Achieve Unpredictability on FGPAs Programmable circuits, like FPGAs, have a hierarchical layout. It is thus convenient to organize the PUF with two levels, namely with a structure of n clusters of m cells each2. For this technology, it is rather easy to copy / paste exactly the logic of one cluster to generate all of them, in an indistinguishable manner (logically, not physically). Thus the Loop PUF can be easily constructed from a set of n clusters of m cells just by replicating the base cluster of m cells. As the routing inside a cluster between the m elements is not constrained, the PUF designer can easily port this structure to any FPGA family. Now, let us consider an MCWC(n, w1; n, w2; \u00b7 \u00b7 \u00b7 ; n, wm; d) and choose a control word u = (u11, u12, . . . , u1n, u21, u22, . . . , u2n, . . . , um1, um2, . . . , umn). Let dij(uij) be the resulting delay of the ith delay element in the jth line and hence, the total measured delay D(u) due to u is given by Pm i=1 Pn j=1 dij(uij). Ideally, dij(uij) = \u00b5 + \u01ebij(uij), where \u01ebij is a small timing variation on the jth delay element on the ith line caused by technological dispersion and \u00b5 is the average delay that is independent of the position on the circuit. However, the latter is not true due to manufacturing constraints. In particular, a designer has no control about the routing within an FPGA cluster and hence, it is hardly possible to get balanced delay elements within a cluster. But fortunately due to copy / paste operation, the internal routing of a cluster can be faithfully reproduced from one cluster to another (see Figure 2). 2Logic Array Block (LABs) for ALTERA and Con\ufb01gurable Logic Blocks (CLBs) for XILINX \f6 . . . . . . . . . \u00b7 \u00b7 \u00b7 n clusters / delay chains m cells / delay elements copy / paste d11 d21 dm1 d12 d22 dm2 d1n d2n dmn Fig. 2. Delay chain layout In other words, we have dij(uij) = \u00b5i(uij) + \u01ebij(uij), where \u01ebij is a small timing variation and \u00b5i is the average delay dependent on the controlled bit and the position of the delay element. We compute the total delay due to u, and we have D(u) = m X i=1 n X j=1 dij(uij) = m X i=1 n X j=1 \u00b5i(uij) + \u01ebij(uij) =  m X i=1 (n \u2212wi)\u00b5i(0) + wi\u00b5i(1) ! +  m X i=1 n X j=1 \u01ebij(uij) ! . (1) The last equality follows from the fact that u belongs to an MCWC(n, w1; n, w2; \u00b7 \u00b7 \u00b7 ; n, wm; d). Furthermore, we observe that all codewords from the MCWC have the same expected response. Therefore, the delay difference between any pair of control words from the MCWC has expectation zero and the sign of the difference is dependent only on \u01ebij\u2019s. In other words, the response depends entirely on the unpredictable physical characteristics of the individual delay elements. C. Hamming Distance to Improve the PUF Reliability The PUF response is very sensitive to environmental noise as the \u01ebij can be very low in comparison to the delays. Hence it is necessary to choose pairs of control words which offer the largest possible difference between their delays. \f7 From (1), we see that D(u) \u2212D(v) = m X i=1 n X j=1 \u01ebij(uij) \u2212\u01ebij(vij) = X uij\u0338=vij \u01ebij(uij) \u2212\u01ebij(vij) . Therefore, the greater the Hamming distance between u and v, the greater the delay difference D(u) \u2212D(v). Hence, by choosing a code of high distance, we improve the reliability of the PUF response. The arguments in this section demonstrate the relevance of MCWC in the design of reliable Loop PUF. In the remaining of the paper, we examine the possible lower and upper bounds for optimal MCWCs, focusing our attention to the case where w1 = w2 = \u00b7 \u00b7 \u00b7 = wm = w. IV. LOWER BOUNDS A. Coding Constructions In this section, we study constructions of MCWCs using known unrestricted codes. Our \ufb01rst construction is based on concatenation. Proposition IV.1. Let q \u2264A(n, d1, w). We have M(m, n, d1d2, w) \u2265Aq(m, d2). Proof: Consider a concatenation scheme [8], [9] where the outer code C is an (m, d2)q code of size Aq(m, d2) over X and the inner code D is a CWC(n, d1, w) of size q. Let \u03c6 : X \u2192D be an injective map. For each codeword u = (u1, u2, . . . , um) in C, we construct the binary codeword (\u03c6(u1), \u03c6(u2), . . . , \u03c6(um)). Then the resulting code is an MCWC(m, n, d1d2, w) of size Aq(m, d2). A special case of concatenation is the product code construction. Recall that if C and D are two binary linear codes then their product C \u2297D is the code of length nm consisting of m by n arrays whose rows belong to C and columns belong to D. If C and D are linear [m, k, d] and [n, l, e] codes, then the code C \u2297D has parameters [nm, kl, de] [9, Lemma 2.8]. We generalize this construction by relaxing certain requirements. In particular, we require only the rows of our arrays to be in C, while not all the columns need to be in D. Formally, consider a systematic CWC(n, d1, w) C of size 2k1 and a systematic (m, d2) code D of size 2k2. Given a binary k2 by k1 matrix M, we replace each column of length k2 of M with its corresponding codeword in D and we obtain a binary m by k1 matrix M\u2032. Next replace each row of length k1 of M\u2032 with its corresponding codeword in C. This results in a binary m by n matrix with constant row weight w. In particular, each row of the matrix belongs to the constant-weight code C while the \ufb01rst k1 columns belong to the code D. Hence, the collection \f8 of all 2k1k2 matrices from this construction results in an MCWC(m, n, d1d2, w). We call this construction a pseudo-product code construction. We remark that as with product construction, the pseudo-product code construction is a special case of concatenation. In addition, the pseudo-product construction coincides with the construction given by Amrani [20, De\ufb01nition 1] in another context. The following proposition follows immediately from the pseudo-product code construction. Proposition IV.2. We have M(m, n, d1d2, w) \u22652s(n,d1,w)s(m,d2) \u2265B(m, d2)s(n,d1,w). Example IV.1. Consider the following systematic constant-weight code {0011, 0101, 1010, 1111} of distance two. Taking its pseudo-product with a binary linear [6, 2, 4] code yields a lower bound of 22\u00b72 = 16 on M(6, 4, 8, 2). We give a simple but robust construction technique for systematic constant-weight codes due to B\u00a8 oinck and van Tilborg. Proposition IV.3 (B\u00a8 oinck and van Tilborg [18, Construction 4.1]). We have S(2n, 2d, n) \u2265S(n, d) \u2265B(n, d). Proof: Let C be a systematic code of size S(n, d). Construct a constant-weight code by the rule D = {(x, x)| x \u2208C}, where the bar denotes complementation. The code D hence has twice the distance of C and is systematic because C is. Example IV.2. Observe that B(2m\u22121, 2m\u22122) = 2m follows from the Plotkin bound and the Reed Muller code RM(1, m\u22121) [21, Chapter 13]. Proposition IV.3 therefore yields S(2m, 2m\u22121, 2m\u22121) \u2265 2m. We extend the code construction in Proposition IV.3 by appending each codeword with a codeword from a suitable constant-weight code. Proposition IV.4. If 2k \u2264A(n, d, w) we have s(n + 2k, d + 2, w + k) \u2265k. Proof: Let C be a constant-weight code of size A(n, d, w). Let \u03c6 : Fk 2 \u2192C be an injective map. Let D = {(x, x, \u03c6(x))| x \u2208Fk 2}, \f9 where the bar denotes complementation. The code D is systematic with information set the \ufb01rst k coordinates and has the required parameters. The next construction generalizes a construction by Zinoviev [10] (see also [22]) and by Etzion [11, Theorem 16] to construct multiply constant-weight codes from q-ary codes. Proposition IV.5. We have M(m, qw, 2d, w) \u2265Aq(mw, d). Proof: Consider an (mw, d)q code of size Aq(mw, d) over the alphabet X . We extend each word of length mw to a word of length qmw by replacing each symbol with a binary word of length q. Speci\ufb01cally, replace each symbol in the codeword with the following characteristic function \u03c6 : X \u2192{0, 1}X, \u03c6(x)y = \uf8f1 \uf8f2 \uf8f3 1, if x = y, 0, otherwise. We check that the new binary word of length qmw comprises m parts each of weight w. It remains to check that the distance. Observe that for any pair of distinct symbols x, y \u2208X , the distance between \u03c6(x) and \u03c6(y) is two. Hence, since the distance between two q-ary codewords is at least d, the distance between the corresponding binary codewords is at least 2d. When q is a prime power and q \u2265mw \u22121, there exists a q-ary Reed Solomon code of length mw and distance d. Hence, Aq(mw, d) = qmw\u2212d+1 and the following corollary is immediate. Corollary IV.1. If q is a prime power and q \u2265mw \u22121, then M(m, qw, 2d, w) \u2265qmw\u2212d+1. On the other hand, when w = 1, we observe that we are able to reverse the construction so as to construct an n-ary codeword of length m from an m by n matrix with constant row weight one. Hence, the following corollary is immediate. Corollary IV.2. We have M(m, n, 2d, 1) = An(m, d). B. Designs Constructions Here, we consider a construction from designs, in particular, resolvable t-designs. A t-(v, k, 1) design, or t-design, is a pair (X, B) such that |X| = v and B is a collection of k-subsets of X, called blocks, with the property that every t-subset of X is contained in exactly one block. A t-design (X, B) is resolvable if the blocks in B can be partitioned into parallel classes, each of which is a partition of X. \f10 Suppose (X, B) is a resolvable t-(v, k, 1) design with M = k \u0000v t \u0001 /v \u0000k t \u0001 parallel classes. Let m = v/k and n = v. For each parallel class, we construct a binary m by n matrix, where the support of each row is given by a corresponding block. Hence, we form a binary m by n matrix with constant row weight k. Since every pair of blocks intersect at most in t \u22121 places, the distance between every pair of binary matrices is at least 2m(k \u2212t + 1). Hence, we obtain an MCWC(m, n, 2m(k \u2212t + 1), k) of size M. We summarize the construction in the following proposition. Proposition IV.6. Suppose there exists a resolvable t-(v, k, 1) design. Then M \u0010v k, n, 2(k \u2212t + 1)v k, k \u0011 \u2265k \u0000v t \u0001 v \u0000k t \u0001. Existence results for resolvable 2-(v, k, 1) design are surveyed by Abel et al. [23, Table 7.35]. When t \u22653, existence results are given by Laue [24] (see also [25]\u2013[27]). V. UPPER BOUNDS Trivially, an MCWC(m, n, d, w) is a CWC(mn, d, mw). Hence, we have our \ufb01rst upper bound. Proposition V.1. We have M(m, n, d, w) \u2264A(nm, d, mw). Next, we extend the techniques of Johnson [6] to obtain the following recursive bounds on T(w1, n1; w2, n2; . . . ; wm, nm; d). Let 1 \u2264i \u2264m. T(w1, n1; w2, n2; . . . ; wm, nm; d) \u2264 \u0016 ni wi T(w1, n1; . . . ; wi \u22121, ni \u22121; . . . ; wm, nm; d) \u0017 , (2) T(w1, n1; w2, n2; . . . ; wm, nm; d) \u2264 \u0016 ni ni \u2212wi T(w1, n1; . . . ; wi, ni \u22121; . . . ; wm, nm; d) \u0017 , (3) T(w1, n1; w2, n2; . . . ; wm, nm; d) \u2264 \u0016 u w2 1/n1 + w2 2/n2 + \u00b7 \u00b7 \u00b7 + w2 m/nm \u2212\u03bb \u0017 , (4) where d = 2u and \u03bb = w1+w2+\u00b7 \u00b7 \u00b7+wm\u2212u. Since M(m, n, d, w) = T(w, n; w, n; . . .; w, n; d) and applying the recursive bounds m times, we obtain the following recursive upper bounds. Proposition V.2. We have M(m, n, d, w) \u2264 \u0016 nm wmM(m, n \u22121, d, w \u22121) \u0017 (5) M(m, n, d, w) \u2264 \u0016 nm (n \u2212w)mM(m, n \u22121, d, w) \u0017 (6) M(m, n, d, w) \u2264 \u0016 d/2 mw2/n \u2212(mw \u2212d/2) \u0017 . (7) \f11 Suppose s = mw \u2212d/2 + 1 \u2264m. Applying (2) for s iterations, we have M(m, n, d, w) \u2264 ns wsT(w \u22121, n \u22121; . . . ; w \u22121, n \u22121; w, n; . . . ; w, n; d) and T(w \u22121, n \u22121; . . . ; w \u22121, n \u2212 1; w, n; . . . , w, n; d) is trivially one. Hence, we obtain the next upper bound. Proposition V.3. If mw \u2212d/2 + 1 \u2264m, then M(m, n, d, w) \u2264 \u0010 n w \u0011mw\u2212d/2+1 . (8) We remark that when w = 1, Proposition V.3 reduces to the classical Singleton bound. Given m, d, w, let i be the smallest integer such that m(w\u2212i)\u2212d/2+1 \u2264m. Then i iterative applications of (5), followed by an application of (8), yields the following corollary. Corollary V.1. Given m, d, w, let i be the smallest integer such that m(w \u2212i) \u2212d/2 + 1 \u2264m and t = m(w \u2212i) \u2212d/2 + 1. Then we have M(m, n, d, w) \u2264 \u0016 nm wm \u0016 (n \u22121)m (w \u22121)m \u00b7 \u00b7 \u00b7 \u0016 (n \u2212i + 1)m (w \u2212i + 1)m \u0016 (n \u2212i)t (w \u2212i)t \u0017\u0017 \u00b7 \u00b7 \u00b7 \u0017\u0017 \u2264 nmw\u2212d/2+1 (w \u2212i)mw\u2212d/2+1. When the m, d and w are \ufb01xed, we establish tightness of the bound given by Corollary V.1. Corollary V.2. Fix m, d and w. Let s = mw \u2212d/2 + 1 and i be the smallest integer such that m(w \u2212i) \u2212d/2 + 1 \u2264m. Consider M(m, n, d, w) as a function of n. We have 1 \u2264lim sup n\u2192\u221e M(m, n, d, w) ns/ws \u2264 ws (w \u2212i)s. (9) In addition, when s \u2264m, n/w \u2265mw \u22121 and n/w is a prime power, we have M(m, n, d, w) = ns ws. Proof: When n/w \u2265mw \u22121 and n/w is a prime power, Corollary IV.2 establishes that lim sup n\u2192\u221e M(m, n, d, w) ns/ws \u22651. Then Corollary V.1 establishes (9). If in addition, when s \u2264m, Proposition V.3 with Corollary IV.2 establishes that M(m, n, d, w) = (n/w)s. VI. ASYMPTOTICS In this section, we consider the asymptotic rate of M(m, n, d, w) when m is large, n is a function of m, d = \u230a\u03b4nm\u230band w = \u230a\u03c9n\u230bfor 0 < \u03b4, \u03c9 < 1. Speci\ufb01cally, we determine the value \f12 \u00b5(\u03b4, \u03c9), where \u00b5(\u03b4, \u03c9) := lim sup m\u2192\u221e log2 M(m, n, \u230a\u03b4mn\u230b, \u230a\u03c9n\u230b) mn . In the following discussion, we make use of the following better known exponents. \u03b1q(\u03b4) := lim sup n\u2192\u221e logq A(n, \u230a\u03b4n\u230b) n , \u03b1(\u03b4) := lim sup n\u2192\u221e log2 A(n, \u230a\u03b4n\u230b) n , \u03b1(\u03b4, \u03c9) := lim sup n\u2192\u221e log2 A(n, \u230a\u03b4n\u230b, \u230a\u03c9n\u230b) n , \u03c3(\u03b4) := lim sup n\u2192\u221e log2 S(n, \u230a\u03b4n\u230b) n , \u03c3(\u03b4, \u03c9) := lim sup n\u2192\u221e log2 S(n, \u230a\u03b4n\u230b, \u230a\u03c9n\u230b) n First, we reduce the problem of determining \u00b5(\u03b4, \u03c9) to problem of determining \u03b1(\u03b4, \u03c9). Lemma VI.1. We have A(nm, d, mw) \u2264 \u0000mn mw \u0001 \u0000n w \u0001mM(m, n, d, w). Lemma VI.1 is analogous to Elias-Bassalygo [21, Theorem 33, Chapter 17] by regarding the set of m by n matrices with constant row weight w as a subset of the set of words of length mn with constant-weight mw. As the proof requires some graph theoretical techniques, its proof is deferred to Section VI-B. Proposition VI.1. We have \u03b1(\u03b4, \u03c9) \u2264\u00b5(\u03b4, \u03c9). Proof: Observe that lim n\u2192\u221elog \u0000mn mw \u0001 \u0000n w \u0001m = mnH(\u03c9) \u2212mnH(\u03c9) = 0. Then applying limits on n, m and taking logarithms for Lemma VI.1, we have our result. The asymptotic version of Proposition V.1 is then Proposition VI.2. We have \u00b5(\u03b4, \u03c9) \u2264\u03b1(\u03b4, \u03c9). The proof is immediate and omitted. Combining both Propositions VI.1 and VI.2, we have that the asymptotic exponent of M(m, n, d, w) is equal to the asymptotic exponent of A(mn, d, mw). \f13 Proposition VI.3. We have \u00b5(\u03b4, \u03c9) = \u03b1(\u03b4, \u03c9). Unfortunately, the value of \u03b1(\u03b4, \u03c9) is in general not known. Estimates of \u03b1(\u03b4, \u03c9) are provided by McEliece et al. [28] and Ericson and Zinoviev [29]. In the following subsection, we focus on the case where \u03c9 = 1 2 and evaluate the asymptotic behavior of the constructions given in Section IV-A. A. Asymptotics for \u03c9 = 1 2 The next result follows from the best known upper bound on \u03b1(\u03b4, \u03c9) due to McEliece et al. Proposition VI.4 (McEliece et al. [28, eq. (2.16)]). We have \u00b5(\u03b4, \u03c9) \u2264g(u2), with g(x) = H((1 \u2212\u221a1 \u2212x)/2), and u = \u2212\u03b4 + p \u03b42 \u22122\u03b4 + 4\u03c9(1 \u2212\u03c9). In particular, \u00b5(\u03b4, 1/2) \u2264H(1/2 \u2212 p \u03b4(1 \u2212\u03b4). (10) Our \ufb01rst construction is based on Proposition IV.1, using geometric Goppa codes as outer codes. In particular, \ufb01x q to be a prime power and a square, and \ufb01x 0 \u2264\u03b4 \u22641\u2212 1 \u221aq\u22121. Tsfasman et al. [30] exhibited the existence of a family of geometric codes with relative distance \u03b4 and rate \u03b1q(\u03b4) \u22651 \u2212\u03b4 \u2212 1 \u221aq \u22121. Suppose we pick a CWC(n, d, n/2) of size q as the inner code. For the outer code, we pick a Goppa (m, \u230a\u03b4mn/d\u230b)q code of rate at least 1 \u2212n\u03b4/d \u22121/(\u221aq \u22121). Applying Proposition IV.1, we obtain an MCWC(m, n, \u230a\u03b4mn\u230b, n/2) of size at least qm(1\u2212n\u03b4/d\u22121/(\u221aq\u22121)) Taking logarithm, we have our \ufb01rst lower bound for \u00b5(\u03b4, 1/2). Theorem VI.1. If there exists a CWC(n, d, n/2) of size q, then for \u03b4 \u2264d/n(1 \u22121/(\u221aq \u22121)), \u00b5(\u03b4, 1/2) \u2265log q d \u0012d n \u0012 1 \u2212 1 \u221aq \u22121 \u0013 \u2212\u03b4 \u0013 . Searching through the online table of lower bounds for A(n, d, w) [16], we pick the following constant-weight codes as inner codes: (i) a CWC(12, 4, 6) of size 112, (ii) a CWC(28, 14, 14) of size 72, \f14 (iii) a CWC(28, 4, 14) of size 12372. Applying Theorem VI.1, we have \u00b5(\u03b4, 1/2) \u2265log 11 6 \u0012 3 10 \u2212\u03b4 \u0013 , (11) \u00b5(\u03b4, 1/2) \u2265log 7 14 \u0012 5 12 \u2212\u03b4 \u0013 , (12) \u00b5(\u03b4, 1/2) \u2265log 1237 14 \u00121235 8652 \u2212\u03b4 \u0013 . (13) Our next construction makes use of the pseudo-product code construction given by Proposition IV.2. The asymptotic version of this proposition is as follows. Proposition VI.5. We have \u00b5(\u03b4, \u03c9) \u2265\u03c3(\u03b41, \u03c9)\u03c3(\u03b42), where 0 < \u03b41, \u03b42 < 1 with \u03b4 = \u03b41\u03b42. Theorem VI.2. We have for \u03b4 \u22641/4, \u00b5(\u03b4, 1/2) \u2265(1 \u2212H( \u221a \u03b4))2/2. (14) Proof: By applying Varshamov-Gilbert (VG) bound [21, Theorem 30, Chapter 17] to systematic codes, we get \u03c3(\u03b42) \u22651 \u2212H(\u03b42). Combining VG bound for linear codes with Proposition IV.3 we get \u03c3(\u03b41, 1/2) \u2265(1 \u2212H(\u03b41))/2. Using Proposition VI.5 with \u03b41 = \u03b42 = \u221a \u03b4, the result follows. Our \ufb01nal construction follows from setting q = 2 in Proposition IV.5. Theorem VI.3. We have for \u03b4 \u22641/2, \u00b5(\u03b4, 1/2) \u22651 \u2212H(\u03b4). (15) Proof: Setting q = 2 in Proposition IV.5 and applying VG bound, we have M(m, 2w, 2d, w) \u2265A(mw, d) \u22652mw(1\u2212H(d/mw)). Taking logarithms, we obtain (15). Coincidentally, (15) can be obtained directly by observing that \u00b5(\u03b4, 1/2) = \u03b1(\u03b4, 1/2) = \u03b1(\u03b4). We summarize all the constructions given in this subsection in Figure 3. The top graph \f15 0 0.1 0.2 0.3 0.4 0.5 Relative Distance 0 0.2 0.4 0.6 0.8 1 Rate Upper Bound Eq. (11) Eq. (12) Eq. (13) 0 0.1 0.2 0.3 0.4 0.5 Relative Distance 0 0.2 0.4 0.6 0.8 1 Rate Upper Bound Eq. (11) Eq. (14) Eq. (15) Fig. 3. Upper and lower bounds for \u03c9 = 1/2. \f16 compares the lower bounds resulting from Theorem VI.1 with various constant-weight codes as inner codes, while the bottom graph compares the lower bounds resulting from Theorem VI.1, Theorem VI.2 and Theorem VI.3. We observe that the construction given by Proposition IV.5 (or Theorem VI.3) provides the best lower bound. B. Proof of Lemma VI.1 El Rouayheb and Georghiades [31] generalized the methods of Elias-Bassalygo using graph theoretical methods. Below we introduce certain concepts necessary for the proof of Lemma VI.1. Given two graphs G = (VG, EG) and H = (VH, EH), a mapping \u03c6 : VG \u2192VH is called a graph homomorphism if u, v are adjacent in G implies that \u03c6(u), \u03c6(v) are adjacent in H. When G = H and \u03c6 is a bijection, then \u03c6 is called an automorphism of G. Observe that the set of all automorphisms of G is a group under composition; it is called the automorphism group of G. A graph is then vertex transitive if the action of its automorphism group on its vertex set is transitive. Given a graph G, a subset X of the vertices is said to be independent if every pair of vertices in X is not adjacent in G. The independence number of G, denoted by \u03b1(G), the maximum size of an independent set in G. The following theorem gives the relation between the independence numbers of two graphs that are related by a graph homomorphism (see also [32, Chapter 7]). Theorem VI.4 (El Rouayheb and Georghiades [31, Theorem 4]). If H is vertex transitive and there is a graph homomorphism from G to H, then \u03b1(H) \u2264V (H) V (G) \u03b1(G). Therefore, Lemma VI.1 is a straightforward application of Theorem VI.4. Let G be the graph whose vertices are the m by n arrays with constant row weight w and two vertices are adjacent if the distance between the corresponding arrays are less than d. It is then not dif\ufb01cult to observe that an independent set in G corresponds to a multiply constant-weight code of distance d and hence, \u03b1(G) = M(m, n, d, w). Similarly, let H be the graph whose vertices are codewords of length mn with constant row weight mw and two vertices are adjacent if the distance between the corresponding arrays are less than d. We also have \u03b1(H) = A(mn, d, mw). Finally, observe that G is a subgraph of H and hence, we have a graph homomorphism from G to H. Since H is vertex transitive, we apply Theorem VI.4 to obtain Lemma VI.1. VII."
+                },
+                {
+                    "url": "http://arxiv.org/abs/1304.0278v3",
+                    "title": "Generalized Balanced Tournament Packings and Optimal Equitable Symbol Weight Codes for Power Line Communications",
+                    "abstract": "Generalized balance tournament packings (GBTPs) extend the concept of\ngeneralized balanced tournament designs introduced by Lamken and Vanstone\n(1989). In this paper, we establish the connection between GBTPs and a class of\ncodes called equitable symbol weight codes. The latter were recently\ndemonstrated to optimize the performance against narrowband noise in a general\ncoded modulation scheme for power line communications. By constructing classes\nof GBTPs, we establish infinite families of optimal equitable symbol weight\ncodes with code lengths greater than alphabet size and whose narrowband noise\nerror-correcting capability to code length ratios do not diminish to zero as\nthe length grows.",
+                    "authors": "Yeow Meng Chee, Han Mao Kiah, Alan C. H. Ling, Chengmin Wang",
+                    "published": "2013-04-01",
+                    "updated": "2014-02-10",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO",
+                        "94B99, 94C30, 05B30"
+                    ],
+                    "main_content": "Introduction Power line communications (PLC) is a technology that enables the transmission of data over electric power lines. It was started in the 1910\u2019s for voice communication [27], and used in the 1950\u2019s in the form of ripple control for load and tari\ufb00management in power distribution. With the emergence of the Internet in the 1990\u2019s, research into broadband PLC gathered pace as a promising technology for Internet access and local area networking, since the electrical grid infrastructure provides \u201clast mile\u201d connectivity to premises and capillarity within premises. Recently, there has been a renewed interest in high-speed narrowband PLC due to applications in sustainable energy strategies, speci\ufb01cally in smart grids (see [14, 16, 24, 33]). However, power lines present a di\ufb03cult communications environment and overcoming permanent narrowband disturbance has remained a challenging problem [2, 25, 29]. Vinck [29] addressed this problem through the use of a coded modulation scheme based on permutation codes. More recently, Chee et al. [7] extended Vinck\u2019s analysis to general block codes and motivated the study of equitable symbol weight codes (ESWCs). Relatively little is known about optimal ESWCs, other than those that correspond to permutation codes, injection codes and frequency permutation arrays. In particular, only six in\ufb01nite families of optimal ESWCs with code length greater than alphabet size are known. These have all been constructed by Ding and Yin [12], and Huczynska and Mullen [17] as frequency permutation arrays and they meet the Plotkin bound. One drawback with the code parameters of these families is that the narrowband noise error-correcting capability to length ratio diminishes as length grows. In this paper, we construct in\ufb01nite families of optimal ESWCs whose code lengths are larger than alphabet size and whose narrowband noise error-correcting capability to length ratios tend to a positive constant as code length grows. These families of codes all attain the generalized Plotkin bound. Our results are based on the construction of equivalent combinatorial designs called generalized balanced tournament packings (GBTPs). GBTPs extend the concept of generalized balanced tournament designs (GBTDs) introduced by Lamken and Vanstone [18]. GBTDs have been extensively studied [8,10,19\u201321,32] and is useful in the constructions of resolvable, near-resolvable, doubly resolvable, and doubly near-resolvable balanced incomplete block designs [19,22,23]. Using the classical correspondence given by Semakov and Zinoviev [28] (see also [4,11,32]), we construct optimal families of ESWCs from certain families of GBTPs. We establish existence results for these families of GBTPs by borrowing standard recursion and direct construction methods from combinatorial design theory. The paper is organized as follows. In Section 2, we introduce ESWCs and survey the known results on optimal codes. In Section 3, we introduce GBTPs and establish the equivalence between GBTPs and ESWCs. At the end of the section, we establish two classes of GBTPs that correspond to optimal equitable symbol weight codes. In Sections 4 to 7, we settle the existence of these two classes of GBTPs. Section 4 outlines the general strategy, while Section 5 and Section 6 provide recursive and direct constructions respectively. Some of the results of the paper have been initially reported at IEEE International Symposium on Information Theory 2012 [6], and the present paper contains detailed proofs and includes a new existence result on a family of GBTPs with block size two and three. 2 \f2 Preliminaries 2.1 Notations For positive integer m and prime power q, denote the ring Z/mZ by Zm and the \ufb01nite \ufb01eld of q elements by Fq. Let Z>0 denote the set of positive integers. Let [m] denote the set {1, 2, . . . , m}. We use angled brackets (\u27e8and \u27e9) for multisets. Disjoint set union is depicted using \u2294. For sets A and B, an element (a, b) \u2208A \u00d7 B is sometimes written as ab for succinctness. A set system is a pair S = (X, A), where X is a \ufb01nite set of points and A \u22862X. Elements of A are called blocks. The order of S is the number of points in X, and the size of S is the number of blocks in A. Let K be a set of nonnegative integers. The set system (X, A) is said to be K-uniform if |A| \u2208K for all A \u2208A. 2.2 Equitable Symbol Weight Codes Let \u03a3 be a set of q symbols. A q-ary code of length n over the alphabet \u03a3 is a subset C \u2286\u03a3n. Elements of C are called codewords. The size of C is the number of codewords in C. For i \u2208[n], the ith coordinate of a codeword u \u2208C is denoted ui, so that u = (u1, u2, . . . , un). Denote the frequency of symbol \u03c3 \u2208\u03a3 in codeword u \u2208\u03a3n by w\u03c3(u), that is, w\u03c3(u) = |{ui = \u03c3 : i \u2208[n]}|. An element u \u2208\u03a3n is said to have equitable symbol weight if w\u03c3(u) \u2208{\u230an/q\u230b, \u2308n/q\u2309} for any \u03c3 \u2208\u03a3. If all the codewords of C have equitable symbol weight, then the code C is called an equitable symbol weight code (ESWC). Consider the usual Hamming distance de\ufb01ned on codewords and codes and let d denote the minimum distance of a code C. In addition, consider the following parameter. De\ufb01nition 2.1. Let C be a q-ary code with minimum distance d. The narrowband noise errorcorrecting capability of C is c(C) = min{e : EC(e) \u2265d}, where EC is a function EC : [q] \u2192[n], given by EC(e) = max \u0393\u2286\u03a3 |\u0393|=e max c\u2208C (X \u03c3\u2208\u0393 w\u03c3(c) ) . Chee et al. [7] established that a code C can correct up to c(C) \u22121 narrowband noise errors and demonstrated that an ESWC maximizes the quantity c(C), for \ufb01xed n, d and q. Henceforth, only ESWCs are considered. A q-ary ESWC of length n having minimum distance d is denoted ESWC(n, d)q. Denote the maximum size of an ESWC(n, d)q by AESW q (n, d). Any ESWC(n, d)q of size AESW q (n, d) is said to be optimal. Taken as a q-ary code of length n, an optimal ESWC(n, d)q satis\ufb01es the generalised Plotkin bound [3]. Theorem 2.2 (Generalised Plotkin Bound). If there is a ESWC(n, d)q C of size M, then \u0012M 2 \u0013 d \u2264n q\u22122 X i=0 q\u22121 X j=i+1 MiMj, (1) where Mi = \u230a(M + i)/q\u230b. If q divides M and \u0000M 2 \u0001 d = n \u0000q 2 \u0001 (M/q)2, then C is optimal. In the rest of this paper, ESWCs whose sizes attain the generalised Plotkin bound are constructed. In particular, the following is established. 3 \fTheorem 2.3. The following holds. (i) AESW q (2q \u22121, 2q \u22122) = ( 3, q = 2, 2q, q \u22653. (ii) AESW q (3q \u22121, 3q \u22123) = ( 4, q = 2, 3q, q \u22653. (iii) AESW q (4q \u22121, 4q \u22124) = ( 4q \u22121, q = 2, 3, 4q, q \u22654. (iv) If q \u226562 or q \u2208{5 \u221218, 30, 42, 46, 48 \u221250, 54 \u221257}, AESW q (5q \u22121, 5q \u22125) = 5q. (v) If q is an odd prime power, AESW q (q2 \u22121, q2 \u2212q) = q2. (vi) AESW q \u00123q \u22121 2 , 3q \u22123 2 \u0013 = ( 4q \u22126, q = 3, 5, 3q, q \u22657 is odd. (vii) AESW q (2q \u22123, 2q \u22124) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 6q \u221212, q = 3, 4, 14, q = 5, 6, 2q + 1, q \u22657, except possibly q \u2208{12, 13}. Observe that any ESWC C with the above parameters must have c(C) = q \u22121. In Table 1, we verify that c(C)/n tends to a positive constant as q grows. In the same table, we compare with known families of optimal ESWC(n, d)q. In particular, only six in\ufb01nite nontrivial families of optimal codes with n > q are known. However, code parameters for these six families are such that their relative narrowband noise errorcorrecting capability to length ratios diminish to zero as q grows. This is undesirable for narrowband noise correction for PLC. Hence, Theorem 2.3 provides in\ufb01nite families of optimal ESWCs with code lengths are larger than alphabet size and whose relative narrowband noise capability to length ratios tend to a positive constant as length grows. These optimal ESWCs are constructed from GBTPs using the classical correspondence given by Semakov and Zinoviev [28]\u2217. We remark that GBTPs extend the concept of GBTDs and consequently Theorem 2.3 (i) to (v) follows directly from known classes of GBTDs. We explain the connection in detail in the next section. \u2217Bogdanova et al. [4] gave a survey of connection between equidistant codes and designs. Using this correspondence, Ding and Yin [11] constructed optimal constant-composition codes, while Yin et al. [32] constructed near-constantcomposition codes. 4 \fTable 1: In\ufb01nite families of optimal ESWC(n, d)q ESWC(n, d)q C |C| c(C) lim q\u2192\u221ec(C)/n Remarks (n, n)q for q \u22652 q min{n, q} \u2013 easy (3, 2)q for q \u22653 q(q \u22121) 2 0 injection code [13] (4, 2)q for q \u22654, q \u0338= 7 q(q \u22121)(q \u22122) 2 0 injection code [13] (n, 1)q for n < q q(q \u22121) \u00b7 \u00b7 \u00b7 (q \u2212n + 1) 1 1/n injection code, easy (qn, 2)q for q \u22652 (qn)!/(n!)q 2 0 frequency permutation array, easy (q, 3)q for q \u22653 q!/2 3 0 permutation code, easy (q, q \u22121)q for prime powers q q(q \u22121) q \u22121 1 injection code [9] (n, n \u22121)q for q su\ufb03ciently large and n \u2264q q(q \u22121) n \u22121 1 \u22121/n injection code [13] (q, q \u22122)q for prime powers q \u22121 q(q \u22121)(q \u22122) q \u22122 1 permutation code [15] \u0000q(q + 1), q2\u0001 q for prime powers q q2 q 0 frequency permutation array [11] \u0010 q(kq2\u22121) k\u22121 , kq2(q\u22121) k\u22121 \u0011 q for prime powers q, 2 \u2264k \u22645, (k, q) \u0338= (5, 9) kq2 q 0 frequency permutation array [12] \u0010 \u00b5qs\u2212t(q2s\u2212t\u22121) qt\u22121 , \u00b5q2s\u2212t(qs\u2212t\u22121) qt\u22121 \u0011 qs\u2212t for prime powers q, 1 \u2264t < s, \u00b5 = Qt\u22121 i=1 qs\u2212i\u22121 qi\u22121 q2s\u2212t qs\u2212t 0 frequency permutation array [12] (qs(q2s+c \u22121), q2s+c(qs \u22121))qs, for prime powers q, and s, c \u22651 q2s+c qs 0 frequency permutation array [12] ( \u0000kq k \u0001 , kq\u2212k kq\u22121 \u0000kq k \u0001 )q for q, k \u22651 kq q \u22121 0 frequency permutation array [17] (2q2 \u2212q, 2q2 \u22122q)q for even q, q / \u2208{2, 6} 2q q 0 frequency permutation array [17] (2q \u22121, 2q \u22122)q for q \u22653 2q q \u22121 1/2 Theorem 2.3 (3q \u22121, 3q \u22123)q for q \u22653 3q q \u22121 1/3 Theorem 2.3 (4q \u22121, 4q \u22124)q for q \u22654 4q q \u22121 1/4 Theorem 2.3 (5q \u22121, 5q \u22125)q for q \u226562 5q q \u22121 1/5 Theorem 2.3 (q2 \u22121, q2 \u2212q)q for q \u22654 q2 q \u22121 0 Theorem 2.3 \u0000 3q\u22121 2 , 3q\u22123 2 \u0001 q for q \u22657 and q odd 3q q \u22121 2/3 Theorem 2.3 (2q \u22123, 2q \u22124)q for q \u226514 2q + 1 q \u22122 1/2 Theorem 2.3 5 \f3 Constructions of Equitable Symbol Weight Codes We \ufb01rst determine AESW q (n, d) for small values of n, q and d. With the exception of AESW 6 (9, 8), an exhaustive computer search established the following values of AESW q (n, d). For AESW 6 (9, 8), an ESWC(9, 8)6 of size 14 was found via computer search. Since an ESWC(9, 8)6 of size 15 cannot exist by the generalized Plotkin bound, it follows that AESW 6 (9, 8) = 14. We record the results of the computations in the following proposition and the corresponding optimal codes can be found at [5]. Proposition 3.1. The following holds: AESW 2 (3, 2) = 3 AESW 2 (5, 3) = 4 AESW 2 (7, 4) = 7 AESW 3 (3, 2) = 6 AESW 3 (4, 3) = 6 AESW 3 (11, 8) = 11 AESW 4 (5, 4) = 12 AESW 5 (7, 6) = 14 AESW 6 (9, 8) = 14. The rest of the paper establishes the remaining values in Theorem 2.3. To do so, we de\ufb01ne a class of combinatorial designs that is equivalent to ESWCs. 3.1 Equitable Symbol Weight Codes and Generalized Balanced Tournament Packings Let \u03bb, v be positive integers and K be a set of nonnegative integers. A (v, K, \u03bb)-packing is a Kuniform set system of order v such that every pair of distinct points is contained in at most \u03bb blocks. The value \u03bb is called the index of the packing. A parallel class (or resolution class) of a packing is a subset of the blocks that partitions the set of points X. If the set of blocks can be partitioned into parallel classes, then the packing is resolvable, and denoted by RP(v, K, \u03bb). An RP(v, K, \u03bb) is called a maximum resolvable packing, denoted by MRP(v, K, \u03bb), if it contains maximum possible number of parallel classes. Furthermore, an MRP(v, {k}, \u03bb) is called a resolvable (v, {k}, \u03bb)-balanced incomplete block design, or RBIBD(v, k, \u03bb) in short, if every pair of distinct points is contained in exactly \u03bb blocks. A simple computation gives the size of an RBIBD(v, k, \u03bb) to be \u03bbv(v\u22121) k(k\u22121) . We de\ufb01ne the combinatorial object of study in this paper. We note that this de\ufb01nition is a generalization of GBTDs to packings and various indices. De\ufb01nition 3.2. Let (X, A) be an RP(v, K, \u03bb) with n parallel classes. Then (X, A) is called a generalized balanced tournament packing if the blocks of A are arranged into an m \u00d7 n array satisfying the following conditions: (i) every point in X is contained in exactly one cell of each column, (ii) every point in X is contained in either \u2308n/m\u2309or \u230an/m\u230bcells of each row. We denote such a GBTP by GBTP\u03bb(K; v, m \u00d7 n). Unless otherwise stated, the rows of a GBTP\u03bb(K; v, m\u00d7n) are indexed by [m] and the columns by [n]. In a GBTP\u03bb(K; v, m \u00d7 n), given point x and column j, there is a unique row that contains the point x in column j. Hence, for each point x \u2208X of a GBTP\u03bb(K; v, m \u00d7 n) (X, A), we may 6 \fcorrespond the codeword c(x) = (r1, r2, . . . , rn) \u2208[m]n, where rj is the row in which point x appears in column j. It is obvious that C = {c(x) : x \u2208X} is an m-ary code of length n over the alphabet [m]. We note that this correspondence is precisely the one used by Semakov and Zinoviev [28] to show the equivalence between equidistant codes and resolvable balanced incomplete block designs. For distinct points x, y \u2208X, the distance between c(x) and c(y) is the number of columns for which x and y are not both contained in the same row. Since there are at most \u03bb blocks containing both x and y, and that no two such blocks can occur in the same column of the GBTP\u03bb(K; v, m\u00d7n), the distance between c(x) and c(y) is at least n \u2212\u03bb. Next, we determine wi(c(x)), for x \u2208X and i \u2208[m]. From the construction of c(x), the number of times a symbol i appears in c(x) is the number of cells in row i that contains x. By the de\ufb01nition of a GBTP\u03bb(K; v, m \u00d7 n), this number belongs to {\u230an/m\u230b, \u2308n/m\u2309}. Hence, C is an ESWC of size v. Finally, this construction of an ESWC from a GBTP can easily be reversed. We record these observations as: Theorem 3.3. Let K be set of nonnegative integers. Then a GBTP\u03bb(K; v, m \u00d7 n) exists if and only if an ESWC(n, n \u2212\u03bb)m of size v exists. We note that the correspondence between GBTPs and ESWCs was observed by Yin et al. [32, Theorem 2.2]. However, in the latter paper, the class of codes constructed is called near-constantcomposition codes (NCCCs). Indeed, an NCCC is a special class of ESWC and one observes that an ESWC(n, d)q is an NCCC when n + 1 \u22610 mod q. Example 3.1. Consider the GBTP1({2, 3}, 6, 3 \u00d7 4) below. {1, 4} {2, 6} {3, 5} {1, 2, 3} {2, 5} {3, 4} {1, 6} {4, 5, 6} {3, 6} {1, 5} {2, 4} Each point x \u2208[6] gives a codeword c(x) = (r1, r2, . . . , r5), where rj is the row in which point x appears in column j. Hence, we have c(1) = (2, 1, 3, 2), c(2) = (2, 2, 1, 3), c(3) = (2, 3, 2, 1), c(4) = (3, 1, 2, 3), c(5) = (3, 2, 3, 1), c(6) = (3, 3, 1, 2). The code C = {c(1), c(2), c(3), c(4), c(5), c(6)} is an ESWC(4, 3)3 of size six. Theorem 3.3 sets up the equivalence between GBTPs and ESWCs. In general, a GBTP may not correspond to an optimal ESWC. However, in the following, we look at speci\ufb01c K\u2019s to derive families of optimal ESWCs. 3.2 Optimal Equitable Symbol Weight Codes from Generalized Balanced Tournament Designs A GBTP\u03bb \u0010 {k}; km, m \u00d7 \u03bb(km\u22121) k\u22121 \u0011 is called a generalized balanced tournament design (GBTD), denoted by GBTD\u03bb(k, m). In this case, we check that each pair of distinct points is contained in exactly \u03bb blocks and every point is contained in either l \u03bb(km\u22121) m(k\u22121) m or j \u03bb(km\u22121) m(k\u22121) k cells of each row. Applying Theorem 3.3, a ESWC \u0010 \u03bb(km\u22121) k\u22121 , \u03bbk(m\u22121) k\u22121 \u0011 m of size km exists and the corresponding code is optimal by generalized Plotkin bound. So, we have the following. 7 \fTheorem 3.4. A GBTD\u03bb(k, m) exists if and only if an optimal ESWC \u0010 \u03bb(km\u22121) k\u22121 , \u03bbk(m\u22121) k\u22121 \u0011 m of size km exists and attains the generalized Plotkin bound. We remark that our de\ufb01nition of a GBTD extends that of Lamken and Vanstone [18], which corresponds in our de\ufb01nition to the case when \u03bb = k \u22121. The following summarizes the state-ofthe-art results on the existence of GBTDk\u22121(k, m). Theorem 3.5 (Lamken [18\u201321], Yin et al. [32], Chee et al. [8], Dai et al. [10]). The following holds. (i) A GBTD1(2, m) exists if and only if m = 1 or m \u22653. (ii) A GBTD2(3, m) exists if and only if m = 1 or m \u22653. (iii) A GBTD3(4, m) exists if and only if m = 1 or m \u22654. (iv) A GBTD4(5, m) exists if m \u226562 or m \u2208{5 \u221218, 30, 42, 46, 48 \u221250, 54 \u221257}. (v) A GBTDk\u22121(k, k) exists if k is an odd prime power. Theorem 2.3 (i) to (v) is now an immediate consequence of Theorem 3.4, 3.5 and Proposition 3.1. The existence of GBTD\u03bb(k, m) when \u03bb \u0338= k \u22121 has not been previously investigated. The smallest open case is when k = 3 and \u03bb = 1, which is the case dealt with in this paper. It follows readily from the fact that a GBTD1(3, m) is also an RBIBD(3m, 3, 1), that a necessary condition for a GBTD1(3, m) to exist is that m must be odd. We note from Proposition 3.1 that AESW 3 (4, 3) = 6 and AESW 5 (7, 6) = 14, which do not meet the Plotkin bound. Hence, the corresponding designs GBTD1(3, 3) and GBTD1(3, 5) do not exist by Theorem 3.4. Hence, a GBTD1(3, m) can exist only if m is odd and m / \u2208{3, 5}. In Sections 4 to 7, we prove that this necessary condition is also su\ufb03cient for the existence of GBTD1(3, m). A direct consequence of this is Theorem 2.3 (vi). 3.3 Optimal Equitable Symbol Weight Codes from GBTP1({2, 3\u2217}; 2m + 1, m \u00d7 (2m \u22123)) Theorem 3.4 constructs optimal ESWCs from GBTDs. In this subsection, we make slight variations to obtain another in\ufb01nite family of optimal ESWCs. Consider a GBTP1({2, 3}; v, m\u00d7n). If there is exactly one block of size three in each resolution class, then we denote the GBTP by GBTP1({2, 3\u2217}; v, m \u00d7 n). A simple computation then shows v = 2m + 1. Now we establish the following construction for optimal ESWCs. Theorem 3.6. Let m \u22657. If there exists a GBTP1({2, 3\u2217}; 2m + 1, m \u00d7 (2m \u22123)), then there exists an optimal ESWC(2m \u22123, 2m \u22124)m of size 2m + 1 which attains the generalized Plotkin bound. Proof. By Theorem 3.3, we have a ESWC(2m \u22123, 2m \u22124)m of size 2m + 1. It remains to verify its optimality. Suppose otherwise that there exists an ESWC(2m \u22123, 2m \u22124)m of size 2m + 2. Consider (1) in Theorem 2.2. On the left hand side, we have \u00122m + 2 2 \u0013 \u00b7 (2m \u22124) = 4m3 \u22122m2 \u221210m \u22124. 8 \fSince \u0004 2m+2+i m \u0005 = 2 for 0 \u2264i \u2264m \u22123 and j 2m+2+(m\u22122) m k = j 2m+2+(m\u22121) m k = 3, the term on the right hand is (2m \u22123)   m\u22123 X i=0 4(m \u22123 \u2212i) + 12 ! + 9 ! = (2m \u22123)(4m(m \u22122) \u22122(m \u22123)(m \u22122) + 9) = 4m3 \u22122m2 \u221212m + 9 But for m \u22657, 4m3 \u22122m2 \u221210m \u22124 > 4m3 \u22122m2 \u221212m + 9, contradicting (1). Hence, an ESWC(2m \u22123, 2m \u22124)m of size 2m + 2 does not exist and the result follows. In the rest of this paper, we construct a GBTP1({2, 3\u2217}; 2m+1, m\u00d7(2m\u22123)) for m \u22654, except possibly m \u2208{12, 13}. This with Theorem 3.6 and Proposition 3.1 gives Theorem 2.3 (vii). 4 Proof Strategy of Theorem 2.3 (vi) and Theorem 2.3 (vii) For the rest of the paper, we determine with \ufb01nite possible exceptions the existence of GBTD1(3, m) and GBTP1({2, 3\u2217}; 2m + 1, m \u00d7 (2m \u22123)). Our proof is technical and rather complex. However, it follows the general strategy of the previous work [10, 20, 32]. This section outlines the general strategy used, and introduces some required combinatorial designs. As with most combinatorial designs, direct construction to settle their existence is often di\ufb03cult. Instead, we develop a set of recursive constructions, building big designs from smaller ones. Direct methods are used to construct a large enough set of small designs on which the recursions can work to generate all larger designs. For our recursion techniques to work, the GBTPs must possess more structure than stipulated in its de\ufb01nition. First, we consider GBTD1(3, m)s that are \u2217colorable which are de\ufb01ned below. 4.1 c-\u2217colorable Generalized Balanced Tournament Designs We generalize the notion of factored GBTDs (FGBTDs) introduced by Lamken [21]. FGBTDs are crucial in the k-tupling construction for GBTDs of index k \u22121. However, when the index is one, we extend this notion to \u2217-colorability. De\ufb01nition 4.1. Let c be positive. A c-\u2217colorable RBIBD(v, k, \u03bb) is an RBIBD(v, k, \u03bb) with the property that its \u03bbv(v\u22121) k(k\u22121) blocks can be arranged in a v k \u00d7 \u03bb(v\u22121) k\u22121 array, and each block can be colored with one of c colors so that (i) each point appears exactly once in each column, and (ii) in each row, blocks of the same color are pairwise disjoint. De\ufb01nition 4.2. A GBTD\u03bb(k, m) is c-\u2217colorable if each of its blocks can be colored with one of c colors so that in each row, blocks of the same color are pairwise disjoint. 9 \f0001\u221e\u2663 204031 \u2663 614010 \u2666 211161 \u2663 511020 \u2665 413111 \u2665 514121 \u2666 615131 \u2663 1011\u221e\u2663 305041 \u2663 3031\u221e\u2666 500061 \u2666 601001 \u2666 002011 \u2665 103021 \u2663 016141 \u2663 2021\u221e\u2666 406051 \u2663 116030 \u2666 5051\u221e\u2665 311050 \u2666 412060 \u2663 513000 \u2663 110151 \u2666 015020 \u2665 4041\u221e\u2666 210040 \u2665 6061\u221e\u2665 114050 \u2663 215060 \u2666 316000 \u2663 410010 \u2666 312101 \u2665 612030 \u2663 013040 \u2666 Figure 1: A 3-\u2217colorable RBIBD(15, 3, 1) (X, A), where X = (Z7 \u00d7 Z2) \u222a{\u221e}. The set of colors used is {\u2663, \u2666, \u2665}. (X, A) has property \u03a0 as 10 is a witness for \u2663and \u221eis a witnesses for both \u2666 and \u2665in row 1. For succintness, a block {x, y, z} is written xyz De\ufb01nition 4.3. A k-\u2217colorable RBIBD(v, k, 1) is k-\u2217colorable with property \u03a0 if there exists a row r such that for each color i, there exists a point (called a witness for i) that is not contained in any block in row r that is colored i. A GBTD1(k, m) that is c-\u2217colorable with property \u03a0 is similarly de\ufb01ned. Example 4.1. The RBIBD(15, 3, 1) in Fig. 1 is 3-\u2217colorable with property \u03a0. Proposition 4.4. If an RBIBD(v, k, 1) is (k \u22121)-\u2217colorable, then it is k-\u2217colorable with property \u03a0. Proof. Consider a (k \u22121)-\u2217colorable RBIBD(v, k, 1) with colors c1, c2, \u00b7 \u00b7 \u00b7 , ck\u22121. There must exists a point, say x, that appears only once in the \ufb01rst row. Recolor the block that contains this point with color ck. This new coloring shows that the RBIBD(v, k, 1) is k-\u2217colorable with property \u03a0, since for the \ufb01rst row, the point x is a witness for colors c1, c2, . . . , ck\u22121, and any point not in the block colored by ck is a witness for ck. Example 4.2. The GBTD1(3, 9) in Fig. 2 is 2-\u2217colorable and is therefore 3-\u2217colorable with property \u03a0 by Proposition 4.4. We note that a 3-\u2217colorable RBIBD and a 3-\u2217colorable RBIBD with property \u03a0 are crucial in the tripling construction of a GBTD1(3, m) and a special GBTD(1(3, m) respectively (see Proposition 5.1). This is an adaptation of the k-tupling construction for GBTDs with index k \u22121 [21, Theorem 3.1]. However, we note certain di\ufb00erences. An FGBTD by de\ufb01nition is necessary a GBTD, while \u2217-colorability and property \u03a0 are de\ufb01ned for RBIBDs. Hence, we do not need a smaller GBTD to seed the recursion in Proposition 5.1. We make use of this fact to yield a special GBTD1(3, 15) in Lemma 7.1. 4.2 Incomplete Generalized Balanced Tournament Packings Incomplete designs are ubiquitous in combinatorial design theory and crucial in \u201c\ufb01lling in the holes\u201d constructions described in section 5. Suppose that (X, A) is a (v, K, \u03bb)-packing. Let W \u2282X with |W| = w. Furthermore, we call (X, W, A) is an incomplete resolvable packing, denoted by IRP(v, K, \u03bb; w), if it satis\ufb01es the following conditions: (i) any pair of points from W occurs in no blocks of A, 10 \f(ii) the blocks in A can be partitioned into parallel classes and partial parallel classes X \\ W. De\ufb01nition 4.5. Let (X, W, A) be an IRP(v, K, \u03bb; w). Then (X, W, A) is called an incomplete generalized balanced tournament packing (IGBTP) if the blocks of A are arranged into an m \u00d7 n array A, with rows and columns indexed by R and C respectively, satisfying the following conditions: (i) there exist a P \u2282R with |P| = m\u2032 and a Q \u2282C with |Q| = n\u2032 such that the cell (r, c) is empty if r \u2208P and c \u2208Q; (ii) for any row r \u2208P, every point in X \\ W is contained in either \u2308n/m\u2309or \u230an/m\u230bcells and the points in W do not appear; for any row r \u2208R \\ P, every point in X is contained in either \u2308n/m\u2309or \u230an/m\u230bcells; (iii) the blocks in any column c \u2208Q form a partial parallel class of X \\ W and the blocks in any column c \u2208C \\ Q forms a parallel class of X. Denote such an IGBTP by IGBTP\u03bb(K, v, m \u00d7 n; w, m\u2032 \u00d7 n\u2032). Example 4.3. An IGBTP1({2, 3\u2217}, 29, 14 \u00d7 25; 9, 4 \u00d7 5) is given in Fig. 3. Consider an IGBTP1({k}, km, m\u00d7 km\u22121 k\u22121 ; k, 1\u00d71). Then its corresponding array has one empty cell and we \ufb01ll this cell with the block W to obtain a GBTD1(k, m). A GBTD1(k, m) obtained in this way is called a special GBTD1(k, m) and the cell occupied by W is said to be special. Example 4.4. The GBTD1(3, 9) in Fig. 2 is a special GBTD1(3, 9) with special cell (1, 5). A few more classes of auxiliary designs are also required. 4.3 Group Divisible Designs and Transversal Designs De\ufb01nition 4.6. Let (X, A) be a set system and let G = {G1, G2, . . . , Gs} be a partition of X into subsets, called groups. The triple (X, G, A) is a group divisible design (GDD) when every 2-subset of X not contained in a group appears in exactly one block, and |A \u2229G| \u22641 for A \u2208A and G \u2208G. We denote a GDD (X, G, A) by K-GDD if (X, A) is K-uniform. The type of a GDD (X, G, A) is the multiset \u27e8|G| : G \u2208G\u27e9. For convenience, the exponential notation is used to describe the type of a GDD: a GDD of type gt1 1 gt2 2 \u00b7 \u00b7 \u00b7 gts s is a GDD with exactly ti groups of size gi, i \u2208[s]. De\ufb01nition 4.7. A transversal design TD(k, n) is a {k}-GDD of type nk. The following result on the existence of transversal designs (see [1]) is sometimes used without explicit reference throughout this paper. Theorem 4.8. Let TD(k) denote the set of positive integers n such that there exists a TD(k, n). Then, we have (i) TD(4) \u2287Z>0 \\ {2, 6}, (ii) TD(5) \u2287Z>0 \\ {2, 3, 6, 10}, (iii) TD(6) \u2287Z>0 \\ {2, 3, 4, 6, 10, 14, 18, 22}, (iv) TD(7) \u2287Z>0 \\ {2, 3, 4, 5, 6, 10, 14, 15, 18, 20, 22, 26, 30, 34, 38, 46, 60}, 11 \fA B where A is the array 1071\u221e2 \u26636032\u221e1 \u2663004162 \u2663607102 \u2666700112 \u2663505152 \u26661142\u221e0 \u2663 5022\u221e1 \u26662001\u221e2 \u26637042\u221e1 \u2663105172 \u2663404142 \u2666001122 \u26637051\u221e2 \u2666 3162\u221e0 \u26634172\u221e0 \u26663011\u221e2 \u26630052\u221e1 \u2663206102 \u2663307112 \u2666102132 \u2663 304152 \u2663101112 \u26665102\u221e0 \u26634021\u221e2 \u26665031\u221e2 \u26632072\u221e1 \u2663400122 \u2663 602142 \u2666405162 \u2663506172 \u26666112\u221e0 \u26631062\u221e1 \u26666041\u221e2 \u26633002\u221e1 \u2663 000102 \u2666502102 \u2663407132 \u2663201162 \u26637122\u221e0 \u2666106142 \u2663003172 \u2663 706132 \u2663703152 \u2666603112 \u2663500142 \u2663302172 \u26630132\u221e0 \u2666207152 \u2663 205112 \u2663007142 \u2663202122 \u2666704122 \u2663601152 \u2663403102 \u2663606162 \u2666 401172 \u2663306122 \u2663100152 \u2663303132 \u2666005132 \u2663702162 \u2663504112 \u2663 where B is the array 0061\u221e2 \u2666 406152 \u2663102040 \u2666203050 \u2663227202 \u2663213151 \u2663 2152\u221e0 \u2663 507162 \u2663314161 \u2663011131 \u2666304060 \u2663320212 \u2663 203142 \u2666 600172 \u2663421222 \u2663415171 \u2663516101 \u2666405070 \u2663 707172 \u2666 002112 \u2663025262 \u2666607010 \u2663623242 \u2663617111 \u2663 501132 \u2663 103122 \u2663710121 \u2663522232 \u2666700020 \u2663724252 \u2663 605122 \u2663 204132 \u26663070\u221e0 \u2663126272 \u26665212\u221e2 \u26634101\u221e1 \u2663 104102 \u2663 305142 \u26661151\u221e1 \u26634000\u221e0 \u2663112141 \u26666222\u221e2 \u2663 300162 \u2663\u221e0\u221e1\u221e2 \u26663272\u221e2 \u26632161\u221e1 \u26635010\u221e0 \u2663001030 \u2666 4012\u221e1 \u2666 701102 \u2666506000 \u26664202\u221e2 \u26633171\u221e1 \u26636020\u221e0 \u2663 Figure 2: A 2-\u2217colorable special GBTD1(3, 9) (X, A), where X = (Z8 \u00d7 Z3) \u222a{\u221e0, \u221e1, \u221e2} and colors {\u2663, \u2666}. The cell (1, 5), occupied by the block 700112, is special. For succinctness, a set {x, y, z} is written xyz. (v) TD(k) \u2287{q : q \u2265k \u22121 is a prime power}. De\ufb01nition 4.9. A doubly resolvable TD(k, n), denoted by DRTD(k, n), is a TD(k, n) whose blocks can be arranged in an n \u00d7 n array such that each point appears exactly once in each row and once in each column. The following proposition describes the relationship between DRTDs and TDs Proposition 4.10 (Folklore, see [9]). There exists a TD(k + 2, n) if and only if there exists a DRTD(k, n). Corollary 4.11. A DRTD(3, n) exists for all n \u22654 and n \u0338\u2208{6, 10}. Proof. A TD(5, n) exists if n \u22654 and n \u0338\u2208{6, 10} by Theorem 4.8. 12 \fA B where A is the array \u2013 \u2013 \u2013 \u2013 \u2013 2, 13 3, 14 4, 15 5, 16 6, 17 7, 18 8, 19 9, 0 \u2013 \u2013 \u2013 \u2013 \u2013 12, 16 13, 17 14, 18 15, 19 16, 0 17, 1 18, 2 19, 3 \u2013 \u2013 \u2013 \u2013 \u2013 15, 18 16, 19 17, 0 18, 1 19, 2 0, 3 1, 4 2, 5 \u2013 \u2013 \u2013 \u2013 \u2013 1, 3 2, 4 3, 5 4, 6 5, 7 6, 8 7, 9 8, 10 0, 10 2, 7 12, 17 4, 16 14, 6 4, 5, 11 i, 18 h, 1 g, 12 f, 18 e, 13 d, 16 c, 13 1, 11 3, 8 13, 18 5, 17 15, 7 a, 0 5, 6, 12 i, 19 h, 2 g, 13 f, 19 e, 14 d, 17 2, 12 4, 9 14, 19 6, 18 16, 8 b, 7 a, 1 6, 7, 13 i, 0 h, 3 g, 14 f, 0 e, 15 3, 13 5, 10 15, 0 7, 19 17, 9 c, 6 b, 8 a, 2 7, 8, 14 i, 1 h, 4 g, 15 f, 1 4, 14 6, 11 16, 1 8, 0 18, 10 d, 10 c, 7 b, 9 a, 3 8, 9, 15 i, 2 h, 5 g, 16 5, 15 7, 12 17, 2 9, 1 19, 11 e, 8 d, 11 c, 8 b, 10 a, 4 9, 10, 16 i, 3 h, 6 6, 16 8, 13 18, 3 10, 2 0, 12 f, 14 e, 9 d, 12 c, 9 b, 11 a, 5 10, 11, 17 i, 4 7, 17 9, 14 19, 4 11, 3 1, 13 g, 9 f, 15 e, 10 d, 13 c, 10 b, 12 a, 6 11, 12, 18 8, 18 10, 15 0, 5 12, 4 2, 14 h, 19 g, 10 f, 16 e, 11 d, 14 c, 11 b, 13 a, 7 9, 19 11, 16 1, 6 13, 5 3, 15 i, 17 h, 0 g, 11 f, 17 e, 12 d, 15 c, 12 b, 14 where B is the array 10, 1 11, 2 12, 3 13, 4 14, 5 15, 6 16, 7 17, 8 18, 9 19, 10 0, 11 1, 12 0, 4 1, 5 2, 6 3, 7 4, 8 5, 9 6, 10 7, 11 8, 12 9, 13 10, 14 11, 15 3, 6 4, 7 5, 8 6, 9 7, 10 8, 11 9, 12 10, 13 11, 14 12, 15 13, 16 14, 17 9, 11 10, 12 11, 13 12, 14 13, 15 14, 16 15, 17 16, 18 17, 19 18, 0 19, 1 0, 2 b, 15 a, 9 14, 15, 1 i, 8 h, 11 g, 2 f, 8 e, 3 d, 6 c, 3 b, 5 a, 19 c, 14 b, 16 a, 10 15, 16, 2 i, 9 h, 12 g, 3 f, 9 e, 4 d, 7 c, 4 b, 6 d, 18 c, 15 b, 17 a, 11 16, 17, 3 i, 10 h, 13 g, 4 f, 10 e, 5 d, 8 c, 5 e, 16 d, 19 c, 16 b, 18 a, 12 17, 18, 4 i, 11 h, 14 g, 5 f, 11 e, 6 d, 9 f, 2 e, 17 d, 0 c, 17 b, 19 a, 13 18, 19, 5 i, 12 h, 15 g, 6 f, 12 e, 7 g, 17 f, 3 e, 18 d, 1 c, 18 b, 0 a, 14 19, 0, 6 i, 13 h, 16 g, 7 f, 13 h, 7 g, 18 f, 4 e, 19 d, 2 c, 19 b, 1 a, 15 0, 1, 7 i, 14 h, 17 g, 8 i, 5 h, 8 g, 19 f, 5 e, 0 d, 3 c, 0 b, 2 a, 16 1, 2, 8 i, 15 h, 18 12, 13, 19 i, 6 h, 9 g, 0 f, 6 e, 1 d, 4 c, 1 b, 3 a, 17 2, 3, 9 i, 16 a, 8 13, 14, 0 i, 7 h, 10 g, 1 f, 7 e, 2 d, 5 c, 2 b, 4 a, 18 3, 4, 10 Figure 3: An IGBTP1({2, 3}, 29, 14 \u00d7 25; 9, 4 \u00d7 5) (X, A), where X = Z20 \u222a{a, b, c, d, e, f, g, h, i} and W = {a, b, c, d, e, f, g, h, i}. For succinctness, a block {x, y, z} is written x, y, z. 4.4 Frame Generalized Balanced Tournament Design Let (X, G, A) be a {k}-GDD with G = {G1, G2, . . . , Gs} and |Gi| \u22610 mod k(k \u22121) for all i \u2208[s]. Let R = 1 k Ps i=1 |Gi| and C = 1 k\u22121 Ps i=1 |Gi|. Suppose there exists a partition [R] = Fs i=1 Ri and a 13 \fA B where A is the array \u2013 \u2013 \u2013 401070 411171 421272 603090 613191 623292 \u2013 \u2013 \u2013 607280 617081 627182 809200 819001 829102 208110 218211 228012 \u2013 \u2013 \u2013 4182\u221e4 4280\u221e3 4081\u221e5 627231 607032 617130 \u2013 \u2013 \u2013 9112\u221e5 9210\u221e4 9011\u221e3 400130 410231 420032 113091 123192 103290 \u2013 \u2013 \u2013 829251 809052 819150 426182 406280 416081 \u2013 \u2013 \u2013 602150 612251 622052 8112\u221e0 8210\u221e1 8011\u221e2 315011 325112 305210 021271 001072 011170 2031\u221e1 2132\u221e2 2230\u221e0 628102 608200 618001 804170 814271 824072 2290\u221e2 2091\u221e0 2192\u221e1 0132\u221e0 0230\u221e1 0031\u221e2 223291 203092 213190 3241\u221e3 3042\u221e5 3140\u221e4 4051\u221e1 4152\u221e2 4250\u221e0 006190 016291 026092 2162\u221e4 2260\u221e3 2061\u221e5 4210\u221e2 4011\u221e0 4112\u221e1 425211 405012 415110 7192\u221e5 7290\u221e4 7091\u221e3 5261\u221e3 5062\u221e5 5160\u221e4 where B is the array 805010 815111 825212 007030 017131 027232 209050 219151 229252 001220 011021 021122 203240 213041 223142 405260 415061 425162 6230\u221e2 6031\u221e0 6132\u221e1 4172\u221e0 4270\u221e1 4071\u221e2 911071 921172 901270 7281\u221e3 7082\u221e5 7180\u221e4 8091\u221e1 8192\u221e2 8290\u221e0 224162 204260 214061 6102\u221e4 6200\u221e3 6001\u221e5 8250\u221e2 8051\u221e0 8152\u221e1 6192\u221e0 6290\u221e1 6091\u221e2 1132\u221e5 1230\u221e4 1031\u221e3 9201\u221e3 9002\u221e5 9100\u221e4 0011\u221e1 0112\u221e2 0210\u221e0 \u2013 \u2013 \u2013 8122\u221e4 8220\u221e3 8021\u221e5 0270\u221e2 0071\u221e0 0172\u221e1 \u2013 \u2013 \u2013 3152\u221e5 3250\u221e4 3051\u221e3 1221\u221e3 1022\u221e5 1120\u221e4 517031 527132 507230 \u2013 \u2013 \u2013 0142\u221e4 0240\u221e3 0041\u221e5 820122 800220 810021 \u2013 \u2013 \u2013 5172\u221e5 5270\u221e4 5071\u221e3 2152\u221e0 2250\u221e1 2051\u221e2 719051 729152 709250 \u2013 \u2013 \u2013 6071\u221e1 6172\u221e2 6270\u221e0 022142 002240 012041 \u2013 \u2013 \u2013 Figure 4: An FrGBTD1(3, 66) (X, G, A), where X = (Z10 \u00d7 Z3) \u222a{\u221ei : i \u2208Z6} and G = {{t0, t1, t2, (5 + t)0, (5 + t)1, (5 + t)2} : t \u2208Z5} \u222a{\u221ei : i \u2208Z6}. For succinctness, a set {x, y, z} is written xyz. partition [C] = Fs i=1 Ci such that for each i \u2208[s], we have |Ri| = |Gi|/k and |Ci| = |Gi|/(k \u22121). We say that (X, G, A) is a frame generalized balanced tournament design (FrGBTD) if its blocks can be arranged in an R \u00d7 C array such that the following conditions hold: (i) the cell (r, c) is empty when (r, c) \u2208Ri \u00d7 Ci for i \u2208[s], (ii) for any row r \u2208Ri, each point in X \\ Gi appears either once or twice and the points in Gi do not appear, 14 \f(iii) for any column c \u2208Ci, each point in X \\ Gi appears exactly once. Denote this FrGBTD by FrGBTD(k, T), where T = \u27e8|Gi| : i \u2208[s]\u27e9. Example 4.5. An FrGBTD(3, 66) is given in Fig. 4. 5 Recursive Constructions In this section, we develop the necessary recursive constructions. We note that these are straightforward adaptions of methods in previous work [10,20,21,32]. 5.1 Recursive Constructions for GBTPs First, for block size three, we have the following tripling construction for GBTDs. This is an adaption of k-tupling construction for the case of GBTDs with index k \u22121 [21, Theorem 3.1] and the doubling construction for balanced tournament designs [26]. Proposition 5.1 (Tripling Construction). Suppose there exists a 3-\u2217colorable RBIBD(m, 3, 1) and a DRTD(3, m). Then there exists a 2-\u2217colorable GBTD1(3, m). Suppose further that the RBIBD(m, 3, 1) is 3-\u2217colorable with property \u03a0. Then the GBTD1(3, m) is a special GBTD1(3, m). Proof. Consider a 3-\u2217colorable RBIBD(m, 3, 1) (X, A) with colors from Z3 and let X\u2032 = {xi : x \u2208X and i \u2208Z3}. Make three copies of the 3-\u2217colorable RBIBD(m, 3, 1) as follows: for the jth copy, j \u2208{1, 2, 3}, each block {x, y, z} of color i in the 3-\u2217colorable RBIBD(m, 3, 1) is replaced by block {xi+j, yi+j, zi+j}, where arithmetic in the subscripts is performed modulo three. Stacking these three m 3 \u00d7 m\u22121 2 arrays together gives an m \u00d7 m\u22121 2 array A with the property that (i) each point in X\u2032 appears exactly once in each column, (ii) each point in X\u2032 appears at most once in each row. Now take a DRTD(3, m) (X\u2032, G, A), where G = {{xi : x \u2208X} : i \u2208Z3}, and adjoin it to A. This gives an m \u00d7 3m\u22121 2 array, which we claim is a GBTD1(3, m). Indeed it is easy to see that in this array, each point in X\u2032 appears exactly once in each column and either once or twice in each row. It remains to show that this array is a BIBD(3m, 3, 1). To see this, observe that any pair of points contained in a group of the DRTD(3, m) is contained in a block of one of the copies of the 3-\u2217colorable RBIBD(m, 3, 1). This GBTD1(3, m) is 2-\u2217colorable by giving the blocks from the DRTD(3, m) one color and the remaining blocks (from the three copies of the RBIBD(m, 3, 1)) another color. If, in addition, the RBIBD(m, 3, 1) is 3-\u2217colorable with property \u03a0, and that in row r of this RBIBD(m, 3, 1), the points x, y, z (not necessarily distinct) are witnesses for colors 0, 1, 2, respectively, then we assume that the DRTD(3, m) used has the block {x0, y1, z2} and that this block can be made to appear in row r, by permuting rows if necessary. The cell that contains {x0, y1, z2} is a special cell of the GBTD1(3, m). 15 \fCorollary 5.2. Let m > 3 and suppose an RBIBD(m, 3, 1) that is 3-\u2217colorable with property \u03a0 exists. Then there exists a special GBTD1(3, 3km), for all k \u22650. Proof. First note that m \u22613 mod 6 since this is a necessary condition for the existence of an RBIBD(m, 3, 1). Hence, there exists a DRTD(3, m) by Corollary 4.11. By Proposition 5.1, there exists a 2-\u2217colorable special GBTD1(3, m), which may be regarded as an RBIBD(3m, 3, 1) that is 3-\u2217colorable with property \u03a0. The corollary then follows by induction. The following propositions are simple generalizations of the standard \u201c\ufb01lling in the hole\u201d construction to construct GBTPs or GBTDs using IGBTPs and FrGBTDs. Proposition 5.3 (IGBTP Construction for GBTP). If an IGBTP\u03bb(K, v, m \u00d7 n; w, m\u2032 \u00d7 n\u2032) and a GBTP\u03bb(K, w, m\u2032 \u00d7 n\u2032) exists, then a GBTP\u03bb(K, v, m \u00d7 n) exists. Proof. Let (X, A) be an IGBTP\u03bb(K, v, m \u00d7 n; w, m\u2032 \u00d7 n\u2032). Fill in the empty subarray of this IGBTP with an a GBTP\u03bb(K, w, m\u2032 \u00d7 n\u2032), (X\u2032, A\u2032). The resulting array is a GBTP\u03bb(K, v, m \u00d7 n), (X, A \u222aA\u2032). FrGBTD is a useful tool to construct larger GBTPs from smaller ones. Proposition 5.4 (FrGBTD Construction for GBTP). Let k \u2208K. Suppose there exists an FrGBTD(k, T) (X, G, A), where G = {G1, G2, . . . , Gs}, and let ri = |Gi|/k and ci = |Gi|/(k \u22121), for i \u2208[s]. If there exists an IGBTP1(K, |Gi| + w, (ri + m) \u00d7 (ci + n); w, m \u00d7 n) for all i \u2208[s], then there exists an IGBTP1(K, Ps i=1 |Gi| + w, (Ps i=1 ri + m) \u00d7 (Ps i=1 ci + n); w, m \u00d7 n). Furthermore, if a GBTP1(K, w, m \u00d7 n) exists, then an GBTP1(K, Ps i=1 |Gi| + w, (Ps i=1 ri + m) \u00d7 (Ps i=1 ci + n)) exists. Proof. We use the notations as in the de\ufb01nition of FrGBTD in Section 4.4, and assume that the blocks of the FrGBTD(k, T) are arranged in an R \u00d7 C array, with rows and columns indexed by [R] and [C], respectively. Let P and Q be two sets satisfying |P| = m, |Q| = n, P \u2229[R] = \u2205, Q \u2229[C] = \u2205. For each i \u2208[s], consider an IGBTP1(K, |Gi| + w, (ri + m) \u00d7 (ci + n); w, m \u00d7 n) (Xi, Ai), where Xi = Gi \u222a{\u221e1, \u221e2, \u00b7 \u00b7 \u00b7 , \u221ew}, and whose rows and columns are indexed by P \u222aRi and Q \u222aCi, respectively. It can be veri\ufb01ed that (X\u2032, A\u2032), where X\u2032 = X \u222a{\u221e1, \u221e2, \u00b7 \u00b7 \u00b7 , \u221ew}, A\u2032 = A \u222a(\u222as i=1Ai), is an IRP(Ps i=1 |Gi| + w, K, 1). Arrange the blocks of (X\u2032, A\u2032) into an (R + m\u2032) \u00d7 (C + n\u2032) array A, whose rows and columns are indexed by P \u222a[R] and Q \u222a[C], respectively, such that each block in A that appears in cell (i, j) of either the FrGBTD or the IGBTP, is placed in cell (i, j) of A. The de\ufb01nition conditions of an FrGBTD ensures that no cells are occupied by two blocks. It is also easily checked that every point in X\u2032 appears exactly once in each column and either once or twice in each row. In addition, the m \u00d7 n subarray indexed by P \u00d7 Q is empty. This gives an IGBTP1(K, Ps i=1 |Gi| + w, (Ps i=1 ri + m) \u00d7 (Ps i=1 ci + n); w, m \u00d7 n). The last statement follows from Proposition 5.3. Since a GBTD is an instance of GBTP, we have the following recursive construction for GBTDs. 16 \fCorollary 5.5 (FrGBTD Construction for GBTD). Suppose an FrGBTD(k, T) exists with groups {G1, G2, . . . , Gs}. Let gi = |Gi|/k, for i \u2208[s]. If there exists a special GBTD1(k, gi + 1) for all i \u2208[s], then there exists a special GBTD1(k, Ps i=1 gi + 1). When the groups are of the same size, we have the following corollary. Corollary 5.6. If there exists an FrGBTD(3, (3g)t) and a special GBTD1(3, g + 1), then there exists a special GBTD1(3, gt + 1). For Proposition 5.3 and Corollary 5.5 to be useful, we require large classes of FrGBTDs. We give three recursive constructions for FrGBTDs next. 5.2 Recursive Constructions for FrGBTDs We adapt the standard direct product construction. Proposition 5.7 (In\ufb02ation). Suppose an FrGBTD(k, T) and a DRTD(k, n) exists. Then there exists an FrGBTD(k, nT). Proof. Let (X, G, A) be an FrGBTD(k, T) arranged in an R \u00d7 C array A, with rows and columns indexed by [R] and [C], respectively. De\ufb01ne X\u2032 = X \u00d7 [n], G\u2032 = {G \u00d7 [n] : G \u2208G}, and for each block A \u2208A, let XA = A \u00d7 [n], GA = {{x} \u00d7 [n] : x \u2208A}. and let (XA, GA, BA) be a DRTD(k, n) whose blocks are arranged in an n \u00d7 n array with rows and columns both indexed by [n]. Let A\u2032 = \u222aA\u2208ABA and the blocks in A\u2032 can be arranged, as follows, in an Rn \u00d7 Cn array, whose rows and columns are indexed by [R] \u00d7 n and [C] \u00d7 n, respectively: a block B \u2208BA is placed in cell ((i, a), (j, b)) if A appears in cell (i, j) of the FrGBTD(k, T) and B appears in cell (a, b) of the DRTD(k, n). Hence, (X\u2032, G\u2032, A\u2032) gives an FrGBTD(k, nT). Wilson\u2019s Fundamental Construction for GDDs [31] can also be modi\ufb01ed to construct FrGBTDs. Fig. 5 describes this construction. Proposition 5.8 (Fundamental Construction). Suppose there exists a (master) GDD (X, G, A) of type T and let w : X \u2192Z\u22650 be a weight function. If for each A \u2208A, there exists an (ingredient) FrGBTD(k, \u27e8w(a) : a \u2208A\u27e9), then there exists an FrGBTD(k, \u27e8P x\u2208G w(x) : G \u2208G\u27e9). Proof. The Fundamental Construction in Fig. 5 constructs the desired FrGBTD from the master GDD and ingredient FrGBTDs. Proposition 5.8 admits the following specialization. Proposition 5.9 (PBD Closure). Suppose there exists a PBD(v, K) (X, A), and for each block A \u2208A, there exists an FrGBTD(3, g|A|). Then there exists an FrGBTD(3, gv). 17 \fInput: (master) GDD D = (X, G, A); weight function w \u2192Z\u22650; (ingredient) FrGBTD(k, TA) DA = (XA, GA, BA) for each A \u2208A, where TA = \u27e8w(x) : x \u2208A\u27e9, XA = \u222ax\u2208A({x} \u00d7 [w(a)]), GA = {{x} \u00d7 [w(x)] : x \u2208A}, and the blocks in BA are arranged in a 1 k P x\u2208A w(x) \u00d7 1 k\u22121 P x\u2208A w(x) array, whose rows and columns are indexed by \u222ax\u2208A({x} \u00d7 [w(x)/k]) and \u222ax\u2208A({x} \u00d7 [w(x)/(k \u22121)]), respectively. Output: FrGBTD(k, \u27e8P x\u2208G w(x) : G \u2208G\u27e9) D\u2217= (X\u2217, G\u2217, A\u2217), where X\u2217= \u222ax\u2208X({x} \u00d7 [w(x)]), G\u2217= {\u222ax\u2208G({x} \u00d7 [w(x)]) : G \u2208G}, A\u2217= \u222aA\u2208ABA, and the blocks in A\u2217are arranged in a 1 k P x\u2208X w(x) \u00d7 1 k\u22121 P x\u2208X w(x) array, whose rows and columns are indexed by \u222ax\u2208X({x} \u00d7 [w(x)/k]) and \u222ax\u2208X({x} \u00d7 [w(x)/(k \u22121)]), respectively, by placing a block B \u2208BA in cell (i, j) of D\u2217if it appears in cell (i, j) of DA. Note: By convention, for x \u2208X, {x} \u00d7 [w(x)] = \u2205if w(x) = 0. Figure 5: Fundamental Construction for FrGBTDs Proof. Consider the PBD as a (master) GDD of type 1v and weight function w(x) = g for all x \u2208X. Now apply the fundamental construction. Proposition 5.10 (FrGBTD from Truncated TD). Let s > 0. Suppose there exists a TD(u+s, m), and g1, g2, . . . , gs are nonnegative integers at most m. If there exists an FrGBTD(k, gt) for each t \u2208{u, u+1, . . . , u+s}, then there exists an FrGBTD(k, T), where T = (g\u00b7m)u(g\u00b7g1)(g\u00b7g2) \u00b7 \u00b7 \u00b7 (g\u00b7gs). Proof. For each i \u2208[s], delete m \u2212gi points from the ith group of the TD(u + s, m). This results in a {u, u + 1, . . . , u + s}-GDD of type mug1g2 \u00b7 \u00b7 \u00b7 gs. Use this as the master GDD and apply the fundamental construction with weight function w that assigns weight g to all points. 6 Direct Constructions This section constructs some small GBTDs and FrGBTDs that are required to seed the recursive constructions given in Section 5. Our main tools are starters and the method of di\ufb00erences . Starter-adder constructions are ubiquitous in the constructions for GBTDs with index k \u22121, associated frames and other types of similar designs (see for example, [8, 10, 20, 21, 32]). Unlike previous work and due to the lack of symmetry in our arrays, we \ufb01x the positions of the starters in our arrays and \u2018develop\u2019 the blocks in a variety of \u2018directions\u2019 (see Figures 6, 7 and 8). This removes the use of adders and surprisingly a careful analysis of the starter conditions allows a prime power construction that is given in Proposition 6.3. First, we recall certain concepts with regards to the method of di\ufb00erences. Let \u0393 be an additive abelian group and let n be a positive integer. For a set system (\u0393, S), the di\ufb00erence list of S is the multiset \u2206S = \u27e8x \u2212y : x, y \u2208A, x \u0338= y, and A \u2208S\u27e9. 18 \fFor a set-system (\u0393 \u00d7 [n], S) and i, j \u2208[n], the multiset \u2206ijS = \u27e8x \u2212y : xi, yj \u2208A, xi \u0338= yj, and A \u2208S\u27e9 is called a list of pure di\ufb00erences when i = j, and called a list of mixed di\ufb00erences when i \u0338= j. 6.1 Direct Constructions for GBTDs De\ufb01nition 6.1 (Starter for GBTD). Let m be an odd positive integer, \u0393 be an additive abelian group of size m. Let T be an index set of size (m \u22121)/2. Let (\u0393 \u00d7 [3], S) be a {3}-uniform set system of size (3m \u22121)/2, where S = {A\u03b1 : \u03b1 \u2208\u0393} \u222a{Bt : t \u2208T}. S is called a (\u0393 \u00d7 [3])-GBTD-starter if the following conditions hold: (i) \u2206iiS = \u0393 \\ {0}, for i \u2208[3], (ii) \u2206ijS = \u0393, for i, j \u2208[3], i \u0338= j, (iii) \u222a\u03b1\u2208\u0393A\u03b1 = \u0393 \u00d7 [3], (iv) {j : \u03b1j \u2208Bt for some \u03b1 \u2208\u0393} = [3], for t \u2208T, (v) each element in \u0393 \u00d7 [3] appears either once or twice in the multiset R =  [ \u03b1\u2208\u0393 A\u03b1 \u2212\u03b1 ! \u222a  [ t\u2208T Bt ! . Furthermore, S is said to be special if (vi) each element in A0 appears exactly once in R. Also, S is said to be 3-\u2217colorable with property \u03a0 if each of the blocks in {A\u03b1 \u2212\u03b1 : \u03b1 \u2208\u0393} and {Bt : t \u2208T}, can be colored with one of three colors so that (vii) blocks of the same color are pairwise disjoint, (viii) for each color c, there exists a point (a witness for c) that is not contained in any block assigned color c. Proposition 6.2. If a (\u0393 \u00d7 [k])-GBTD-starter exists, then a GBTD1(k, m) exists. Similarly, if there exists a special (\u0393 \u00d7 [3])-GBTD-starter, then there exists a special GBTD1(3, m); and if there exists a 3-\u2217colorable (\u0393\u00d7[3])-GBTD-starter with property \u03a0, then there exists a 3-\u2217colorable GBTD1(3, m) with property \u03a0. 19 \fA B where A is the array A0 A\u2212\u03b11 + \u03b11 A\u2212\u03b12 + \u03b12 \u00b7 \u00b7 \u00b7 A\u2212\u03b1m\u22121 + \u03b1m\u22121 A\u03b11 A0 + \u03b11 A\u03b11\u2212\u03b12 + \u03b12 \u00b7 \u00b7 \u00b7 A\u03b11\u2212\u03b1m\u22121 + \u03b1m\u22121 . . . . . . . . . ... . . . A\u03b1m\u22121 A\u03b1m\u22121\u2212\u03b11 + \u03b11 A\u03b1m\u22121\u2212\u03b12 + \u03b12 \u00b7 \u00b7 \u00b7 A0 + \u03b1m\u22121 and B is the array B1 B2 \u00b7 \u00b7 \u00b7 B(m\u22121)/(k\u22121) B1 + \u03b11 B2 + \u03b11 \u00b7 \u00b7 \u00b7 B(m\u22121)/(k\u22121) + \u03b11 . . . . . . ... . . . B1 + \u03b1m\u22121 B2 + \u03b1m\u22121 \u00b7 \u00b7 \u00b7 B(m\u22121)/(k\u22121) + \u03b1m\u22121 . Figure 6: A GBTD1(k, m) from (\u0393 \u00d7 [k])-GBTD-starter S = {A\u03b1 : \u03b1 \u2208\u0393} \u222a{Bt : t \u2208T}, where \u0393 = {0, \u03b11, . . . , \u03b1m\u22121} and T = [(m \u22121)/(k \u22121)]. Proof. Let X = \u0393\u00d7[k], and suppose S = {A\u03b1 : \u03b1 \u2208\u0393}\u222a{Bt : t \u2208T} is an (\u0393\u00d7[k])-GBTD-starter. Let A = [ A\u2208S {A + \u03b1 : \u03b1 \u2208\u0393}. Then (X, A) is a BIBD(km, k, 1), whose blocks can be arranged in an m \u00d7 (km\u22121) k\u22121 array, whose rows and columns are indexed by \u0393 and \u0393 \u222aT, respectively, as follows: \u2022 for \u03b1, \u03b2 \u2208\u0393, the block A\u03b1 + \u03b2 is placed in cell (\u03b1 + \u03b2, \u03b2), and \u2022 for t \u2208T and \u03b1 \u2208\u0393, the block Bt + \u03b1 is placed in cell (\u03b1, t). Fig. 6 depicts the placement of blocks in the array. For \u03b2 \u2208\u0393, the set of blocks occupying column \u03b2 is {A\u03b1 + \u03b2 : \u03b1 \u2208\u0393}, which form a resolution class by condition (iii) of De\ufb01nition 6.1. Similarly, for t \u2208T, the set of blocks occupying column t is {Bt + \u03b1 : \u03b1 \u2208\u0393}, which form a resolution class by condition (iv) in De\ufb01nition 6.1. The set of blocks occupying row 0 is given by R, and by condition (v) of De\ufb01nition 6.1, each point in X appears either once or twice in row 0. Since the blocks occupying row \u03b1 (\u03b1 \u2208\u0393) are exactly the translates of the blocks in R by \u03b1, every point in X also appears either once or twice in row \u03b1. Suppose S = {A\u03b1 : \u03b1 \u2208\u0393} \u222a{Bt : t \u2208T} is a special (\u0393 \u00d7 [3])-GBTD-starter. Then condition (vi) of 6.1 ensures that the cell (0, 0) is special. On the other hand, if S be a 3-\u2217colorable (\u0393 \u00d7 [3])-GBTD-starter and let ci be the color assigned to ( Ai \u2212i, if i \u2208\u0393, Bi, otherwise. 20 \fFor \u03b1, \u03b2 \u2208\u0393 and t \u2208T, assign the block A\u03b1 + \u03b2 color c\u03b1 and the block Bt + \u03b2 color ct. Then conditions (vii) and (viii) of De\ufb01nition 6.1 ensure that the GBTD1(3, m) is 3-\u2217colorable with property \u03a0. Proposition 6.3. Let q \u22611 mod 6. Then there exists a special (Fq \u00d7 [3])-GBTD starter that is 3-\u2217colorable with property \u03a0. Proof. Let s = (q \u22121)/6 and \u03c9 be a primitive element of Fq. Consider \u03b3 \u2208Fq that satis\ufb01es the following conditions (note that \u03c92s has order three): (A) \u03b3 / \u2208{0, \u22121, \u2212\u03c92s, \u2212\u03c94s}; (B) \u03b3 / \u2208 \u001a\u03c92is \u2212\u03c9t+2js \u03c9t \u22121 : i \u0338= j \u2208[3], t \u2208[s \u22121] \u001b . The existence of \u03b3 is guaranteed since the cardinality of the union of sets in (A) and (B) is at most 4 + 6(s \u22121) < 6s + 1 = q. De\ufb01ne \u039b to be \b \u2212\u03b3\u03c9t\u22121+2(j\u22121)s : t \u2208[s], j \u2208[3] \t and construct the following q +3s = (3q \u22121)/2 blocks. For \u03b1 \u2208Fq, let A\u03b1 = \uf8f1 \uf8f2 \uf8f3 \b\u0000\u03c9t\u22121+2(j\u22121)s\u0001 i : j \u2208[3] \t , if \u03b1 = \u2212\u03b3\u03c9t\u22121+2(i\u22121)s where t \u2208[s], i \u2208[3], n\u0010 \u2212\u03b1 \u03b3 \u03c92(i\u22121)s\u0011 i : i \u2208[3]) o , otherwise. For (t, j) \u2208[s] \u00d7 [3], let B(t,j) = n\u0010 \u03c9t\u22121+2(j\u22121)s \u0010 \u03c92(i\u22121)s + \u03b3 \u0011\u0011 i : i \u2208[3] o Let S = {A\u03b1 : \u03b1 \u2208Fq} \u222a{B(t,j) : (t, j) \u2208[s] \u00d7 [3]} and we claim that S is the desired starter. De\ufb01ne D = {{\u03c9t\u22121+2(j\u22121)s : j \u2208[3]} : t \u2208[s]}, and Wilson [30] showed that the blocks in D are mutually disjoint and \u2206D = Fq \\ {0}. Hence, for condition (i) of De\ufb01nition 6.1, we check for i \u2208[3], \u2206iiS = \u2206ii{A\u03b1 : \u03b1 = \u2212\u03b3\u03c9t\u22121+2(i\u22121)s, t \u2208[s], i \u2208[3]} = \u2206D = Fq \\ {0}. For condition (ii), we verify for i \u0338= i\u2032 \u2208[3], \u2206ii\u2032S = [ \u03b1/ \u2208\u039b \u0012 \u2212\u03b1 \u03b3 \u0010 \u03c92(i\u22121)s \u2212\u03c92(i\u2032\u22121)s\u0011\u0013 \u222a [ (t,j)\u2208[s]\u00d7[3] \u03c9t\u22121+2(j\u22121)s \u0010 \u03c92(i\u22121)s \u2212\u03c92(i\u2032\u22121)s\u0011 = \u0010 \u03c92(i\u22121)s \u2212\u03c92(i\u2032\u22121)s\u0011 \uf8eb \uf8ed[ \u03b1/ \u2208\u039b \u2212\u03b1 \u03b3 \u222a [ (t,j)\u2208[s]\u00d7[3] \u03c9t\u22121+2(j\u22121)s \uf8f6 \uf8f8 = \u0010 \u03c92(i\u22121)s \u2212\u03c92(i\u2032\u22121)s\u0011 Fq = Fq. 21 \fFor condition (iii) of De\ufb01nition 6.1, since the number of points in S \u03b1\u2208Fq A\u03b1 is kq, it su\ufb03ces to check that each point \u03b2i \u2208Fq \u00d7[k] belongs to some block A\u03b1. Indeed, if \u03b2/\u03c92(i\u22121)s = \u03c9(t\u22121)+2(j\u22121)s for some (t, j) \u2208[s] \u00d7 [3], then let \u03b1 = \u2212\u03b3\u03c9t\u22121+2(i\u22121)s and so, \u03b2i = \u0000\u03c9t\u22121+2(i+j\u22122)s\u0001 i belongs to A\u03b1. Otherwise, \u2212\u03b3\u03b2/\u03c92(i\u22121)s / \u2208\u039b. Let \u03b1 = \u2212\u03b3\u03b2/\u03c92(i\u22121)s and \u03b2i \u2208A\u03b1 as desired. Condition (iv) of De\ufb01nition 6.1 is clearly true from the de\ufb01nition of B(t,j). We establish condition (v) of De\ufb01nition 6.1 through the following claims: Claim 6.1. The blocks in S \u03b1/ \u2208\u039b(A\u03b1 \u2212\u03b1) \u222aS (t,j)\u2208[s]\u00d7[3] B(t,j) form a resolution class. As above, it su\ufb03ces to check that each point \u03b2i \u2208Fq \u00d7 [3] belongs to some block in S \u03b1/ \u2208\u039b(A\u03b1 \u2212 \u03b1) \u222aS (t,j)\u2208[s]\u00d7[k] B(t,j) as the total number of points is kq. Indeed, if \u03b2/(\u03c92(i\u22121)s + \u03b3) = \u03c9t\u22121+2(j\u22121)s for some (t, j) \u2208[s] \u00d7 [k], then \u03b2i \u2208B(t,j). Otherwise, \u2212\u03b3\u03b2/(\u03c92(i\u22121)s +\u03b3) / \u2208\u039b. Let \u03b1 = \u2212\u03b3\u03b2/(\u03c92(i\u22121)s +\u03b3) (note that \u03b1 is well-de\ufb01ned by Condition (A)) and \u03b2i \u2208A\u03b1 \u2212\u03b1. Claim 6.2. Each point in Fq \u00d7 [k] appears at most once in S \u03b1\u2208\u039b (A\u03b1 \u2212\u03b1). Note that the blocks are of the form n\u0010 \u03c9t\u22121+2(j\u22121)s + \u03b3\u03c9t\u22121+2(i\u22121)s\u0011 i : j \u2208[3] o for (t, i) \u2208[s] \u00d7 [3]. Suppose otherwise that a point appears twice. That is, there exist j, j\u2032 \u2208[3], (t, i), (t\u2032, i) \u2208[s] \u00d7 [3] with t > t\u2032 such that \u03c9t\u22121+2(j\u22121)s + \u03b3\u03c9t\u22121+2(i\u22121)s = \u03c9t\u2032\u22121+2(j\u2032\u22121)s + \u03b3\u03c9t\u2032\u22121+2(i\u22121)s. Hence, \u03b3 = \u03c92(j\u2032\u2212i)s \u2212\u03c92(j\u2212i)s+(t\u2212t\u2032) \u03c9t\u2212t\u2032 \u22121 . Since t \u0338= t\u2032, we have t \u2212t\u2032 \u2208[s \u22121]. If j \u0338= j\u2032, this contradicts Condition (B). Otherwise j = j\u2032 implies \u03b3 = \u2212\u03c92(j\u2212i)s contradicting (A). Next, observe that A0 = {(0, i) : i \u2208[3]}. By Claim 6.1, to establish condition (vi) of De\ufb01nition 6.1, it su\ufb03ces to show that 0i / \u2208A\u03b1 \u2212\u03b1 for \u03b1 \u2208\u039b and i \u2208[3]. Suppose otherwise. Then there exists (t, j) \u2208[s] \u00d7 [3] and i \u2208[3] such that (\u03c9(j\u22121)s + \u03b3)\u03c9t+(i\u22121)s = 0, contradicting (A). Finally, we exhibit that S is 3-\u2217colorable with property \u03a0 by assigning the block A0 color \u2663, the blocks A\u03b1 \u2212\u03b1 for \u03b1 / \u2208\u039b and Bt for t \u2208T color \u2665and the blocks A\u03b1 \u2212\u03b1 for \u03b1 \u2208\u039b color \u2666. Then this assignment satis\ufb01es condition (vii) of De\ufb01nition 6.1. In addition, 01 is a witness for both \u2665and \u2666and \u03b11 is a witness for \u2663for some \u03b1 \u0338= 0, satisfying condition (viii) of De\ufb01nition 6.1. Corollary 6.4. Let q \u22611 mod 6. Then a 3-\u2217colorable GBTD1(3, m) with property \u03a0 exists. Proof. This follows from Proposition 6.2 and Proposition 6.3. Corollary 6.5. A special GBTD1(3, m) exists for m \u2208{1, 17, 29, 35, 47, 53, 55}, a 3-\u2217colorable special GBTD1(3, m) with property \u03a0 for m \u2208{9, 11, 23} and a 3-\u2217colorable RBIBD(15, 3, 1) with property \u03a0. 22 \fW B B + 01 A C C + 01 where W is a (w \u22121)/2 \u00d7 (w \u22124) empty array, A is an m \u00d7 (w \u22124) array, {00, 01} A1 A1 + 01 A2 A2 + 01 \u00b7 \u00b7 \u00b7 A(w\u22125)/2 A(w\u22125)/2 + 01 {10, 11} A1 + 10 A1 + 11 A2 + 10 A2 + 11 \u00b7 \u00b7 \u00b7 A(w\u22125)/2 + 10 A(w\u22125)/2 + 11 . . . . . . . . . . . . . . . ... . . . . . . {(m \u22121)0, (m \u22121)1} A1 \u221210 A1 \u221211 A2 \u221210 A2 \u221211 \u00b7 \u00b7 \u00b7 A(w\u22125)/2 \u221210 A(w\u22125)/2 \u221211 , B and C are the following (w \u22121)/2 \u00d7 m and m \u00d7 m arrays, B1 B1 + 10 \u00b7 \u00b7 \u00b7 B1 \u221210 B2 B1 + 10 \u00b7 \u00b7 \u00b7 B1 \u221210 . . . . . . ... . . . B(w\u22121)/2 B(w\u22121)/2 + 10 \u00b7 \u00b7 \u00b7 B(w\u22121)/2 \u221210 , C0 Cm\u22121 + 10 \u00b7 \u00b7 \u00b7 C1 \u221210 C1 C0 + 10 \u00b7 \u00b7 \u00b7 C2 \u221210 . . . . . . ... . . . Cm\u22121 Cm\u22122 + 10 \u00b7 \u00b7 \u00b7 C0 \u221210 . Figure 7: An IGBTP1({2, 3\u2217}, 2m + w, (m + (w \u22121)/2) \u00d7 (2m + w \u22124); w, (w \u22121)/2 \u00d7 (w \u22124)) from a ((Zm \u00d7 Z2) \u222aWw)-GBTP-starter. Proof. A special GBTD1(3, 1) exists trivially. In addition, a 3-\u2217colorable special GBTD1(3, 9) with property \u03a0 is given by Example 4.4, and a 3-\u2217colorable RBIBD(15, 3, 1) with property \u03a0 is given by Example 4.1. For m \u2208{11, 17, 23, 29, 35, 47, 53, 55}, apply Proposition 6.2 with special (Zm \u00d7 [3])-GBTDstarters and 3-\u2217colorable special (Zm \u00d7 [3])-GBTD-starters with property \u03a0 given in [5]. 6.2 Direct Constructions for an IGBTP1({2, 3\u2217}, 2m + w, (m + (w \u22121)/2) \u00d7 (2m + w \u22124); w, (w \u22121)/2 \u00d7 (w \u22124)) As with GBTDs, we use a set of starters to construct GBTPs. To construct this starters, we need the notion of in\ufb01nite elements, or intransitive starters. Given an abelian group \u0393, we augment the point set with in\ufb01nite elements, denoted by \u221ei where i belongs to some index set I. The in\ufb01nite elements are \ufb01xed under addition by elements in \u0393. That is, \u221ei + \u03b3 = \u221ei for \u03b3 \u2208\u0393. Let w be a positive integer and Ww = {\u221ei : i \u2208[w]}. So, given a block A \u2282\u0393 \u222aWw and \u03b3 \u2208\u0393, A + \u03b3 = {a + \u03b3 : a \u2208A \\ Ww} \u222a(A \u2229Ww). We also extend the de\ufb01nition of di\ufb00erence lists. For a set system (\u0393\u222aWw, S), then the di\ufb00erence list of S is given by the multiset \u2206S = \u27e8x \u2212y : x, y \u2208A \\ Ww, x \u0338= y, A \u2208S\u27e9. De\ufb01nition 6.6. Let m be an odd integer with m \u226511 Let (Zm \u00d7 Z2 \u222aWw, S) be a {2, 3}-uniform set system of size w \u22123 + m, where S = {Ai : i \u2208[(w \u22125)/2]} \u222a{Bi : i \u2208[(w \u22121)/2]} \u222a{Ci : i \u2208Zm}. 23 \fW B B + 01 B + 02 B + 03 A C D + 01 C + 02 D + 03 D C + 01 D + 02 C + 03 where W is a 4 \u00d7 5 empty array, A is a 2m \u00d7 5 array, {00, 01} {x0, x2} {y0, y3} A A + 02 {10, 11} {(x + 1)0, x2} {(y + 1)0, (y + 1)3} A + 10 A + 12 . . . . . . . . . . . . . . . {(m \u22121)0, (m \u22121)1} {(x \u22121)0, x2} {(y \u22121)0, (y \u22121)3} A + (m \u22121)0 A + (m \u22121)2 {02, 03} {x1, x3} {y1, y2} A + 01 A + 03 {12, 13} {(x + 1)1, x3} {(y + 1)1, (y + 1)2} A + 11 A + 13 . . . . . . . . . . . . . . . {(m \u22121)2, (m \u22121)3} {(x \u22121)1, x3} {(y \u22121)1, (y \u22121)2} A + (m \u22121)1 A + (m \u22121)3 , B, C and D are the following 4 \u00d7 m, m \u00d7 m and m \u00d7 m arrays respectively, B1 B1 + 10 \u00b7 \u00b7 \u00b7 B1 \u221210 B2 B2 + 10 \u00b7 \u00b7 \u00b7 B2 \u221210 B3 B3 + 10 \u00b7 \u00b7 \u00b7 B3 \u221210 B4 B4 + 10 \u00b7 \u00b7 \u00b7 B4 \u221210 , C0 Cm\u22121 + 10 \u00b7 \u00b7 \u00b7 C1 \u221210 C1 C0 + 10 \u00b7 \u00b7 \u00b7 C2 \u221210 . . . . . . ... . . . Cm\u22121 Cm\u22122 + 10 \u00b7 \u00b7 \u00b7 C0 \u221210 , D0 Dm\u22121 + 10 \u00b7 \u00b7 \u00b7 D1 \u221210 D1 D0 + 10 \u00b7 \u00b7 \u00b7 D2 \u221210 . . . . . . ... . . . Dm\u22121 Dm\u22122 + 10 \u00b7 \u00b7 \u00b7 D0 \u221210 . Figure 8: An IGBTP1({2, 3\u2217}, 4m + 9, (2m + 4) \u00d7 (4m + 5); 9, 4 \u00d7 5) from a ((Zm \u00d7 Z4) \u222aW9)GBTP-starter. 24 \fsatisfying |Ai| = 2 for i \u2208[(w \u22125)/2], |Bi| = 2 for i \u2208[(w \u22121)/2], |C0| = 3, and |Ci| = 2 for i \u2208Zm \\ {0}. S is called a ((Zm \u00d7 Z2) \u222aWw)-IGBTP-starter if the following conditions hold: (i) \u2206S = Zm \u00d7 Z2 \\ {00, 01}, (ii) {j : aj \u2208Ai} = Z2 for i \u2208[(w \u22125)/2], (iii) {Bi : i \u2208[(w \u22121)/2]} \u222a{Cj : j \u2208Zm} = (Zm \u00d7 Z2) \u222aWw, (iv) |Ci \u2229Ww| \u22641 for i \u2208Zm, (v) each element in (Zm \u00d7 Z2) \u222aWw appears either once or twice in the multiset R = {00, 01} \u222a \uf8eb \uf8ec \uf8ec \uf8ed [ i\u2208[(w\u22125)/2] j\u2208Z2 Ai + 0j \uf8f6 \uf8f7 \uf8f7 \uf8f8\u222a \uf8eb \uf8ed [ ij\u2208Zm\u00d7Z2 Ci \u2212ij \uf8f6 \uf8f8. Proposition 6.7. Suppose there exists a ((Zm \u00d7 Z2) \u222aWw)-IGBTP-starter.Then there exists an IGBTP1({2, 3\u2217}, 2m + w, (m + (w \u22121)/2) \u00d7 (2m + w \u22124); w, (w \u22121)/2 \u00d7 (w \u22124)). Proof. Let X = Zm \u00d7 Z2 \u222aWw, A = {S + j : S \u2208S and j \u2208Zm \u00d7 Z2} \u222a{{i0, i1} : i \u2208Zm}. Then (X, Ww, A) is an IRP(2m+w, K, 1; w), whose blocks can be arranged in an (m+(w\u22121)/2)\u00d7 (2m + w \u22124) array as in Figure 8. We index the rows by [(w \u22121)/2] \u222aZm and the columns by [w \u22124] \u222a(Zm \u00d7 Z2). First, check that the cell (r, c) is empty for (r, c) \u2208[(w \u22121)/2] \u00d7 [w \u22124]. For j \u2208[w \u22124], the set of blocks occupying column j is Zm \u00d7 Z2 by condition (ii) of De\ufb01nition 6.6. For j \u2208Zm \u00d7 Z2, \ufb01rst observe that the set of the blocks occupying the column 00 by condition (iii) of De\ufb01nition 6.6 is (Zm \u00d7 Z2) \u222aWw. Since the blocks of column j are translates (by j) of the blocks in column 00, the union of the blocks in column j is also (Zm \u00d7 Z2) \u222aWw. For i \u2208[(w \u22121)/2], each element in Zm \u00d7 Z2 appears exactly twice in row i by construction. For i \u2208Zm, let Ri denote the multiset containing all the points appearing in the blocks of row i. Then R0 = R and Ri = R0 + i0, for all i \u2208Zm. Hence, it su\ufb03ces each element in X appears either once or twice in R, which follows immediately from conditions (v) in De\ufb01nition 6.6. De\ufb01nition 6.8. Let m be an odd integer with m \u226511. Let ((Zm \u00d7 Z4) \u222aW9, S) be a {1, 2, 3}uniform set system of size 7 + 2m, where S = {x0} \u222a{y0} \u222aA \u222a{Bi : i \u2208[4]} \u222a{Ci : i \u2208Zm} \u222a{Di : i \u2208Zm}. satisfying |A| = 2, |Bi| = 2 for i \u2208[4], |C0| = 3, |Ci| = 2 for i \u2208Zm \\ {0} and |Di| = 2 for i \u2208Zm. S is called a ((Zm \u00d7 Z4) \u222aW9)-IGBTP-starter if the following conditions hold: (i) \u2206S = (Zm \u00d7 Z4) \\ {00, 01, 02, 03}, 25 \f(ii) {j : aj \u2208A} = {0, 2}, (iii) {Bi : i \u2208[(w \u22121)/2]} \u222a{Ci : i \u2208Zm} \u222a{Di : i \u2208Zm} = (Zm \u00d7 Z4) \u222aW9, (iv) |Ci \u2229W9| \u22641 and |Di \u2229W9| \u22641 for i \u2208Zm, (v) each element in (Zm \u00d7 Z4) \u222aW9 appears either once or twice in the multisets R\u25e6= {00, 01, x0, x2, y0, y3} \u222aA \u222aA + 02 \u222a \uf8eb \uf8ed [ i\u2208Zm,j\u2208{0,2} Ci \u2212ij \uf8f6 \uf8f8\u222a \uf8eb \uf8ed [ i\u2208Zm,j\u2208{1,3} Di \u2212ij \uf8f6 \uf8f8, R\u2022 = {02, 03, x1, x3, y1, y2} \u222aA + 01 \u222aA + 03 \u222a \uf8eb \uf8ed [ i\u2208Zm,j\u2208{1,3} Ci \u2212ij \uf8f6 \uf8f8\u222a \uf8eb \uf8ed [ i\u2208Zm,j\u2208{0,2} Di \u2212ij \uf8f6 \uf8f8. Proposition 6.9. Suppose there exists a (Zm \u00d7 Z4 \u222aW9)-IGBTP-starter. Then there exists an IGBTP1({2, 3\u2217}, 4m + 9, (2m + 4) \u00d7 (4m + 5); 9, 4 \u00d7 5). Proof. Let X = (Zm \u00d7 Z4) \u222aW9, A = {S + j : S \u2208S, |S| \u0338= 1, j \u2208Zm \u00d7 Z2} \u222a{{i0, i1} : i \u2208Zm} \u222a{{i2, i3} : i \u2208Zm} \u222a{{(x + i)0, (x + i)2} : i \u2208Zm} \u222a{{(x + i)1, (x + i)3} : i \u2208Zm} \u222a{{(y + i)0, (y + i)3} : i \u2208Zm} \u222a{{(y + i)1, (y + i)2} : i \u2208Zm}. Then (X, W9, A) is an IRP(4m + 9, K, 1; 9), whose blocks can be arranged in a (2m + 4) \u00d7 (4m + 5) array as in Figure 7. We index the rows by [4] \u222a(Zm \u00d7 {\u25e6, \u2022}) and the columns by [5] \u222a(Zm \u00d7 Z4). First, check that the cell (r, c) is empty for (r, c) \u2208[4] \u00d7 [5]. For j \u2208[5], the set of blocks occupying column j is Zm \u00d7 Z4 by condition (ii) of De\ufb01nition 6.8. For j \u2208Zm \u00d7 Z4, \ufb01rst observe that the set of the blocks occupying the column 00 by condition (iii) of De\ufb01nition 6.8 is (Zm \u00d7 Z4) \u222aW9. Since the blocks of column j are translates (by j) of the blocks in column 00, the union of the blocks in column j is also (Zm \u00d7 Z4) \u222aW9. For i \u2208[4], each element in Zm \u00d7 Z4 appears exactly twice in row i by construction. For (i, \u2217) \u2208Zm \u00d7 {\u25e6, \u2022}, let R(i,\u2217) denote the multiset containing all the points appearing in the blocks of row (i, \u2217). Then R(0,\u2217) = R\u2217and R(i,\u2217) = R(0,\u2217) + i0, for all i \u2208Zm. Hence, it su\ufb03ces each element in X appears either once or twice in R\u2217, which follows immediately from conditions (v) in De\ufb01nition 6.8. Corollary 6.10. An IGBTP1({2, 3\u2217}, 2m + 9, (m + 4) \u00d7 (2m + 5); 9, 4 \u00d7 5) exists for m \u2208{s : 10 \u2264 s \u226445}\u222a{47, 49, 53, 57, 77} \\ {16, 20, 24, 28, 36, 40, 44}, and an IGBTP1({2, 3\u2217}, 2m + 11, (m + 5) \u00d7 (2m + 7); 11, 5 \u00d7 7) exists for m \u2208{15, 19, 23, 27, 31, 35, 45, 49}. Proof. The required ((Zm\u00d7Z2)\u222aW9)-IGBTP-starter for m \u2208{s : 11 \u2264s \u226449, s odd}\u222a{53, 57, 77} and ((Zm \u00d7 Z4) \u222aW9)-IGBTP starter for m \u2208{s : 5 \u2264s \u226421, s odd} is given in [5] and we apply Proposition 6.7 and Proposition 6.9 to obtain the corresponding IGBTP. Similarly, to construct an IGBTP1({2, 3\u2217}, 2m + 11, (m + 5) \u00d7 (2m + 7); 11, 5 \u00d7 7) for m \u2208 {15, 19, 23, 27, 31, 35, 45, 49}, we apply Proposition 6.7 to (Zm \u00d7 Z2 \u222aW11)-IGBTP starters listed in [5]. 26 \fIt remains to construct an IGBTP1({2, 3\u2217}, 33, 16 \u00d7 29; 9, 4 \u00d7 5). Consider ((Z3 \u00d7 Z8) \u222aW9, S), a {2, 3}-uniform set system of size 36, where S comprise the blocks below: A1 = {10, 12} A2 = {11, 15} A3 = {00, 04} A4 = {13, 16} A5 = {03, 05} A6 = {11, 13} A7 = {14, 17} A8 = {01, 06} A9 = {00, 05} A10 = {02, 04} A11 = {14, 16} A12 = {10, 13} A13 = {02, 05} A14 = {12, 17} A15 = {01, 07} A16 = {15, 17} A17 = {02, 06} A18 = {03, 07} A19 = {11, 14} A20 = {10, 16} B1 = {00, 01} B2 = {05, 15} B3 = {11, 24} B4 = {07, 13} C1 0 = {10, 21, 26} C1 1 = {10, 21} C1 2 = {10, 21} C2 0 = {02, \u221e1} C2 1 = {04, \u221e2} C2 2 = {12, \u221e3} C3 0 = {20, \u221e4} C3 1 = {23, \u221e5} C3 2 = {16, \u221e6} C4 0 = {27, \u221e7} C4 1 = {22, \u221e8} C4 2 = {25, \u221e9}. Let X = (Z3 \u00d7 Z8) \u222aW A = {S + j : S \u2208S, j \u2208Z3 \u00d7 Z8}. Then (X, W, A) is an IRP(33, {2, 3\u2217}, 1; 9), whose blocks can be arranged in a 16 \u00d7 29 array as in Figure 9. It can be readily veri\ufb01ed that this arrangement results in an IGBTP1({2, 3\u2217}, 33, 16 \u00d7 29; 9, 4 \u00d7 5). 6.3 Direct Constructions for FrGBTDs Lemma 6.11. There exists an FrGBTD(2, 2t) for t \u2208{4, 5}. Proof. The desired FrGBTDs are given in Fig. 10 and Fig. 11. De\ufb01nition 6.12. Let t be a positive integer, and let I = [t \u22121] \u00d7 [2]. Let (Z3t \u00d7 [2], S) be a 3uniform set system of size 2(t\u22121), where S = {Ai : i \u2208I}. S is called a (Z3t \u00d7[2])-FrGBTD-starter if the following conditions hold: (i) \u2206ijS = Z3t \\ {0, t, 2t} for i, j \u2208[2], (ii) \u222ai\u2208IAi = (Z3t \\ {0, t, 2t}) \u00d7 [2], (iii) for j \u2208[2], each element in (Zt \\ {0}) \u00d7 [2] appears either once or twice in the multiset Rj = t\u22121 [ i=1 A(i,j) \u2212i mod t, (iv) r \u2208(Zt \\ {0}) \u00d7 [2] for each r \u2208R1 \u222aR2. Proposition 6.13. If a (Z3t \u00d7 [2], 6t)-FrGBTD-starter exists, then an FrGBTD(3, 6t) exists. 27 \fW B B + 01 B + 02 B + 03 B + 04 B + 05 B + 06 B + 07 A C1 C4 + 01 C3 + 02 C2 + 03 C1 + 04 C4 + 05 C3 + 06 C2 + 07 C2 C1 + 01 C4 + 02 C3 + 03 C2 + 04 C1 + 05 C4 + 06 C3 + 07 C3 C2 + 01 C1 + 02 C4 + 03 C3 + 04 C2 + 05 C1 + 06 C4 + 07 C4 C3 + 01 C2 + 02 C1 + 03 C4 + 04 C3 + 05 C2 + 06 C1 + 07 where W is a 4 \u00d7 5 empty array, A is a 12 \u00d7 5 array, A1 A2 A3 A4 A5 A1 + 10 A2 + 10 A3 + 10 A4 + 10 A5 + 10 A1 + 20 A2 + 20 A3 + 20 A4 + 20 A5 + 20 A6 A7 A8 A9 A10 A6 + 10 A7 + 10 A8 + 10 A9 + 10 A10 + 10 A6 + 20 A7 + 20 A8 + 20 A9 + 20 A10 + 20 A11 A12 A13 A14 A15 A11 + 10 A12 + 10 A13 + 10 A14 + 10 A15 + 10 A11 + 20 A12 + 20 A13 + 20 A14 + 20 A15 + 20 A16 A17 A18 A19 A20 A16 + 10 A17 + 10 A18 + 10 A19 + 10 A20 + 10 A16 + 20 A17 + 20 A18 + 20 A19 + 20 A20 + 20 , B is a 4 \u00d7 3 array, B1 B1 + 10 B1 + 20 B2 B2 + 10 B2 + 20 B3 B3 + 10 B3 + 20 B4 B4 + 10 B4 + 20 , Ci for i \u2208[4] is a 3 \u00d7 3 array, Ci 0 Ci 2 + 10 Ci 1 + 20 Ci 1 Ci 0 + 10 Ci 2 + 20 Ci 2 Ci 1 + 10 Ci 0 + 20 . Figure 9: An IGBTP1({2, 3\u2217}, 33, 16 \u00d7 29; 9, 4 \u00d7 5). 28 \f\u2014 \u2014 {2,7} {6,3} {7,1} {3,5} {5,6} {1,2} {2,3} {6,7} \u2014 \u2014 {3,0} {7,4} {0,2} {4,6} {5,7} {1,3} {3,4} {7,0} \u2014 \u2014 {4,1} {0,5} {1,6} {5,2} {6,0} {2,4} {4,5} {0,1} \u2014 \u2014 Figure 10: An FrGBTD1(2, 24) (X, G, A), where X = Z8 and G = {{i, 4 + i} : i \u2208Z4}. \u2014 \u2014 {7,9} {2,4} {3,4} {8,9} {6,2} {1,7} {1,8} {6,3} {7,4} {2,9} \u2014 \u2014 {8,0} {3,5} {4,5} {9,0} {7,3} {2,8} {3,9} {8,4} {8,5} {3,0} \u2014 \u2014 {9,1} {4,6} {5,6} {0,1} {1,2} {6,7} {4,0} {9,5} {9,6} {4,1} \u2014 \u2014 {0,2} {5,7} {6,8} {1,3} {2,3} {7,8} {5,1} {0,6} {0,7} {5,2} \u2014 \u2014 Figure 11: An FrGBTD1(2, 25) (X, G, A), where X = Z10 and G = {{i, 5 + i} : i \u2208Z5}. Proof. Let X = Z3t \u00d7 [2], G = {Gi = {i1, (t + i)1, (2t + i)1, i2, (t + i)2, (2t + i)2} : i \u2208Zt}, A = {Ai + j : i \u2208I and j \u2208Z3t}. Then (X, G, A) is a {3}-GDD of type 6t, whose blocks can be arranged in a 2t\u00d73t array, with rows and columns indexed by Zt \u00d7 [2] and Z3t, respectively, as follows: the block A(i,j) + k is placed in cell ((i + k, j), k). The set of blocks occupying column zero are {Ai : i \u2208I} and by condition (ii) of De\ufb01nition 6.12, S i\u2208I Ai = X \\ G0. For other j \u2208Z3t, observe that the blocks occupying column j are translates (by j) of the blocks in column zero, and hence the union of the blocks in column j is X \\Gj\u2032, where j\u2032 \u2261j mod t. For (i, j) \u2208Zt \u00d7 [2], let R(i,j) denote the multiset containing all the points appearing in the blocks of row (i, j). Then R(i,j) = R(0,j) + i, for all i \u2208Zt. Hence, it su\ufb03ces to check that each element of X \\G0 appears either once or twice in R(0,j) and the elements of R(0,j) belong to X \\G0 for j \u2208[2]. This, however, follows immediately from conditions (iii) and (iv) in De\ufb01nition 6.12, since R(0,j) = Rj \u222a(Rj + t) \u222a(Rj + 2t) for j \u2208[2]. Corollary 6.14. There exist an FrGBTD(3, 6t) for all t \u2208{5, 6, 7, 8}, an FrGBTD(3, 24t) for all t \u2208{5, 8} and an FrGBTD(3, 30t) for all t \u2208{5, 7}. Proof. An FrGBTD1(3, 66) is given by Example 4.5. An FrGBTD(3, 6t) for t \u2208{5, 7} exists by applying Proposition 6.13 with FrGBTD-starters given in [5]. The existence of an FrGBTD(3, 24t), t \u2208{5, 8} follows by applying Proposition 5.7 with an FrGBTD(3, 6t) (constructed in this proof) and a DRTD(3, 4), whose existence is provided by Corollary 4.11. The existence of an FrGBTD(3, 30t), t \u2208{5, 7} follows by applying Proposition 5.7 similarly. To prove the existence of an FrGBTD(3, 68), consider (Z48, S), a {3}-uniform set system of size 7, where S comprise the blocks below: A1 = {2, 3, 5} A2 = {4, 14, 31} A3 = {9, 22, 45} A4 = {15, 34, 43} A5 = {20, 35, 42} A6 = {13, 17, 47} A7 = {1, 6, 12}. 29 \fObserve that S satis\ufb01es the following conditions: (i) \u2206S = Z48 \\ {0, 8, 16, 24, 32, 40}, (ii) \u222ai\u2208[7]Ai mod 24 = Z24 \\ {0, 8, 16}, (iii) each element in Z16 \\ {0, 8} appears either once or twice in the multiset R = [ i\u2208[7] Ai \u2212i mod 16, (iv) r \u2208Z16 \\ {0, 8} for each r \u2208R. Further, let X = Z48, G = {{i + 8k : k \u2208Z6} : i \u2208Z8}, A = {Ai + j : i \u2208[7] and j \u2208Z48}. Then (X, G, A) is a {3}-GDD of type 68, whose blocks can be arranged in a 16 \u00d7 24 array, with rows and columns are indexed by Z16 and Z24, respectively, as follows: the block Ai + j is placed in cell (i + j, j). This array can be veri\ufb01ed to be an FrGBTD(3, 68). 7 Existence of GBTDs and GBTPs We apply recursive constructions in Section 5 with small designs directly constructed in Section 6 to completely settle the existence of GBTD1(3, m) and GBTP1({2, 3\u2217}; 2m + 1, m \u00d7 (2m \u22123)). 7.1 Existence of GBTD1(3, m) Lemma 7.1. There exists a special GBTD1(3, 3rq) for all r \u22650 and q \u2208Q, where Q = {q : q \u22611 mod 6 is a prime power} \u222a{5, 9, 11, 23}, except when (r, q) = (0, 5). Proof. Existence of a special GBTD1(3, q) for all q \u2208Q \\ {5} is provided by Corollary 6.4 and 6.5. These GBTDs are all 3-\u2217colorable with property \u03a0. The lemma then follows by considering these GBTDs as RBIBDs and applying Corollary 5.2. Lemma 7.2. Let s \u2208[2] and suppose there exists a TD(5 + s, n). If 0 \u2264gi \u2264n, i \u2208[s] and that there exists a special GBTD1(3, m) for all m \u2208{2n + 1} \u222a{2gi + 1 : i \u2208[s]}, then there exists a special GBTD1(3, 10n + 1 + 2 Ps i=1 gi). Proof. By Corollary 6.14, there exists an FrGBTD(3, 6t) for all t \u2208{5, 6, 7}. By Proposition 5.10, there exists an FrGBTD(3, (6n)5(6g1) \u00b7 \u00b7 \u00b7 (6gs)). Now apply Corollary 5.5 to obtain a special GBTD1(3, 10n + 1 + 2 Ps i=1 gi). Lemma 7.3. A special GBTD1(3, m) exists for odd m \u22657. 30 \fTable 2: Existence of special GBTD1(3, m) Authority m Corollary 6.5 9, 11, 17, 23, 29, 35, 47, 53, 55 Lemma 7.1 7, 13, 15, 19, 21, 25, 27, 31, 33, 37, 39, 43, 45, 49, 57, 61, 63, 67, 69, 73, 75 Corollary 5.6 with (g, t) in {(8, 5), (5, 10), (8, 8), (7, 10)} 41, 51, 65, 71 Lemma 7.2 with n = 5, g1 = 4 59 Lemma 7.2 with n = 7, g1, g2 \u2208{0} \u222a{t : 3 \u2264t \u22647} {s : 77 \u2264s \u226495, s odd} Table 3: Existence of IGBTP1({2, 3\u2217}, 2m + 9, (m + 4) \u00d7 (2m + 5); 4 \u00d7 5) Authority m Corollary 6.10 {s : 10 \u2264s \u226457}\\{16, 20, 24, 28, 32, 36, 40, 44, 48, 50, 52, 54, 55, 56} Lemma 7.4 with (n, g) \u2208{(10, 0), (11, 0), (12, 0), (13, 0), (11, 10), (11, 11), (14, 0)} 40, 44, 48, 52, 54, 55, 56 Proof. First, a special GBTD1(3, m) can be constructed for odd m, 7 \u2264m \u226495. Details are provided in Table 2. We then prove the lemma by induction on m \u226597. Let E = {t : t \u22659} \\ {10, 14, 15, 18, 20, 22, 26, 30, 34, 38, 46, 60}. By Theorem 4.8, a TD(7, n) exists for any n \u2208E. If there exists a special GBTD1(3, m\u2032) for odd m\u2032, 7 \u2264m\u2032 \u22642n+1, then apply Lemma 7.2 with 3 \u2264g1, g2 \u2264n to obtain a special GBTD1(3, m) for odd m, 10n+7 \u2264m \u226414n+1. Hence, take n = 9 to obtain a special GBTD1(3, 97). Suppose there exists a GBTD1(3, m\u2032) for all odd m\u2032 < m. Then there exists n \u2208E with 10n+7 \u2264m \u226414n+1. Suppose otherwise. Then there exists n1 \u2208E such that 14n1 +1 < 10n2 +7 for all n2 > n1 and n2 \u2208E. This, together with the fact that n1 \u22659, implies that n2 \u2212n1 > 3 for all n2 \u2208E and n2 > n1. However, a quick check on E gives a contradiction. Since n \u2208E and there exists a special GBTD1(3, m\u2032) for all m\u2032 \u22642n + 1 < 10n + 7 \u2264m (induction hypothesis), there exists a special GBTD1(3, m) and induction is complete. Lemma 7.3 shows that a GBTD1(3, m) exists for all odd m \u0338= 3, 5. Theorem 2.3 (vi) now follows. 7.2 Existence of GBTP1({2, 3\u2217}; 2m + 1, m \u00d7 (2m \u22123)) Lemma 7.4. Suppose there exists a TD(5, n). Suppose 0 \u2264g \u2264n and that there exists an IGBTP1({2, 3\u2217}, 2m + 9, (m + 4) \u00d7 (2m + 5); 9, 4 \u00d7 5) for m \u2208{n, g}. Then there exists an IGBTP1({2, 3\u2217}, 2M + 9, (M + 4) \u00d7 (2M + 5); 9, 4 \u00d7 5), where M = 4n + g. Proof. By Lemma 6.11, there exists an FrGBTD(2, 2t) for all t \u2208{4, 5}. By Proposition 5.10, there exists an FrGBTD(2, (2n)4(2g)). Now apply Proposition 5.4 to obtain an IGBTP1({2, 3\u2217}, 2M + 9, (M + 4) \u00d7 (2M + 5); 9, 4 \u00d7 5). Lemma 7.5. There exists an IGBTP1({2, 3\u2217}, 2m+9, (m+4)\u00d7(2m+5); 9, 4\u00d75) for any m \u226510, except possibly for m \u2208{16, 20, 24, 28, 32, 36, 46, 50}. 31 \fProof. Let E = {16, 20, 24, 28, 32, 36, 46, 50}. An IGBTP1({2, 3\u2217}, 2m+9, (m+4)\u00d7(2m+5); 9, 4\u00d7 5) can be constructed for 10 \u2264m \u226457 and m / \u2208E \u222a{51}. Details are provided in Table 3. When m = 51, consider a TD(5, 11) and delete four points from a block to form a {4, 5}-GDD of type 10411. Proposition 5.8 yields an FrGBTD(2, 20422) and hence, Proposition 5.4 yields an IGBTP1({2, 3\u2217}, 2m + 9, (m + 4) \u00d7 (2m + 5); 9, 4 \u00d7 5) with m = 51. We then prove the lemma by induction on m \u226557. Let E\u2032 = {4n + g : n \u2208E, 10 \u2264g \u226413} and assume the lemma is true for n < m. When m / \u2208E\u2032, then write m = 4n + g with 13 \u2264n < m, n / \u2208E and g \u2208{10, 11, 12, 13}. Since a TD(5, n) which exists by Theorem 4.8, applying Lemma 7.4 with the corresponding n and g, we obtain the desired IGBTP. When m \u2208E\u2032, we have two cases. \u2022 If m = 77, the required IGBTP is given by Corollary 6.10. \u2022 Otherwise, apply Lemma 7.4 with (n, g) taking values in {(15, 14), (15, 15), (19, 0), (18, 18), (19, 15), (23, 0), (19, 17), (22, 18), (22, 19), (27, 0), (22, 21), (25, 22), (25, 23), (31, 0), (25, 25), (29, 22), (29, 23), (35, 0), (29, 25), (31, 30), (31, 31), (39, 0), (33, 25), (39, 38), (39, 39), (49, 0), (40, 37), (42, 42), (43, 39), (43, 40), (43, 41)}. This completes the induction. Lemma 7.6. A GBTP1({2, 3\u2217}, 2m + 1, m \u00d7 (2m \u22123)) exists for m \u22654, except possibly for m \u2208{12, 13}. Proof. A GBTP1({2, 3\u2217}; 2m + 1, m \u00d7 (2m \u22123)) can be found via computer search for 4 \u2264m \u226411. The GBTPs are listed in [5]. For m \u2208{20, 24, 28, 32, 36, 40, 50, 54}, set M = m \u22125 and we apply Proposition 5.3 with the GBTP1({2, 3\u2217}, 11, 5\u00d77) and the IGBTP1({2, 3\u2217}, 2M+11, (M+5)\u00d7(2M+7); 11, 5\u00d77) constructed in Corollary 6.10. Finally, for m \u226514 and m / \u2208{20, 24, 28, 32, 36, 40, 50, 54}, set M = m\u22124 and apply Proposition 5.3 with GBTP1({2, 3\u2217}, 9, 4 \u00d7 5) and the IGBTP1({2, 3\u2217}, 2M + 9, (M + 4) \u00d7 (2M + 5); 9, 4 \u00d7 5) constructed in Lemma 7.5. Lemma 7.6 shows that a GBTP1({2, 3\u2217}, 2m + 1, m \u00d7 (2m \u22123)) exists for all m \u22654, except possibly for m \u2208{12, 13}. Theorem 2.3 (vii) now follows. 8"
+                },
+                {
+                    "url": "http://arxiv.org/abs/0712.3895v1",
+                    "title": "An Enumeration of Graphical Designs",
+                    "abstract": "Let $\\Psi(t,k)$ denote the set of pairs $(v,\\lambda)$ for which there exists\na graphical $t$-$(v,k,\\lambda)$ design. Most results on graphical designs have\ngone to show the finiteness of $\\Psi(t,k)$ when $t$ and $k$ satisfy certain\nconditions. The exact determination of $\\Psi(t,k)$ for specified $t$ and $k$ is\na hard problem and only $\\Psi(2,3)$, $\\Psi(2,4)$, $\\Psi(3,4)$, $\\Psi(4,5)$, and\n$\\Psi(5,6)$ have been determined. In this paper, we determine completely the\nsets $\\Psi(2,5)$ and $\\Psi(3,5)$. As a result, we find more than 270000\ninequivalent graphical designs, and more than 8000 new parameter sets for which\nthere exists a graphical design. Prior to this, graphical designs are known for\nonly 574 parameter sets.",
+                    "authors": "Yeow Meng Chee, Petteri Kaski",
+                    "published": "2007-12-23",
+                    "updated": "2007-12-23",
+                    "primary_cat": "math.CO",
+                    "cats": [
+                        "math.CO"
+                    ],
+                    "main_content": "INTRODUCTION For a \ufb01nite set X and a nonnegative integer t, the set of all t-subsets of X is denoted \u0000X t \u0001 . A k-uniform set system is a pair (X, B), where X is a \ufb01nite set of elements called points and B \u2286 \u0000X k \u0001 . Elements of B are called blocks. The order of (X, B) is the number of points, |X|. A design with parameters t-(v, k, \u03bb) is a k-uniform set system (X, B) of order v such that every T \u2208 \u0000X t \u0001 is contained in exactly \u03bb blocks c \u20dd John Wiley & Sons, Inc. CCC Journal of Combinatorial Designs Vol. , () 1 arXiv:0712.3895v1  [math.CO]  23 Dec 2007 \f2 CHEE AND KASKI of B. To avoid triviality, we impose the following restrictions on a t-(v, k, \u03bb) design (X, B): (i) t \u22652, (ii) t < k, (iii) B \u0338= \u00d8, and B \u0338= \u0000X k \u0001 . For two designs, (X, A) and (Y, B), an isomorphism of (X, A) onto (Y, B) is a bijection \u03c3 : X \u2192Y such that \u03c3(A) = B. An automorphism of a design is an isomorphism of the design onto itself. The set of all automorphisms of a design D forms a group under functional composition. This group is called the automorphism group of D and is denoted by Aut(D). A subgroup H \u2264Aut(D) is a group of automorphisms of D. Let V be a set of cardinality n and consider the induced action of the symmetric group Sn = Sym(V ) on the set X = \u0000V 2 \u0001 . This de\ufb01nes an embedding of Sn into S( n 2) = Sym(X) with image group S[2] n . By canonical extension, S[2] n also acts on \u0000X k \u0001 . A t-(v, k, \u03bb) design (X, B) is graphical if it has a group of automorphisms that is permutation isomorphic to S[2] n with v = \u0000n 2 \u0001 . In particular, B is then a union of orbits of S[2] n on \u0000X k \u0001 . The term \u201cgraphical design\u201d is motivated by the following alternative perspective. Considering the complete graph Kn with vertex set V , we may view X as the edge set of Kn, in which case the orbits of S[2] n on \u0000X k \u0001 are in a one-to-one correspondence with the isomorphism classes of spanning k-edge subgraphs of Kn. Thus, we may view the block set B of a graphical design as a set of spanning kedge subgraphs of Kn, closed under isomorphism of graphs, such that every t-edge subgraph of Kn is a subgraph of \u03bb graphs in B. Although the de\ufb01nition of a graphical design does not explicitly assume this graphical structure, a required group of automorphisms induces the structure (in a canonical manner for n \u0338= 4) because one of the orbits of S[2] n corresponds to the line graph of Kn, from which one can recover the sets of edges having a vertex in common when n \u0338= 4. Two graphical designs (X, A) and (Y, B), with individualized required groups of automorphisms, H and K, respectively, are equivalent if there exists an isomorphism \u03c3 of (X, A) onto (Y, B) such that \u03c3H\u03c3\u22121 = K. The \ufb01rst example of a graphical design has been attributed to R. M. Wilson by Kramer and Mesner [12]: Example 1.1. A graphical 3-(10, 4, 1) design is obtained by taking as blocks all spanning 4-edge subgraphs of K5 isomorphic to one of the following graphs: However, Betten et al. [1] have reported that already in 1970, M. H. Klin has described graphical designs when he determined the overgroups of S[2] n . But Klin\u2019s result was unpublished, except for a short note that appeared in a less well known journal [10]. Further examples of graphical designs were given by Driessen [9]. The \fAN ENUMERATION OF GRAPHICAL DESIGNS 3 \ufb01rst systematic approach to determining the existence of graphical designs was undertaken by Chouinard et al. [8], who determined all graphical t-(v, k, \u03bb) designs with \u03bb = 1 and \u03bb = 2. These results led Chouinard [6] to make the following conjecture, which remains open. Conjecture 1.2 (Chouinard). For any \ufb01xed \u03bb, there exist only \ufb01nitely many graphical t-(v, k, \u03bb) designs. Partial progress on this conjecture has been obtained by Chouinard [7]. Computers were brought to bear in the early nineties, which resulted in further progress in the construction of graphical t-(v, k, \u03bb) designs. Kreher et al. [13] used the LLL algorithm to construct many examples of graphical t-(v, k, \u03bb) designs. Chee [2, 3] used symbolic computational methods to \ufb01nd all graphical 2-(v, 3, \u03bb), 2-(v, 4, \u03bb), 3-(v, 4, \u03bb), and 4-(v, 5, \u03bb) designs. Further sporadic examples were also obtained by Kramer [11] and Chee [4]. In the late nineties, more graphical t-(v, k, \u03bb) designs were discovered by Betten et al. [1] using an improved implementation of the LLL algorithm. This is the state-of-the-art. Despite that more than twenty years have passed since the introduction of graphical designs, only a small \ufb01nite number of them are known. Let \u039b(t, k, v) denote the set of \u03bb \u22641 2 \u0000v\u2212t k\u2212t \u0001 for which a graphical t-(v, k, \u03bb) design exists, and let \u03a8(t, k) = {(v, \u03bb) : \u03bb \u2208\u039b(t, k, v)}. The reason for restricting \u03bb \u22641 2 \u0000v\u2212t k\u2212t \u0001 is to avoid duplication by complementation, since if (X, B) is a (graphical) t-(v, k, \u03bb) design, then its complement, (X, \u0000X k \u0001 \\ B), is a (graphical) t-(v, k, \u0000v\u2212t k\u2212t \u0001 \u2212\u03bb) design. The parameters of all graphical designs known are given in Appendix A, where Table I presents those sets \u03a8(t, k) which we have complete knowledge of, and Table II lists known elements of some \u03a8(t, k) which we have yet to completely determine. The authority for these tables are [1, 2, 3, 4, 11, 13] (cf. [5]). In total, there are only 574 parameter sets for which we know there exist graphical designs. Indeed, results in the literature are either on construction of sporadic examples, on nonexistence, or on the \ufb01niteness of the number of graphical designs with certain parameters. The purpose of this paper is to improve this state of knowledge by determining completely the sets \u03a8(2, 5) and \u03a8(3, 5). With this result, the sets \u03a8(t, k) are now completely known for 2 \u2264t < k \u22645. As a by-product, we give more than 8000 new parameter sets for which there exists a graphical design, substantially improving on the number of graphical designs known thus far. Our results also correct some minor errors in [1]. 2. KRAMER\u2013MESNER MATRICES AND OUTLINE OF APPROACH Suppose we wish to construct a t-(v, k, \u03bb) design (X, B) with a group of automorphisms \u0393. Then B is a union of orbits of \u0393 on \u0000X k \u0001 . Let O(t) 1 , O(t) 2 , . . . , O(t) N(t) and O(k) 1 , O(k) 2 , . . . , O(k) N(k) be the orbits of \u0393 on \u0000X t \u0001 and on \u0000X k \u0001 , respectively. De\ufb01ne an N(t) \u00d7 N(k) integer matrix Wt,k(X|\u0393) by the rule that the (i, j)-entry is |{K \u2208O(k) j : K \u2287T}|, where T \u2208O(t) i can be chosen arbitrarily. Such Wt,k(X|\u0393) \f4 CHEE AND KASKI matrices are called Kramer\u2013Mesner matrices, after Kramer and Mesner [12] who observed the following. Theorem 2.1 (Kramer and Mesner). There exists a t-(v, k, \u03bb) design with a group of automorphisms \u0393 if and only if there exists a {0, 1}-vector u such that Wt,k(X|\u0393)u = \u03bb(1, . . . , 1)T. (1) Based on Theorem 2.1, our approach to determining \u03a8(2, 5) and \u03a8(3, 5) is to \ufb01nd all solutions to the equation Wt,k(X|S[2] n )u = \u03bb(1, . . . , 1)T for (t, k) = (2, 5) and (t, k) = (3, 5). More precisely, we perform the following steps: (i) determine a bound n0 so that no graphical t-(v, k, \u03bb) design exists for n \u2265n0; and (ii) enumerate all graphical t-( \u0000n 2 \u0001 , k, \u03bb) designs for n < n0 by determining all solutions to Wt,k(X|S[2] n )u = \u03bb(1, . . . , 1)T. The \ufb01rst step is accomplished via a combinatorial analysis and the second step is accomplished via computation. It is not hard to see that distinct {0, 1}-vectors u satisfying (1) give inequivalent graphical designs. Betten et al. [1] have computed the matrices W2,5(X|S[2] n ) and W3,5(X|S[2] n ). These take the forms given in Figs. 1 and 2, where nk denotes the falling factorial n(n \u22121) \u00b7 \u00b7 \u00b7 (n \u2212k + 1). Observe that the matrices have constant row sum \u0000( n 2)\u2212t k\u2212t \u0001 . A list of orbit representatives indexing the rows and columns of W2,5(X|S[2] n ) and W3,5(X|S[2] n ) is given in Appendix B. 3. UPPER BOUNDS FOR EXISTENCE Our subsequent proofs of the nonexistence of graphical designs for n large enough in the cases (t, k) = (2, 5) and (t, k) = (3, 5) are quantitative versions of the proof of a \ufb01niteness theorem of Betten et al. [1]. The orbit of a graph G under the action of S[2] n is denoted by Orb(G). 3.1 Upper Bound for Existence of Graphical 2-(v, 5, \u03bb) Designs We prove in this section that no graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) design exists if n \u2265538. Let (X, B) be a graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) design, where n \u2265538. We may assume without loss of generality that B \u2287Orb(G(5) 26 ), since otherwise we can consider the complement of the design. Let \u00b5i denote the sum of all entries of degree i (as a polynomial in n) in row two of W2,5(X|S[2] n ). Then we have \u00b56 = 1 48(n \u22124)6, \u00b55 = 3 4(n\u22124)5, \u00b54 = 55 6 (n\u22124)4, \u00b53 = 275 6 (n\u22124)3, and \u00b52 = 89(n\u22124)2. De\ufb01ne the integers \u03bb6 = \u00b56 and \u03bbi = \u03bbi+1 + \u00b5i for i = 2, 3, 4, 5. By considering the number of blocks in Orb(G(5) 26 ) containing G(2) 2 , we see that \u03bb \u2265\u03bb6. (2) \fAN ENUMERATION OF GRAPHICAL DESIGNS 5 index 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 4(n \u22123) 4 5 2 (n \u22123)3 10(n \u22124)2 6(n \u22123)2 16(n \u22124) 7(n \u22123)2 12(n \u22124) 4(n \u22123)2 4(n \u22124) 2 3 (n \u22123)3 4(n \u22124)2 1 8 (n \u22123)4 7 6 (n \u22124)3 2 3 (n \u22123)4 4(n \u22124)3 5(n \u22123)3 20(n \u22124)2 3 2 (n \u22123)3 4(n \u22124)2 3 2 (n \u22123)4 14(n \u22124)3 4(n \u22123)3 24(n \u22124)2 1 4 (n \u22123)5 4(n \u22124)4 6(n \u22123)2 16(n \u22124) (n \u22123)2 4(n \u22124) 3 2 (n \u22123)4 14(n \u22124)3 5(n \u22123)3 20(n \u22124)2 2(n \u22123)4 12(n \u22124)3 7 3 (n \u22123)3 4(n \u22124)2 1 48 (n \u22123)6 3 4 (n \u22124)5 1 4 (n \u22123)5 4(n \u22124)4 1 8 (n \u22123)5 7 6 (n \u22124)4 1 2 (n \u22123)3 3(n \u22124)2 1 4 (n \u22123)4 2 3 (n \u22124)3 1 6 (n \u22123)3 0 0 1 48 (n \u22124)6 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 FIG. 1. Transpose of the Kramer\u2013Mesner matrix W2,5(X|S[2] n ) Lemma 3.1. B \u2287Orb(G(5) 20 ). Proof. Suppose that B \u0338\u2287Orb(G(5) 20 ). Then by considering the number of blocks in B containing G(2) 1 , we have \u03bb \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 48(n \u22123)6. The above inequality, together with inequality (2), implies \u03bb6 \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 48(n \u22123)6, \f6 CHEE AND KASKI index 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3(n \u22123) 0 3 3 0 3 2 (n \u22123)3 5(n \u22125) (n \u22124)2 3 2 (n \u22124)2 12 3(n \u22123)2 8 4(n \u22124) 3(n \u22124) 0 3(n \u22123)2 4 5(n \u22124) 6(n \u22124) 0 3 2 (n \u22123)2 1 2(n \u22124) 6(n \u22124) 0 1 2 (n \u22123)3 3(n \u22125) 0 0 0 1 8 (n \u22123)4 1 2 (n \u22125)2 0 0 3(n \u22126) 0 3(n \u22125)2 0 1 2 (n \u22124)3 0 0 12(n \u22125) 3(n \u22124)2 3(n \u22124)2 0 0 2(n \u22125) (n \u22124)2 3 2 (n \u22124)2 0 0 7(n \u22125)2 1 2 (n \u22124)3 0 24(n \u22126) 0 12(n \u22125) 3(n \u22124)2 0 24 0 3 2 (n \u22125)3 0 0 12(n \u22126)2 0 6 6(n \u22124) 3(n \u22124) 0 0 2 n \u22124 0 0 0 5(n \u22125)2 (n \u22124)3 0 36(n \u22126) 0 8(n \u22125) 4(n \u22124)2 3(n \u22124)2 24 0 5(n \u22125)2 (n \u22124)3 3 2 (n \u22124)3 24(n \u22126) 0 2(n \u22125) (n \u22124)2 4(n \u22124)2 0 0 1 8 (n \u22125)4 0 0 7 2 (n \u22126)3 0 (n \u22125)3 1 8 (n \u22124)4 0 15(n \u22126)2 0 1 2 (n \u22125)3 0 1 8 (n \u22124)4 3(n \u22126)2 0 n \u22125 1 2 (n \u22124)2 0 6 0 1 2 (n \u22125)2 0 1 2 (n \u22124)3 0 0 0 0 1 2 (n \u22124)2 0 0 0 0 0 1 8 (n \u22126)4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 FIG. 2. Transpose of the Kramer\u2013Mesner matrix W3,5(X|S[2] n ) giving n6 \u221269n5 + 1085n4 \u22128435n3 + 36642n2 \u221284664n + 80832 \u22640, which is impossible for n \u226551. So B \u2287S i\u2208{20,26} Orb(G(5) i ) and by considering the number of blocks in B containing G(2) 2 , we now have \u03bb \u2265\u03bb5. (3) \fAN ENUMERATION OF GRAPHICAL DESIGNS 7 Lemma 3.2. B \u2287S i\u2208{13,21,22} Orb(G(5) i ). Proof. Suppose that B contains at most two of the orbits Orb(G(5) i ), i \u2208{13, 21, 22}. Then by considering the number of blocks in B containing G(2) 1 , we have \u03bb \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 8(n \u22123)5. The above inequality, together with inequality (3), implies \u03bb5 \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 8(n \u22123)5, giving 3n5 \u2212295n4 + 4475n3 \u221228541n2 + 85198n \u221298184 \u22640, which is impossible for n \u226582. So B \u2287S i\u2208{13,20,21,22,26} Orb(G(5) i ) and by considering the number of blocks in B containing G(2) 2 , we now have \u03bb \u2265\u03bb4. (4) Lemma 3.3. B \u2287S i\u2208{7,8,11,16,18,24} Orb(G(5) i ). Proof. Suppose that B contains at most \ufb01ve of the orbits Orb(G(5) i ), i \u2208{7, 8, 11, 16, 18, 24}. Then by considering the number of blocks in B containing G(2) 1 , we have \u03bb \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 8(n \u22123)4. The above inequality, together with inequality (4), implies \u03bb4 \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 8(n \u22123)4, giving 3n4 \u22121154n3 + 14721n2 \u221264450n + 95256 \u22640, which is impossible for n \u2265372. So B \u2287S i\u2208{7,8,11,13,16,18,20,21,22,24,26} Orb(G(5) i ) and by considering the number of blocks in B containing G(2) 2 , we now have \u03bb \u2265\u03bb3. (5) \f8 CHEE AND KASKI Lemma 3.4. B \u2287S i\u2208{2,6,9,10,12,17,19,23,25} Orb(G(5) i ). Proof. Suppose that B contains at most eight of the orbits Orb(G(5) i ), i \u2208{2, 6, 9, 10, 12, 17, 19, 23, 25}. Then by considering the number of blocks in B containing G(2) 1 , we have \u03bb \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 6(n \u22123)3. The above inequality, together with inequality (5), implies \u03bb3 \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u22121 6(n \u22123)3, giving n3 \u2212546n2 + 4541n \u22129516 \u22640, which is impossible for n \u2265538. So B \u2287S i\u2208{2,6,7,8,9,10,11,12,13,16,17,18,19,20,21,22,23,24,25,26} Orb(G(5) i ) and by considering the number of blocks in B containing G(2) 2 , we now have \u03bb \u2265\u03bb2. (6) Lemma 3.5. B \u2287S i\u2208{3,4,5,14,15} Orb(G(5) i ). Proof. Suppose that B contains at most four of the orbits Orb(G(5) i ), i \u2208{3, 4, 5, 14, 15}. Then by considering the number of blocks in B containing G(2) 1 , we have \u03bb \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u2212(n \u22123)2. The above inequality, together with inequality (6), implies \u03bb2 \u2264 \u0012\u0000n 2 \u0001 \u22122 3 \u0013 \u2212(n \u22123)2, giving n2 \u221259n + 216 \u22640, which is impossible for n \u226556. So B \u2287S i\u2208{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26} Orb(G(5) i ) = \u0000X 5 \u0001 \\ Orb(G(5) 1 ). If B \u0338\u2287Orb(G(5) 1 ), then (X, B) cannot be a 2-( \u0000n 2 \u0001 , 5, \u03bb) design unless 4(n \u22123) = 4, which is impossible for n \u22655. So B \u2287Orb(G(5) 1 ) and hence B = \u0000X k \u0001 , which is excluded from the de\ufb01nition of a design to avoid triviality. We summarize the above results as: Theorem 3.6. No graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) design exists if n \u2265538. \fAN ENUMERATION OF GRAPHICAL DESIGNS 9 3.2 Upper Bound for Existence of Graphical 3-(v, 5, \u03bb) Designs We prove in this section that no graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) design exists if n \u226534. Let (X, B) be a graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) design, where n \u226534. We may assume without loss of generality that B \u2287Orb(G(5) 7 ), since otherwise we can consider the complement of the design. By considering the number of blocks in Orb(G(5) 7 ) containing G(3) 1 , we see that \u03bb \u22651 8(n \u22123)4. (7) Lemma 3.7. B \u2287S i\u2208{20,21,22,26} Orb(G(5) i ). Proof. Suppose that B \u0338\u2287Orb(G(5) 20 ), Then by considering the number of blocks in B containing G(3) 2 , we have \u03bb \u2264 \u0012\u0000n 2 \u0001 \u22123 2 \u0013 \u22121 8(n \u22125)4. The above inequality, together with inequality (7), implies 1 8(n \u22123)4 \u2264 \u0012\u0000n 2 \u0001 \u22123 2 \u0013 \u22121 8(n \u22125)4, giving (n \u22124)(n3 \u221238n2 + 231n \u2212498) \u22640, which is impossible for n \u226532. To show that B \u2287Orb(G(5) i ) for i \u2208{21, 22, 26}, mimic the proof above. It follows that B \u2287S i\u2208{7,20,21,22,26} Orb(G(5) i ). Let A = \u0000X 5 \u0001 \\B and consider the 3-( \u0000n 2 \u0001 , 5, \u03bb\u2032) design (X, A). By considering the number of blocks in A containing G(3) 5 , we see that \u03bb\u2032 \u226412(n \u22126)2 + 84(n \u22126) + 66. (8) Lemma 3.8. A \u0338\u2287Orb(G(5) i ) for i \u2208{2, 6, 11, 13, 16, 18, 24}. Proof. Suppose that A \u2287Orb(G(5) 2 ). Then by considering the number of blocks in Orb(G(5) 2 ) containing G(3) 1 , we have \u03bb\u2032 \u22653 2(n \u22123)3. The above inequality, together with inequality (8), implies 3 2(n \u22123)3 \u226412(n \u22126)2 + 84(n \u22126) + 66, \f10 CHEE AND KASKI giving n3 \u221220n2 + 95n \u2212120 \u22640, which is impossible for n \u226514. To show that A \u0338\u2287Orb(G(5) i ) for i \u2208{6, 11, 13, 16, 18, 24}, mimic the proof above. It follows that B \u2287S i\u2208{2,6,7,11,13,16,18,20,21,22,24,26} Orb(G(5) i ). By considering the number of blocks in A containing G(3) 5 , we now have \u03bb\u2032 \u226454. (9) Lemma 3.9. A \u0338\u2287Orb(G(5) i ) for i \u2208{1, 3, 4, 5, 8, 9, 10, 12, 14, 15, 17, 19, 23, 25}. Proof. Suppose that A \u2287Orb(G(5) 1 ). Then by considering the number of blocks in Orb(G(5) 1 ) containing G(3) 1 , we have \u03bb\u2032 \u22653(n \u22123). The above inequality, together with inequality (9), implies 3(n \u22123) \u226454, which is impossible for n \u226522. To show that A \u0338\u2287Orb(G(5) i ) for i \u2208{3, 4, 5, 8, 9, 10, 12, 14, 15, 17, 19, 23, 25}, mimic the proof above. We can now conclude that B \u2287 \u0000X 5 \u0001 , which is excluded from the de\ufb01nition of a design to avoid triviality. We summarize the above results as: Theorem 3.10. No graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) design exists if n \u226534. 4. COMPUTATION FOR EXISTENCE The symbolic computation approach of Chee [2] can, in theory, be used to \ufb01nd all graphical t-(v, k, \u03bb) designs for given t and k, without the need to establish upper bounds for existence, such as in the previous section. However, in practice, the method becomes infeasible when k becomes large. Already for k = 5 we would have to solve up to 33 million systems of simultaneous Diophantine equations of degree up to six. Fortunately, using the upper bounds from the previous section, a straightforward exhaustive search su\ufb03ces. In both of the cases (t, k) = (2, 5) and (t, k) = (3, 5), there are 26 possible orbits of 5-edge graphs, implying that we can easily enumerate all the 226 = 67108864 candidate designs, represented as {0, 1}vectors u, and \ufb01lter out those candidates that do not constitute a solution to the system \fAN ENUMERATION OF GRAPHICAL DESIGNS 11 Wt,5(X|S[2] n )u = \u03bb(1, . . . , 1)\u22a4, \u03bb \u22641 2 \u0012\u0000n 2 \u0001 \u2212t 5 \u2212t \u0013 . In particular, this system needs to be considered in the two cases t = 2 and t = 3 for all n \u2264537 and n \u226439, respectively. Both authors of this paper independently carried out this computation with the following identical results. 4.1 Existence of Graphical 2-(v, 5, \u03bb) Designs Our computations show that there are no graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) designs for 40 \u2264 n \u2264537. For n \u226439, the number of inequivalent graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) designs is fairly large, and for reasons of space, it is infeasible to give a complete listing within this paper. A complete catalogue of the designs can be found on the \ufb01rst author\u2019s website at \u27e8http://www1.spms.ntu.edu.sg/~ymchee/graphical.htm \u27e9. We record this result as: Theorem 4.1. There are 8619 elements in \u03a8(2, 5) and there exist 271360 inequivalent graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) designs. No graphical 2-( \u0000n 2 \u0001 , 5, \u03bb) design exists if n \u226540. 4.2 Existence of Graphical 3-(v, 5, \u03bb) Designs Our computations show that there are no graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) designs for 10 \u2264 n \u226433. For n \u22649, a complete listing of all inequivalent graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) designs found is presented below. All elements of \u03a8(3, 5) and inequivalent solutions (v, \u03bb) {0, 1}-vectors uT giving Number of inequivalent solutions inequivalent solutions (15, 30) 10010100110000001000001000 1 (21, 3) 00000010000000100000000010 1 (21, 30) 00001100001001000000001100 1 (21, 33) 00001110001001100000001110 1 (21, 39) 00010010100000010000000010 3 01000011010000101000000000 01000011010001100100000000 (21, 48) 10010000001100100010000010 2 10100000000100100110000010 (21, 69) 00011110101001010000001110 5 00101110100000011000001110 00101110100001010100001110 00101111000101010010001110 01001111011001101000001100 \f12 CHEE AND KASKI (21, 75) 01010011110000011000000000 2 01010011110001010100000000 (28, 30) 00000100000011000000001110 1 (28, 150) 00110101010100010110001000 4 00110101011000011010001000 11001010100111100100110110 11001010101011101000110110 (36, 180) 00101010011001000101001000 1 (36, 198) 11000000011001110010001110 1 (36, 258) 10101111000110111011000110 3 10110100011100100110100010 10110100101010101010100010 We record this result as: Theorem 4.2. There are 13 elements in \u03a8(3, 5) and there exist 26 inequivalent graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) designs. No graphical 3-( \u0000n 2 \u0001 , 5, \u03bb) design exists if n \u226510. 5."
+                },
+                {
+                    "url": "http://arxiv.org/abs/0712.2640v1",
+                    "title": "Optimal Memoryless Encoding for Low Power Off-Chip Data Buses",
+                    "abstract": "Off-chip buses account for a significant portion of the total system power\nconsumed in embedded systems. Bus encoding schemes have been proposed to\nminimize power dissipation, but none has been demonstrated to be optimal with\nrespect to any measure. In this paper, we give the first provably optimal and\nexplicit (polynomial-time constructible) families of memoryless codes for\nminimizing bit transitions in off-chip buses. Our results imply that having\naccess to a clock does not make a memoryless encoding scheme that minimizes bit\ntransitions more powerful.",
+                    "authors": "Yeow Meng Chee, Charles J. Colbourn, Alan C. H. Ling",
+                    "published": "2007-12-17",
+                    "updated": "2007-12-17",
+                    "primary_cat": "cs.AR",
+                    "cats": [
+                        "cs.AR",
+                        "cs.DM",
+                        "cs.IT",
+                        "math.IT",
+                        "B.4.3; B.m; E.4"
+                    ],
+                    "main_content": "INTRODUCTION Energy e\ufb03ciency is an important product quality characteristic. For mobile applications, such as handheld and wireless devices, it not only impacts the usability and acceptance directly, but also a\ufb00ects reliability and packaging cost of the product. Consequently, design techniques for minimizing system power consumption are important for achieving high product quality. Power-e\ufb03cient design requires the reduction of power dissipation throughout the design, during all stages of the design process, subject to constraints on system performance and quality of service. In CMOS circuits, most power is dissipated as dynamic power for charging and discharging of internal node capacitances. Thus, researchers have investigated techniques for minimizing the number of transitions inside the circuits. The power dissipation at the input/output (I/O) pads of an integrated circuit (IC) is even higher because o\ufb00-chip buses have switching capacitances that are orders of magnitude greater than those internal to a chip. The power dissipated at the I/O pads of an IC ranges from 10% to 80% of the total power dissipation with a typ\u2217Y. M. Chee is also with Card View Pte. Ltd., 41 Science Park Road, #04-08A The Gemini, Singapore Science Park II, Singapore 117610. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro\ufb01t or commercial advantage and that copies bear this notice and the full citation on the \ufb01rst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speci\ufb01c permission and/or a fee. ICCAD\u201906, November 5-9, 2006, San Jose, CA Copyright 2006 ACM 1-59593-389-1/06/0011...$5.00 ical value of 50% for circuits optimized for low power [?]. The concern of this paper is with low power encoding for o\ufb00-chip buses. Bus encoding is used to reduce power dissipated on the bus lines. It has been shown to be an e\ufb00ective technique for power reduction. Compared to on-chip buses of deep submicron (DSM) circuits, the ratio of inter-wire capacitances (or crosstalk coupling) to substrate capacitance is much lower in o\ufb00-chip buses. The energy model for o\ufb00-chip buses is thus simpler. We follow the model discussed by Catthoor et al. [?]. For a bus with k wires, each one having metal interconnect capacitance of C, and under a supply voltage Vdd, the total energy consumed by the bus for a computation of N cycles can be estimated as Ebus = N \u00b7 k \u00b7 C \u00b7 V 2 dd. (1) The average power can then be obtained by multiplying Ebus by f, the frequency at which the bus operates. However, the derivation of (1) assumes that for each cycle, all k bus wires exhibit a transition that causes the corresponding capacitance to switch. In general, not all the wires switch; hence a cycle-based model is more accurate, as in: Ebus =  N X i=1 ki ! \u00b7 C \u00b7 V 2 dd, (2) where ki is the number of wires switching at cycle i. The only freedom in (2) for reducing energy consumption is ki, since N depends on the application running on the processor, and C and Vdd depend on the technology. Therefore, bus encoding techniques for reducing energy consumption have focused on reducing ki, the bit switching activity of the o\ufb00-chip bus. However, no explicit encoding scheme with provable optimality (with respect to any formal measure) is known. In this paper, we give the \ufb01rst explicit and provably optimal memoryless encoding schemes (with respect to rate and maximum energy consumed per cycle), both stateless and stateful, for o\ufb00-chip buses. 2. MATHEMATICAL PRELIMINARIES 2.1 Codes The Hamming n-space is the set H(n) = {0, 1}n endowed with the Hamming distance dH de\ufb01ned as follows: for u, v \u2208 H(n), dH(u, v) is the number of positions where u and v di\ufb00er. The Hamming weight (for short, weight) of a vector u \u2208H(n) is the number of positions in u with non-zero value, arXiv:0712.2640v1  [cs.AR]  17 Dec 2007 \fand is denoted wH(u). The ith component of u is denoted ui. The support of a vector u \u2208H(n), denoted supp(u), is the set {i : ui = 1}. Any subset of H(n) is called a code of length n. A constant weight code of length n and weight w is any subset of H(n, w) = {u \u2208H(n) : wH(u) = w}. The elements of a code are called codewords. Let C \u2286H(n) be a code. The size of C is |C|, the number of codewords in the code. The diameter of C is diam(C) = maxu,v\u2208C dH(u, v). Given two codes C1, C2 \u2286H(n), the cross diameter of C1 and C2 is crossdiam(C1, C2) = maxu\u2208C1,v\u2208C2 dH(u, v). By de\ufb01nition, crossdiam(C, C) = diam(C). 2.2 Set Systems For integers i < j, the set {i, i + 1, . . . , j} is abbreviated as [i, j]. Moreover, we also write [j] for [1, j]. For a \ufb01nite set X and k \u2264|X|, we de\ufb01ne 2X = {A : A \u2286X} and `X k \u00b4 = {A \u22082X : |A| = k}. The ring Z/nZ is denoted Zn. A set system of order n is a pair (X, A), where X is a \ufb01nite set of n points and A \u22862X. The elements of A are called blocks. A set system is said to be k-uniform if A \u2286 `X k \u00b4 . Let ([n], A) be a set system. The incidence vector of a block A \u2208A is the vector \u03b9(A) \u2208H(n), where \u03b9(A)i = ( 1, if i \u2208A; and 0, otherwise. There is a natural correspondence between the Hamming n-space and the complete set system of order n, ([n], 2[n]): the positions of vectors in H(n) correspond to points in [n], a vector u \u2208H(n) corresponds to the block supp(u), and dH(u, v) = |supp(u)\u2206supp(v)|. From this, it follows that there is a bijection between the set of all codes of length n and the set of all set systems of order n. There is also a bijection between the set of all constant weight codes of length n and weight w and the set of all w-uniform set systems of order n. So we may speak of the set system of a code, or the code of a set system. 3. PROBLEM FORMULATION Let s \u22651 and n \u2265k. An s-state n-bit encoding scheme for a source S \u2286H(k) is a triple E = \u27e8C, E, D\u27e9, where 1. C is a code of length n, 2. E : S \u00d7 Zs \u2192C is an injective map called an encoding function, and 3. D : C \u00d7 Zs \u2192S is a surjective map called a decoding function, such that D(E(u)) = u for all u \u2208S. The encoding function induces a subscode Ci \u2286C for each i \u2208Zs, de\ufb01ned by Ci = {v \u2208C : v = E(u, i) for some u \u2208S}. Conversely, {Ci, i \u2208Zs} uniquely identi\ufb01es the class of functions that E (and hence D) can belong to: E \u2208{f : S \u00d7 Zs \u2192C : f(\u00b7, i) : S \u2192Ci is injective for all i \u2208Zs}. Figure 1: Non-adaptive encoding for o\ufb00-chip bus Figure 2: Adaptive encoding for o\ufb00-chip bus Encoding and decoding modules that implement E and D, respectively, are inserted at the interface of the bus (see Fig. 1). Suppose that u, v \u2208S are two words to be communicated across the bus at steps t and t + 1 of a computation. In the absence of a bus encoding scheme, the number of bit switchings in this computation cycle is |{i : ui \u0338= vi}|. This quantity is precisely dH(u, v), which can be as high as k. If an s-state n-bit encoding scheme is used, then u\u2032 = E(u, i) and v\u2032 = E(v, j) are communicated instead in steps t and t + 1 of the computation, where i = t (mod s) and j = t + 1 (mod s). The resulting number of bit switchings in this computation cycle is therefore dH(u\u2032, v\u2032), which is bounded above by crossdiam(Ci, Cj). To ensure that the encoding scheme guarantees performance under as general a condition as possible, we adopt a worstcase analysis model. Indeed for a given performance guarantee \u03b4 \u2264k, we require that crossdiam(Ci, Ci+1) \u2264\u03b4 for all i \u2208Zs. We call such a family of codes (Ci)i\u2208Zs an (n, \u03b4)s-low power (LP) code. A 1-state encoding scheme is known as a stateless encoding scheme since it does not need to know the state (step number) of the computation in order to encode the word on the bus. In this case, an (n, \u03b4)1-LP code is a code C such that diam(C) \u2264\u03b4. Any s-state encoding scheme with s \u22652 is called stateful. While a stateless encoding scheme does not need access to a clock, a stateful encoding scheme would require access to a clock to know the particular computation cycle. The encoding schemes that we consider are non-adaptive: The choice of codeword to transmit across the bus in step t does not depend on codewords that have been transmitted in steps i, i < t. A diagram depicting adaptive encoding is given in Fig. 2. Both non-adaptive and adaptive encoding schemes for low-power data buses have been considered (see, for example, [?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?]). Adaptive encoding schemes have better rates. However non-adaptive encoding schemes are often simpler to implement since they do not need to have a continuously changing data model, and do not require memory (to track history of transmitted codewords). Thus non-adaptive encoding schemes are also known as memoryless encoding schemes. The focus of this paper is on memoryless encoding schemes. \fThe wire expansion of E, an n-bit encoding scheme for a source S \u2286H(k), is the quantity \u03b1(E) = n/k. Wire expansion should be minimized since it a\ufb00ects the system area. Hence, given \u03b4 \u2264k, we want to \ufb01nd an (n, \u03b4)s-LP code (Ci)i\u2208Zs with the smallest n such that |Ci| \u22652k, for all i \u2208Zs. In this way we arrive at the following equivalent problem: Problem 1. Given n, \u03b4, and s, \ufb01nd an (n, \u03b4)s-LP code (Ci)i\u2208Zs maximizing mini\u2208Zs |Ci|. Let Fs(n, \u03b4) = max (n, \u03b4)s-LP code (Ci)i\u2208Zs min i\u2208Zs |Ci|. An (n, \u03b4)s-LP code (Ci)i\u2208Zs such that mini\u2208Zs |Ci| = Fs(n, \u03b4) is called optimal. Lemma 1. Fs(n, \u03b4) \u2264F2(n, \u03b4) for all s \u22652. Proof. This follows from the property that if (Ci)i\u2208Zs is an (n, \u03b4)s-LP code, then for any A \u2286Zs such that |A| \u22652, (Ci)i\u2208A is also an (n, \u03b4)|A|-LP code. Lemma 1 shows that there is no advantage in using s-state encoding schemes with s > 2. Hence, we restrict our attention to s = 1 (the stateless case) and s = 2 (the stateful case) for the rest of this paper. Problem 2. Given n and \u03b4, construct an optimal (n, \u03b4)1LP code. Problem 3. Given n and \u03b4, construct an optimal (n, \u03b4)2LP code. For u, v \u2208H(n, w), we have dH(u, v) \u2264min{2w, n}. Therefore one way to constrain the number of bit switchings is to limit the code to contain codewords of low weight. This suggests the consideration of constant weight codes. Problem 4. Given n and \u03b4, construct an optimal (n, \u03b4)1LP code of constant weight w. Problem 5. Given n and \u03b4, construct an optimal (n, \u03b4)2LP code of constant weight w. In subsequent sections, we present complete solutions to Problems 2, 3, and 4. Problem 5 is solved for n su\ufb03ciently large. 4. OPTIMAL STATELESS LOW POWER CODES By de\ufb01nition, ([n], A) is the set system of an (n, \u03b4)1-LP code if and only if |A1\u2206A2| \u2264\u03b4 for all A1, A2 \u2208A. The (n, \u03b4)1-LP codes are known as anticodes of length n and maximum distance \u03b4 and were introduced in the coding theory literature by Farrell [?] in 1970. However, set systems ([n], A) satisfying |A1\u2206A2| \u2264\u03b4 for all A1, A2 \u2208A were studied earlier by Kleitman [?], who obtained a complete solution to the problem of determining the maximum number of blocks in such a set system. Theorem 1 (Kleitman). Let N(n, \u03b4) denote the maximum number of blocks in a set system ([n], A) satisfying |A1\u2206A2| \u2264\u03b4 for all A1, A2 \u2208A. Then N(n, \u03b4) = (P\u03b4/2 i=0 `n i \u00b4 , if \u03b4 \u22610 (mod 2); P(\u03b4\u22121)/2 i=0 `n i \u00b4 + ` n\u22121 (\u03b4\u22121)/2 \u00b4 , if \u03b4 \u22611 (mod 2). The extremal set systems are given by A = 8 > < > : S\u03b4/2 i=0 `[n] i \u00b4 , if \u03b4 \u22610 (mod 2); S(\u03b4\u22121)/2 i=0 `[n] i \u00b4 \u222a {A \u222a{x} : A \u2208 `[n]\\{x} (\u03b4\u22121)/2 \u00b4 }, if \u03b4 \u22611 (mod 2), where x is any \ufb01xed element of [n]. The explicit description of optimal (n, \u03b4)1-LP codes provided by Theorem 1 shows that such codes can be constructed in time polynomial in the size of the code. This solves Problem 2. 4.1 Restriction to Constant Weight We now address constant weight (n, \u03b4)1-LP codes. The diameter of any constant weight (n, \u03b4)1-LP code is even, so we need only consider \u03b4 \u22610 (mod 2). In this subsection, we show the equivalence of constant weight (n, \u03b4)1-LP codes and intersecting families. Definition 1. A set system (X, A) is a t-intersecting family if |A1 \u2229A2| \u2265t holds for all A1, A2 \u2208A. Lemma 2. The code C of a set system ([n], A) is an (n, \u03b4)1LP code of constant weight w if and only if ([n], A) is a w-uniform (w \u2212\u03b4/2)-intersecting family. Proof. Since ([n], A) is (w \u2212\u03b4/2)-intersecting, |A1 \u2229A2| \u2265w \u2212\u03b4/2 \u21d4 |A1\u2206A2| \u2264\u03b4 \u21d4 dH(\u03b9(A1), \u03b9(A2)) \u2264\u03b4, for all A1, A2 \u2208A. Hence, C has diameter at most \u03b4. Let M(n, k, t) be the maximum number of blocks in a kuniform t-intersecting family of order n. The determination of M(n, k, t) and the structure of maximum k-uniform tintersecting families was initiated by Erd\u02dd os et al. [?], who proved the celebrated Erd\u02dd os-Ko-Rado Theorem. This result has been improved subsequently by Frankl [?], Wilson [?], and Ahlswede and Khachatrian [?], culminating in the following. \fTheorem 2 (Complete Intersection Theorem). Let 1 \u2264t \u2264k \u2264n and de\ufb01ne F(i) = {F \u2208 `[n] k \u00b4 : |F \u2229 [t + 2i]| \u2265t + i}. Let r = arg maxi\u2208[0,(n\u2212t)/2] |F(i)|. Then M(n, k, t) = |F(r)| and F(r) is a maximum k-uniform tintersecting family of order n. Lemma 2 together with Theorem 2 completely determines the optimal (n, \u03b4)1-LP codes of constant weight w, solving Problem 4. 5. OPTIMAL STATEFUL LOW POWER CODES Two set systems ([n], A) and ([n], B) form a \u03b4-pair if |A\u2206B| \u2264 \u03b4 for all A \u2208A and B \u2208B. Let ([n], A) and ([n], B) be the set systems of C0 and C1, respectively. Then from the de\ufb01nition, ([n], A) and ([n], B) form a \u03b4-pair if and only if (C0, C1) is an (n, \u03b4)2-LP code. Let 0 \u2264r \u2264n. A quasi-sphere of dimension n and radius r is a set Q of the form Q = {A \u2286[n] : 0 \u2264|A| \u2264r}\u222a\u00af Q, where \u00af Q consists of the \ufb01rst L elements of {A \u2286[n] : |A| = r + 1} in lexicographic order, for some L \u22650. Given 0 \u2264N \u22642n, there exists a unique quasi-sphere Q of dimension n such that |Q| = N (see [?] for a proof). We want to determine the maximum of min{|A|, |B|}, where ([n], A) and ([n], B) form a \u03b4-pair. A related result has been obtained by Ahlswede and Katona [?]. Theorem 3 (Ahlswede and Katona). Let 1 \u2264N \u2264 2n, 1 \u2264\u03b4 \u2264n, and let ([n], A) and ([n], B) be a \u03b4-pair with |A| = N. Then max |B| is achieved if A is a quasi-sphere and B = {B : |B\u2206A| \u2264\u03b4 for all A \u2208A}. We assume without loss of generality that \u03b4 < n, since otherwise we may take A = B = 2[n]. Suppose that ([n], A) and ([n], B) form a \u03b4-pair, with |A| = N, such that B has the maximum number of blocks. Write N = Pr i=0 `n i \u00b4 + m, 0 \u2264m \u2264 ` n r+1 \u00b4 , for some r \u2264n. Theorem 3 asserts that A is the quasi-sphere of dimension n and radius r, with |A| = N, and B = {B \u2286[n] : |B\u2206A| \u2264\u03b4 for all A \u2208A}. Suppose that B \u2208B and |B| = k. Then there exists A \u2208([n] \\ B) with |A| = \u03b4 \u2212k + 1 whenever \u03b4 < n. Since |A\u2206B| = \u03b4 + 1, we have A \u0338\u2208A. This is possible only if \u03b4 \u2212k + 1 > r, or k < \u03b4\u2212r+1. It follows that no k-subset of [n], k \u2265\u03b4\u2212r+1, is in B. Hence, |B| \u2264P\u03b4\u2212r i=0 `n i \u00b4 . Now consider the case that \u03b4 \u22610 (mod 2). If r \u2265\u03b4/2, then |B| \u2264P\u03b4/2 i=0 `n i \u00b4 \u2264N(n, \u03b4). If r < \u03b4/2, then |A| \u2264 P\u03b4/2\u22121 i=0 `n i \u00b4 + ` n \u03b4/2 \u00b4 = P\u03b4/2 i=0 `n i \u00b4 \u2264N(n, \u03b4). Hence, we have min{|A|, |B|} \u2264N(n, \u03b4). Next, consider the case that \u03b4 \u22611 (mod 2). If r \u2265(\u03b4 +1)/2, then |B| \u2264P(\u03b4\u22121)/2 i=0 `n i \u00b4 \u2264N(n, \u03b4). If r \u2264(\u03b4 \u22123)/2, then |A| \u2264P(\u03b4\u22123)/2 i=0 `n i \u00b4 + ` n (\u03b4\u22121)/2 \u00b4 \u2264P(\u03b4\u22123)/2 i=0 `n i \u00b4 + ` n\u22121 (\u03b4\u22121)/2 \u00b4 \u2264 N(n, \u03b4). Hence, min{|A|, |B|} \u2264N(n, \u03b4), when r \u0338= (\u03b4 \u2212 1)/2. We deal with the remaining case in which \u03b4 \u22611 (mod 2) and r = (\u03b4 \u22121)/2. If m \u2264 `n\u22121 r \u00b4 , then |A| \u2264P(\u03b4\u22121)/2 i=0 `n i \u00b4 + ` n\u22121 (\u03b4\u22121)/2 \u00b4 \u2264N(n, \u03b4). If m > `n\u22121 r \u00b4 , then by de\ufb01nition of a quasi-sphere, A contains the sets {1}\u222aR, where R \u2208 `[2,n] r \u00b4 . We claim that 1 \u2208B for every B \u2208B such that |B| = r + 1. Suppose not, let R \u2286[n] \\ B such that |R| = r. Then A = {1} \u222aR \u2208A, and |A\u2206B| = 2r + 2 > \u03b4, giving a contradiction. It follows that all subsets of size r + 1 in B must contain the element 1, and hence the number of subsets of size r + 1 in B is at most `n\u22121 r \u00b4 . Hence, |B| \u2264 P(\u03b4\u22121)/2 i=0 `n i \u00b4 + ` n\u22121 (\u03b4\u22121)/2 \u00b4 \u2264N(n, \u03b4). This establishes the following: Theorem 4. For any \u03b4-pair ([n], A) and ([n], B), min{|A|, |B|} \u2264 (P\u03b4/2 i=0 `n i \u00b4 , if \u03b4 \u22610 (mod 2); P(\u03b4\u22121)/2 i=0 `n i \u00b4 + ` n\u22121 (\u03b4\u22121)/2 \u00b4 , if \u03b4 \u22611 (mod 2), with equality if A = B = 8 > < > : S\u03b4/2 i=0 `[n] i \u00b4 , if \u03b4 \u22610 (mod 2) S(\u03b4\u22121)/2 i=0 `[n] i \u00b4 \u222a {A \u222a{x} : A \u2208 ` [n]\\{x} (\u03b4\u22121)/2 \u00b4 }, if \u03b4 \u22611 (mod 2), where x is any \ufb01xed element of [n]. Corollary 1. F1(n, \u03b4) = F2(n, \u03b4); the size of an optimal (n, \u03b4)2-LP code is the same as the size of an optimal (n, \u03b4)1LP code. This solves Problem 3. 5.1 Restriction to Constant Weight Two set systems ([n], A) and ([n], B) are called cross-wise t-intersecting if |A \u2229B| \u2265t for all A \u2208A and B \u2208B. Lemma 3. Let ([n], A) and ([n], B) be the set systems of codes C0 and C1, respectively. Then (C0, C1) is an (n, \u03b4)2-LP code of constant weight w if and only if ([n], A) and ([n], B) are w-uniform and cross-wise (w \u2212\u03b4/2)-intersecting. Proof. Follows from the observation that ([n], A) and ([n], B) are cross-wise (w \u2212\u03b4/2)-intersecting if and only if for all A \u2208A and B \u2208B, |A \u2229B| \u2265w \u2212\u03b4/2 \u21d4 |A\u2206B| \u2264\u03b4 \u21d4 dH(\u03b9(A), \u03b9(B)) \u2264\u03b4. Hence, (C0, C1) is an (n, \u03b4)2-LP code. Let ([n], A) and ([n], B) be two set systems that are wuniform cross-wise t-intersecting, where t = w \u2212\u03b4/2. Suppose that A1, A2 \u2208A. Let B \u2208B, U \u2286A1 \u2229A2 \u2229B, and u = |U|. Then B must contain a further t \u2212u points from each of A1 \\U and A2 \\U. The remaining w\u2212(2t\u2212u) points in B are from the n \u2212(2t \u2212u) points not already contained in B. The total possible number of such blocks B is then   w \u2212u t \u2212u !2  n \u2212(2t \u2212u) w \u2212(2t \u2212u) ! . (3) \fIf u \u2264t \u22121, then for n large enough, (3) is less than M(n, k, t) = `n\u2212t w\u2212t \u00b4 , and hence |B| \u2264M(n, k, t). On the other hand, if u \u2265t, then |A1 \u2229A2| \u2265t, which implies A is t-intersecting. So we also have |A| \u2264M(n, k, t). This gives the following result. Theorem 5. Suppose ([n], A) and ([n], B) are w-uniform cross-wise t-intersecting set systems. Then for large enough n, min{|A|, |B|} \u2264 `n\u2212t w\u2212t \u00b4 , with equality if A = B = {A \u2208 `[n] w \u00b4 : T \u2286A}, where T is any \ufb01xed t-subset of [n]. Corollary 2. For n large enough, the size of an optimal (n, \u03b4)2-LP code of constant weight w is the same as the size of an optimal (n, \u03b4)1-LP code of constant weight w. This solves Problem 5 for n large enough. A solution for all n seems out of reach at the moment. 6. IMPLEMENTATION The encoding schemes introduced can be easily implemented with a look-up table, since our codes are explicit and constructible in polynomial time. However, it is also possible to encode and decode algorithmically, removing the need to store a look-up table of size 2k when the souce is H(k). We illustrate this with an encoding/decoding algorithm for the optimal (n, \u03b4)1-LP code from Section 4. We assume that the source is H(k). First we de\ufb01ne ranking and unranking algorithms. Given a set of objects X, rank and unrank are functions f : X \u2192 {0, . . . , |X|\u22121} and g : {0, . . . , |X|\u22121} \u2192X, such that f and g are bijections satisfying g(f(x)) = x for all x \u2208X. Computing f is ranking, and computing g is unranking. E\ufb03cient ranking and unranking algorithms for H(n, w) (equivalent to w-subsets of an n-set) are well known (see, for example, [?]). A simple and e\ufb03cient ranking and unranking algorithm for H(n, w) using the co-lexicographic ordering is described below. rank(w, x) { {t1, . . . , tw} = supp(x); r = 0; for i = 1 to w r = r + ` ti\u22121 k+1\u2212i \u00b4 ; return r; } unrank(n, w, r) { x = n; for i = 1 to w { while ` x w+1\u2212i \u00b4 > r x = x \u22121; ti = x + 1; r = r \u2212 ` x k+1\u2212i \u00b4 ; } return \u03b9({t1, . . . , tw}); } We now give a high-level overview of the method. We interpret x \u2208H(k) as an integer in the interval [0, 2k \u22121] in the natural way. Let Ni = Pi j=0 `n j \u00b4 \u22121, for 0 \u2264i \u2264n. By convention, N\u22121 = \u22121. An x \u2208[Nt\u22121 + 1, Nt] is encoded as an element of H(n, t) by unranking x\u2212Nt\u22121 \u22121. To decode a received y \u2208H(n), simply rank y. More precisely, to encode a source x \u2208H(k) to a codeword y, the encoder performs the following steps: Step 1: Find t such that Nt\u22121 < x \u2264Nt. Step 2: m = x \u2212Nt\u22121 \u22121. Step 3: y = unrank(n, t, m). The decoding algorithm is even simpler. To decode y \u2208H(n) to a source vector x, the decoder performs the following steps: Step 1: Compute w = wH(y). Step 2: x = rank(w, y). An optimal constant weight (n, \u03b4)1-LP code can be similarly implemented. 7. NOT ALL LIMITED WEIGHT CODES ARE EQUAL In [?], Stan and Burleson introduced the following bus encoding scheme. De\ufb01ne an m-limited weight code (m-LWC) of length n to be a code C \u2286\u222am w=0H(n, w). Suppose that the number of source states to be transmitted across a bus is 2k and that the source states are to be encoded with a code of length n. Let m be the smallest integer such that m X i=0   n m ! \u22652k. Stan and Burleson claimed that an m-LWC code of length n, comprising as codewords all elements in \u222am\u22121 w=0 H(n, w) and the remaining 2k \u2212Pm i=0 ` n m \u00b4 codewords from H(n, m), is \u201coptimal in the sense that any other code with the same length cannot have better statistical properties for low power.\u201d This statement is true since we have shown in Section 4 that an optimal (n, \u03b4)1-LP code is an m-LWC, but we must exercise caution when choosing the m-LWC, since not every m-LWC is optimal, as can be seen in the example below. Example 1. Let k = 5 and n = 6. We give two 3-LWCs of length n and size 2k with di\ufb00erent diameters. Each of the two 3-LWCs of length n and size 2k contains all 22 elements in \u222a2 w=0H(6, w). The \ufb01rst 3-LWC containing the 10 additional codewords 000111 001011 010011 100011 001110 111000 110100 101100 011100 110001 has diameter six, and the second 3-LWC containing the 10 additional codewords 100011 100101 100110 101001 101010 101100 110001 110010 110100 111000 has diameter \ufb01ve. Our results in Section 4 give m-LWCs that are optimal, for every length. \f8."
+                },
+                {
+                    "url": "http://arxiv.org/abs/0712.2619v1",
+                    "title": "A New Lower Bound for A(17,6,6)",
+                    "abstract": "We construct a record-breaking binary code of length 17, minimal distance 6,\nconstant weight 6, and containing 113 codewords.",
+                    "authors": "Yeow Meng Chee",
+                    "published": "2007-12-17",
+                    "updated": "2007-12-17",
+                    "primary_cat": "cs.IT",
+                    "cats": [
+                        "cs.IT",
+                        "cs.DM",
+                        "math.CO",
+                        "math.IT",
+                        "E.4"
+                    ],
+                    "main_content": "Introduction Let A(n, d, w) denote the maximum possible number of codewords in a binary code of length n, minimal distance d and constant weight w. The Nordstrom-Robinson code N16 of length 16, minimal distance 6, and containing 256 codewords has weight enumerator 1 + 112x6 + 30x8 + 112x10 + x16. Hence, taking all the codewords of weight 6 in N16 gives a constant weight code that shows A(16, 6, 6) \u2265112. Since A(17, 6, 6) \u2265A(16, 6, 6), we also have A(17, 6, 6) \u2265112. This is in fact the best lower bound on A(17, 6, 6) known [2]. In this note, we give the \ufb01rst improvement on the lower bound for A(17, 6, 6) since that implied by the 1967 result of Nordstrom and Robinson [3]. We exhibit a new binary code C of length 17, minimal distance 6, constant weight 6, and containing 113 codewords, showing A(17, 6, 6) \u2265113. Our code has no particular structure (its automorphism group is trivial) and is obtained through a combination of search techniques involving simulated annealing [4], length-reduction [1], and local optimization. 1 \fThe support supp(x) of a codeword x = (x1, . . . , xn) is the set of indices of its non-zero coordinates, that is, supp(x) = {i | xi \u0338= 0}. The supports of the codewords in C are listed in the next section. 2 The Code 0 1 2 3 6 15 0 1 2 4 11 16 0 1 2 7 8 9 0 1 2 10 12 13 0 1 3 4 8 10 0 1 3 5 7 12 0 1 3 9 13 16 0 1 4 6 7 13 0 1 5 6 10 16 0 1 5 8 11 13 0 1 6 9 11 12 0 1 7 10 11 15 0 1 8 12 14 15 0 2 3 4 9 12 0 2 3 5 8 16 0 2 3 7 11 13 0 2 4 5 7 10 0 2 4 8 13 15 0 2 5 6 9 13 0 2 5 11 14 15 0 2 6 7 12 16 0 2 6 8 10 11 0 3 4 5 6 11 0 3 4 7 14 16 0 3 5 10 13 15 0 3 6 7 9 10 0 3 6 8 12 13 0 3 8 9 11 15 0 3 10 11 12 14 0 4 5 12 13 14 0 4 6 8 9 16 0 4 6 10 12 15 0 4 7 8 11 12 0 4 9 10 11 13 0 5 6 7 8 15 0 5 8 9 10 14 0 5 9 12 15 16 0 6 11 13 14 16 0 7 8 10 13 16 0 7 9 13 14 15 1 2 3 4 5 13 1 2 3 7 10 14 1 2 3 8 11 12 1 2 4 6 9 10 1 2 4 7 12 15 1 2 5 6 7 11 1 2 5 8 10 15 1 2 5 12 14 16 1 2 6 8 13 16 1 2 9 11 13 15 1 3 4 6 12 16 1 3 4 7 9 11 1 3 5 6 8 14 1 3 5 11 15 16 1 3 6 10 11 13 1 3 7 8 13 15 1 3 9 10 12 15 1 4 5 7 8 16 1 4 5 9 14 15 1 4 5 10 11 12 1 4 6 8 11 15 1 4 8 9 12 13 1 4 10 13 14 16 1 5 6 12 13 15 1 5 7 9 10 13 1 6 7 8 10 12 1 6 7 9 15 16 1 7 11 12 13 16 1 8 9 10 11 16 2 3 4 6 7 8 2 3 4 10 11 15 2 3 5 6 10 12 2 3 5 7 9 15 2 3 6 9 11 16 2 3 8 9 10 13 2 3 12 13 15 16 2 4 5 6 15 16 2 4 5 8 9 11 2 4 6 11 12 13 2 4 7 9 13 16 2 4 8 10 12 14 2 5 7 8 12 13 2 5 10 11 13 16 2 6 7 10 13 15 2 6 8 9 12 15 2 7 8 11 15 16 2 7 9 10 11 12 2 9 10 14 15 16 3 4 5 8 12 15 3 4 5 9 10 16 3 4 6 13 14 15 3 4 7 10 12 13 3 4 8 11 13 16 3 5 6 7 13 16 3 5 7 8 10 11 3 5 9 11 12 13 3 6 7 11 12 15 3 6 8 10 15 16 3 7 8 9 12 16 4 5 6 7 9 12 4 5 6 8 10 13 4 5 7 11 13 15 4 6 7 10 11 14 4 7 8 9 10 15 4 11 12 14 15 16 5 6 8 11 12 16 5 6 9 10 11 15 5 7 9 11 14 16 5 7 10 12 14 15 5 8 13 14 15 16 6 7 8 9 11 13 6 9 10 12 13 16 8 10 11 12 13 15 2"
+                }
+            ],
+            "Jingwen Li": [
+                {
+                    "url": "http://arxiv.org/abs/2201.12833v1",
+                    "title": "Word Segmentation and Morphological Parsing for Sanskrit",
+                    "abstract": "We describe our participation in the Word Segmentation and Morphological\nParsing (WSMP) for Sanskrit hackathon. We approach the word segmentation task\nas a sequence labelling task by predicting edit operations from which\nsegmentations are derived. We approach the morphological analysis task by\npredicting morphological tags and rules that transform inflected words into\ntheir corresponding stems. Also, we propose an end-to-end trainable pipeline\nmodel for joint segmentation and morphological analysis. Our model performed\nbest in the joint segmentation and analysis subtask (80.018 F1 score) and\nperformed second best in the individual subtasks (segmentation: 96.189 F1 score\n/ analysis: 69.180 F1 score).\n  Finally, we analyse errors made by our models and suggest future work and\npossible improvements regarding data and evaluation.",
+                    "authors": "Jingwen Li, Leander Girrbach",
+                    "published": "2022-01-30",
+                    "updated": "2022-01-30",
+                    "primary_cat": "cs.CL",
+                    "cats": [
+                        "cs.CL"
+                    ],
+                    "main_content": "Introduction This paper describes our efforts to solve the tasks proposed by the WSMP hackathon1 associated with Forum for Information Retrieval Evaluation (FIRE) 2021.2 The hackathon focuses on two Sanskrit NLP problems, namely word segmentation and morphological analysis. These tasks are precursors to downstream NLP tasks like dependency parsing. However, characteristics of the Sanskrit language make them non-trivial and even challenging on their own. Sanskrit had a rich oral tradition, knowledge and wisdom were passed on through recitation long before the scriptures could be consolidated in written form. It is then not surprising that its orthography faithfully re\ufb02ects the euphonic modi\ufb01cations 1https://sanskritpanini.github.io/ 2http://fire.irsi.res.in/fire/2021/ home made to reduce articulatory efforts during continuous speech or chanting, which requires stricter rhythm. This phonetic smoothing process over consecutive words, termed sandhi, comes at the cost of easy and unambiguous identi\ufb01cation of word boundaries and word forms. Just by looking at the resulting phonetic stream, which can be written in arbitrary syllabic script or transliteration scheme, it is not enough to directly identify the sequence of words as there exist many lexically and phonetically valid parses. Consider the following simple example taken from Guhe (2008), transliterated from Devan\u00af agar\u00af \u0131 script to IAST:3 (1) kena kena kena path\u00af a path\u00af a path\u00af a bhav\u00af ansakhy\u00af a bhav\u00af an sakhy\u00af a bhav\u00af an sakhy\u00af a sah\u00af agacchat saha \u00af agacchat saha agacchat The italicised \ufb01rst line in 1 is the sandhied sequence, followed by two segmentations with their difference underlined. Note that the spaces (hiatus) in the original sentence do not fully correspond to word boundaries. For someone with knowledge of the language, these are the \ufb01rst two segmentations that come into mind because a human can look at this sentence holistically and \ufb01rst determine the in\ufb02ected verb hidden in the sandhied chunk sah\u00af agacchat. To do so, this person must already have an idea what the sentence is about, combining knowledge of a lexicon, possible sound changes and a grammar. However, a machine can only process the sequence linearly and operate without such linguistic knowledge unless explicitly provided or learned through training. Therefore, determining where sandhi has occurred is already a challenge. But suppose we know for sah\u00af agacchat, \u00af a is the result of applying the sandhi rule of merging similar word-initial and -\ufb01nal vowels into its long form. Here it is still 3International Alphabet of Sanskrit Transliteration arXiv:2201.12833v1  [cs.CL]  30 Jan 2022 \fnon-trivial to obtain the word forms because in the reverse process, \u00af a can be split into any combination of \u00af a and a. The possible positions where sandhi could have occurred and the different ways of splitting amount to a multitude of phonetically possible segmentations. Hence the word segmentation task is reformulated as identifying the semantically most valid segmentation for the given sentence (Krishna et al., 2017). For simplicity, here we only discuss the semantically valid splits for the long vowel \u00af a, in particular, \u00af a + a and a + a. From the segmentations we obtain the following morphological analyses:4 (2) a. ... sakhy\u00af a saha \u00af agacchat a-\u00af agam-t PST-come-3SG.PST.PAR \u2019came with (a male/female) friend\u2019 b. ... sakhy\u00af a saha agacchat a-gam-t PST-go-3SG.PST.PAR \u2019went with (a male/female) friend\u2019 Without any context, 2a and 2b are equally valid. Example 2 also shows that if word segmentation and morphological parsing are performed separately and sequentially, errors from segmentation will propagate into analysis. Moreover, as Krishna et al. (2020) pointed out, Sanskrit is a language characterised by a high degree of syncretism and homonymy, therefore the morphological analysis for a given surface word form is usually not unique. The in\ufb02ected noun sakhy\u00af a in our example sentence, for instance, obtains the following analyses: (3) a. ... sakhy\u00af a sakhi-\u00af a friend.M-INS.SG saha ... \u2019with (a male) friend\u2019 b. ... sakhy\u00af a sakh\u00af \u0131-\u00af a friend.F-INS.SG saha ... \u2019with (a female) friend\u2019 Since the same word is analysed in 3a and 3b with two stems, sakhi and sakh\u00af \u0131, we have a case of 4Glossings used here deviates slightly from the Leipzig Glossing Rules, accessible from: https://www.eva.mpg.de/lingua/resources/ glossing-rules.php. PAR refers to Parasmaipada homonymy. A more extreme case of homonymy is demonstrated by \u00b4 sis .y\u00af ah . , which is the nominative plural of the masculine noun \u00b4 sis .ya (meaning student) and also the second person singular optative parasmaipada of the verb \u00b4 s\u00af as (meaning to command). From the feminine noun stem nau (meaning ship, vessel), n\u00af avah . is the resulting form for: (a) ablative singular (b) genitive singular (c) nominative plural (d) accusative plural and (e) vocative plural, hence an example of syncretism. Notice that the above morphological analyses consider just the individual word without contextual information. It does not realise that bhav\u00af an is actually a form of the honori\ufb01c pronoun bhavat in the second person, so the verb in 2 is actually intended for the second person, conforming to the in\ufb02ection paradigm for the third person. This may also pose challenges to downstream tasks if morphological tags were to be used as inputs. For the simple example sentence in 1, ambiguity at both segmentation and analysis level already produces four distinct readings.5 More ambiguities are expected as the length of the input sentence increases. Using this toy example, we also showed that neither word segmentation nor morphological analysis is to be seen as separated from semantics. Although generation of a sentence is deterministic with a given sequence of stems, their associated morphological properties and a set of sandhi rules, the reverse is non-deterministic at every step. This observation probably encourages postponing decisions at task-level and combining more contextual information to jointly resolve ambiguities. The rest of this paper is structured as follows: Section 2 gives general descriptions of the three tasks of the hackathon, Section 3 introduces the dataset provided, Section 4 introduces our experiment with transliteration schemes, Section 5 describes our approaches to each of the three tasks in terms of data preprocessing and model structure, Section 6 outlines our training and tuning process and the results we obtained. Also, Section 6 contains an extensive error analysis. Finally, in Section 7 we discuss our \ufb01ndings and conclude this paper. 2 Task Descriptions We brie\ufb02y introduce the three tasks here, for more detailed descriptions please visit the of\ufb01cial 5See translations in Example 2 \fhackathon website.6 2.1 Word segmentation For the \ufb01rst task, a sandhied sentence in the form of a single string is given as input, with unambiguous word boundaries marked by spaces. The desired output is a list of words in the sentence in unsandhied form, as shown in Ex. 1, taken from the training dataset. Different from other languages where tokenization is also non-trivial, like Japanese, in Sanskrit, another layer of complexity is added to retrieve the correct segmentation: Tokenized outputs also have to undergo a reverse process of the euphonic changes. The task, therefore, is two-fold: \ufb01nding correct word boundaries and retrieving unsandhied word forms. Input: bhavati c\u00af atra Output: [\u2019bhavati\u2019, \u2019ca\u2019, \u2019atra\u2019] Example 1: Input-output pair for task 1 2.2 Morphological parsing In the second task, the segmentation ground-truth is given as input. Our goal is to predict the stem and morphological tag for each word. The morphological tags used in the provided dataset follow the standards of the Sanskrit Heritage Reader (Goyal and Huet, 2016), where depending on the grammatical category of a word, different numbers of labels are combined to form a single tag string. A \ufb01nite verb receives a morphological tag with at most six labels, corresponding to 1) mood 2) tense 3) conjugation class 4) voice 5) number and 6) person, while a noun receives a triple consisting of 1) case 2) number and 3) gender as tag. Other indeclinables (avyaya) receive their own POS tags as morphological tags. Input: bhavati ca atra Output: [(\u2019bh\u00af u\u2019, \u2019pr. [1] ac. sg. 3\u2019), (\u2019ca\u2019, \u2019conj.\u2019), (\u2019atra\u2019, \u2019adv.\u2019)] Example 2: Input-output pair for task 2 2.3 Combined segmentation and analysis The third task combines task 1 and task 2: Given a sandhied sentence as a single string, we predict 6https://sanskritpanini.github.io/ tasks.html the segmented unsandhied words with their full morphological analysis: stem and morphological tag. The output is a list of triples (word, stem, tag) for each sentence. Input: bhavati c\u00af atra Output: [(\u2019bhavati\u2019, \u2019bh\u00af u\u2019, \u2019pr. [1] ac. sg. 3\u2019), (\u2019ca\u2019, \u2019ca\u2019, \u2019conj.\u2019), (\u2019atra\u2019, \u2019atra\u2019, \u2019adv.\u2019)] Example 3: Input-output pair for task 3 3 Dataset The WSMP dataset provided by the task organizers is separated into training and development sets. The organizers do not release the test set. The training set consists of 90 000 sentences and the development set contains 10 332 sentences. Ground truths for all three tasks are provided alongside the input sentences. Apart from the main datasets, each sentence is also paired with a graphML7 \ufb01le which stores all valid parses of the input sentence generated by Sanskrit Heritage Reader (Goyal and Huet, 2016). These sentences, as described by the organizers, are taken from the Digital Corpus of Sanskrit8 (DCS) (Hellwig, 2010\u20132021), a corpus for digitalised Sanskrit manuscripts of various domains of knowledge and styles. 4 Transliteration There have been opposing views on the effect of using different unit representations in language modeling. Huet (2013) argues that transliteration is completely irrelevant to Sanskrit text processing and the segmentation problem, while Adiga et al. (2021), in a pioneer large-scale study of Sanskrit ASR (Automatic Speech Recognition), favors phonetic based graphemic representations over scripts without one-to-one correspondence to phonemes. For our own investigation, we experimented with two transliteration schemes for all three tasks to see whether the use of different transliteration schemes has in\ufb02uence over model performance. The \ufb01rst one is the given input in IAST, the other is the internal representation9 used by Oliver Hellwig for 7cf. Krishna et al. (2017) for structural speci\ufb01cations of the graphML \ufb01les 8Accessible from: https://github.com/ OliverHellwig/sanskrit/tree/master/dcs/ data 9Character mapping from IAST is de\ufb01ned in: https://github.com/OliverHellwig/ sanskrit/blob/master/papers/2018emnlp/ code/data_loader.py \fthe DCS, which is similar to SLP1.10 Different from IAST, which uses two Unicode characters to represent diphthongs and aspirated consonants, the second transliteration scheme uses one Unicode character, establishing one-to-one correspondence between transliterated character and phonemic unit. 5 Method 5.1 Word segmentation We formulate the word segmentation task as a sequence labelling problem: Given a character sequence s = c1c2 . . . cn, predict exactly 1 edit operation for each character. An edit operation locally modi\ufb01es the input sequence. By applying the predicted edit operations to each character of the source sentence, we arrive at the predicted sequence s\u2032 = c\u2032 1c\u2032 2 . . . c\u2032 n. Then, we retrieve the predicted segmentation of the input sentence by splitting s\u2032 at predicted spaces. In this way, word boundaries and unsandhied word form are determined at the same time. Data preprocessing Our data preprocessing has to achieve two goals: (a) extract the necessary edit operations and (b) align the edit operations with the characters. First, we remove all datapoints where the source sentence is longer than the target sequences, as these represent corrupted datapoints. This affects \u22481% of the dataset. Then, we align each source sequence with the respective target sentence by minimising Levenshtein distance. The aligned sequences decompose into aligned and non-aligned chunks. Aligned chunks are transformed into each other by simple copy operations. Non-overlapping chunks are kept as edit rules, which are represented as tuples of ngrams (SOURCE, TARGET). These rules denote what source ngram is transformed into which target ngram. Each source character in SOURCE gets the corresponding rule as label. For inference, we contract consecutively predicted equal rules. To ensure SOURCE contains at least 1 character, we concatenate TARGETs of rules with empty SOURCE to the TARGET of the left rule. Finally, we introduce a special rule for the case where we simply insert a space after a character. COPY and INSERT-SPACE rules abstract away from the concrete character, thereby reducing the total number of rules. The sequences of edit operations generated for the training set are used as ground-truth 10Sanskrit Library Phonetic basic encoding scheme Figure 1: Visualisation of our word segmentation model (Sec. 5.1): Character convolutions of different \ufb01lter sizes are followed by downsampling and BiLSTM. Finally, rules are predicted for all input characters. Also note the residual connections. labels to train our model. In total, we extract 166 rules from the dataset without transliteration and 187 rules when transliterating Sanskrit inputs. An example of the result of our data preprocessing is in Tab. 4. Model For predicting the label sequence, we use a model that is very similar to the model proposed by Hellwig and Nehrdich (2018): First, we extract ngram features from the input character embeddings by 1d-convolutions with \ufb01lter sizes 2 to 8. Inputs are 0-padded to ensure that sequence lengths remain the same after convolution. Then, we apply a residual block to the ngram features. We concatenate all ngram features and project them to a lower dimension. Finally, we run a 2-layer BiLSTM with residual connection on the sequence and predict the labels for each character. A visualisation of the model is in Fig 1. 5.2 Morphological parsing We also formulate the morphological parsing task as a sequence-labelling task with labels being rules extracted from the train data. These rules describe how to transform a word into the word\u2019s stem. Mor\fp \u00af u r v a s y e t i d i n \u00af a n t a r a k r . t a s y a C C C C C C C a i C C C C C C a a C C C C I C C C C C C C Table 4: Example of our data preprocessing for task 1: Given the input sentence \u201cp\u00af urvasyeti din\u00af antarakr .tasya\u201d with corresponding ground truth segmentation \u201cp\u00af urvasya iti dina antara kr .tasya\u201d, we copy unchanged characters and predict Sandhi rules at word boundaries, if necessary. \u201cC\u201d mean copy and \u201cI\u201d mean insert space (after character). For example, we segment \u201cdin\u00af antara\u201d by predicting the Sandhi rule (\u00af a \u2192a a) for the \ufb01rst \u201c\u00af a\u201d. phological tags are optionally included as part of the rules. If the rules include morphological tags, our model can jointly predict the stem and the tag of a word. Otherwise, the model uses rules only for predicting stems and predicts tags separately. Predicting tags separately turns the task into a multitask learning problem (cf. Gupta et al. (2020)). In general, we \ufb01nd that predicting tags separately yields better performance. One possible reason is that predicting tags separately allows for correct prediction of the tag, even if the wrong rule is predicted for the word. This scenario can arise when the same surface word form has multiple valid morphological analyses. Dataset preprocessing To extract rules, we process the (WORD, STEM) pairs individually. We align WORD and STEM by matching their longest common in\ufb01x. Then, for both WORD and STEM, we extract a pre\ufb01x and a suf\ufb01x. The pre\ufb01x corresponds to the substring preceding the matched in\ufb01x. The suf\ufb01x corresponds to the substring following the matched in\ufb01x. Both pre\ufb01x and suf\ufb01x can be empty. The \ufb01nal rule is thus a quadruple (pre\ufb01xWORD, suf\ufb01xWORD, pre\ufb01xSTEM, suf\ufb01xSTEM). Given a rule and a word, we can apply the rule if the word starts with pre\ufb01xWORD and ends with suf\ufb01xWORD. If this is the case, we can generate the predicted stem by substituting pre\ufb01xWORD with pre\ufb01xSTEM and substituting suf\ufb01xWORD with suf\ufb01xSTEM. An example of alignment and extraction of suf\ufb01x and pre\ufb01x is in Fig. 2. Model For predicting rules (and optionally morphological tags), we \ufb01rst encode each token separately by character convolutions with \ufb01lter sizes 2 to 6. We 0-pad tokens so that their lengths remain unchanged after convolution. The extracted features are combined into a single token embedding by feature-wise max pooling. We also experimented with using the concatenated \ufb01nal hidden states of a BiLSTM run on the character features as token embeddings. However, we found this to perform worse. After having thus calculated an embedding for Figure 2: Visualisation of our stem rule extraction method for task 2 (morphological parsing, see Sec. 5.2): We align the (transliterated) words \u201csaH\u201d and \u201ctad\u201d, where \u201ctad\u201d is the stem of \u201csaH\u201d, by their longest common in\ufb01x, \u201ca\u201d. Then, \u201cs\u201d becomes pre\ufb01xWORD and \u201cH\u201d becomes suf\ufb01xWORD. Likewise, \u201ct\u201d becomes pre\ufb01xSTEM and \u201cd\u201d becomes suf\ufb01xSTEM. each token, we run a sentence level BiLSTM on the token embeddings. The reason is that we want to include context information. However, we include a residual connection by adding each token embedding to the respective BiLSTM output. In this way, the resulting token embeddings encode both subword and contextual information. Rules and morphological tags are predicted from these contextualised token embeddings. A visualisation of the model is in Fig 3. Note that our model can be seen as a modi\ufb01ed version of ELMo (Peters et al., 2018). For inference, we check which rules are applicable to a given token. A rule is applicable to a token iff the token starts with pre\ufb01xWORD and ends with suf\ufb01xWORD. Then, we take the applicable rule with highest prediction probability. If no rules are applicable, we predict the unaltered word. 5.3 Combined segmentation and analysis We propose a pipeline approach to combined segmentation and analysis that is end-to-end trainable for both predicted segmentations and predicted stems and morphological tags: First, we apply the same model as for word segmentation to predict the words (see Sec. 5.1). Then, we convert the predicted labels into token boundaries (i, j), where i \fFigure 3: Visualisation of our morphological parsing model (Sec. 5.2): We encode each word by character convolutions of different \ufb01lter sizes, which are maxpooled to arrive at embeddings for each word. Then, we capture context information by running a BiLSTM on the token embeddings. Finally, rules are predicted for all input words. Note that residual connections are not shown. denotes the index of the \ufb01rst character of the corresponding word in the unsegmented input string and j denotes the index of the last character. For each predicted token, we calculate a token embedding by feature-wise max pooling the corresponding outputs of the BiLSTM model as described in Sec. 5.1. The corresponding outputs are all timesteps of the input sequence starting with index i and ending with index j. Finally, we predict stem rules and morphological tags from the token embeddings. In this case, we always predict stems and morphological tags separately, because this yields better performance in our setup. As noted in Sec. 5.2, stem rules and morphological tags can be predicted jointly. Note in particular that we need to predict the segmentation \ufb01rst in order to proceed with the calculation of token embeddings. Therefore, our approach implements a pipeline. However, by reusing hidden states of the BiLSTM, the predictions for stems and morphological tags are differentiable w.r.t. the input characters. Thereofore, the whole model remains end-to-end trainable. Dataset preprocessing is the same as for task 1 (see Sec. 5.1) and task 2 (see Sec. 5.2). 6 Results 6.1 Training All models are trained by minimising the crossentropy-loss between prediction probabilities and one-hot encoded target labels. We use a vanilla SGD optimizer without weight decay or momentum. The learning rate is set according to a one cycle learning rate scheduler (Smith and Topin, 2019). For our models, we found the combination of SGD and one cycle learning rate scheduler to perform better than Adam (Kingma and Ba, 2015). All models are implemented in PyTorch (Paszke et al., 2019). 6.2 Hyperparameter tuning In order to \ufb01nd the best con\ufb01guration for our models, we perform extensive hyperparameter tuning using the Ray Tune library (Liaw et al., 2018).11 An overview over the tuned hyperparameters and their possible values is in the appendix. We make the following observations: \u2022 Larger embedding sizes always work better. This is surprising, as we even see improvements when the embedding dimension is already larger than the vocabulary size. \u2022 Wider models perform better in all evaluated cases. However, this is not surprising given the abundance of training data. We do not evaluate whether deeper models also perform better. \u2022 The performance of our models already peaks after 15 epochs. We do not observe improved performance when training longer. \u2022 Both the use of transliteration and larger vs. smaller batch sizes do not show a clear trend as to which is better. Instead, this decision is dependent on the particular subtask. \u2022 For predicting stem rules for morphological parsing, including all rules generated from the training data (i.e. not discarding infrequent rules) and predicting morphological stems separately from tags yields better performance. 6.3 Hackathon results Our approaches obtained the second place at the FIRE 2021 hackathon12 in both the word segmen11https://www.ray.io/ray-tune 12https://sanskritpanini.github.io/ index.html \ftation track and the morphological parsing track. Our pipeline model described in Sec. 5.3 obtained the \ufb01rst place in the combined segmentation and analysis track. Since the test set is not released, we report of\ufb01cial results from the hackathon\u2019s codalab page.13 Results for all three tracks are in Tab. 5a, Tab. 5b, and Tab. 5c. Generally, the top results are very close to each other. Due to the black-box nature of the evaluation, we cannot assess whether all differences are signi\ufb01cant. However, we observe that our end-toend trainable pipeline yields best results for the third track, despite our models do not yield best performance on the individual tasks. One possible conclusion is that end-to-end training in this case makes the pipeline more robust. However, the pipeline can still bene\ufb01t from our distinct approaches to the two individual tasks. 6.4 Error analysis Since the test set is not released, we evaluate our model performance on dev data. The scoring script we use is taken from the starting kit provided by the task organisers, which differs from the actual script used for the hackathon14. Segmentation errors For word segmentation we report errors of the following types: \u2022 Where multiple splits are possible, the model does not pick the correct split. This includes not splitting where it should and oversplitting with the INSERT-SPACE operation, which is the most common error type. Errors of this type are expected since not all ambiguities can be resolved. \u2022 The edit operations extracted are themselves problematic due to incorrect ground-truths. For instance, we extracted rules that map single characters to the empty string, which sometimes causes weird vanishing characters in the prediction. Upon inspection, we found that some of the ground-truths are marked with this problem, so we may have extracted such rules from corrupt datapoints in the training data. 13https://competitions.codalab.org/ competitions/35744#results, as of February 1, 2022 14The updated scoring script can be accessed from https://drive.google.com/drive/folders/ 1hxeB_t5ymKsiLm9XQuN4gvf_OS4yzRYy \u2022 We observe false errors due to incorrect ground-truths. The gold segmentations can sometimes be incomplete, either characters or entire chunks of the input sentence are missing. Analysis errors For predicting stems and morphological tagging, our T2 model evaluated on dev set produces 65.51% full match for the sentences. Out of the 68 094 words to be predicted, 92.51% are fully correctly predicted. And 87.69% of the erroneous cases are partially correct, which means either the stem or the tag is correctly predicted. Note that the tag is a single string combining multiple morphological properties and the evaluation we choose to report here uses strict comparison. The hackathon, however, uses a different way to compare the strings, which will be discussed later in 6.5. For morphological tagging, our predictions are wrong most of the time only by one or two subtags. For instance the model predicts \u201cm. sg. acc.\u201d instead of \u201cn. sg. acc.\u201d for the form deham (meaning body), only the gender is incorrect. Since gender is an inherent property of the noun, and according to the declension paradigm of masculine and neuter nouns ending in short vowel -a, both analyses are valid without contextual information or a lexicon. Similarly, for verbs, the conjugate class is associated with the root, we observe cases where only the class is incorrectly predicted. For example the verbs in one particular conjugation class, the tenth, are often confused with causatives. They are sometimes indistinguishable for human as well, hence our model predicted \u201cca. opt. ac. sg. 3\u201d where the correct analysis would be \u201copt. [10] ac. sg. 3\u201d. For recovering word stems, the model does not perform well when sound changes are involved, in particular the retro\ufb02exion of n and s to n . and s ., or one induced by an insertion of a past tense marker a between a preverb and the stem, like for abhidh\u00af av-, when a is inserted between the preverb abhiand the stem dh\u00af av, it changes to ahbyadh\u00af av-, our model prediction fails to remove the tense marker a and restore i from y. 6.5 Reevaluating evaluation Since the hackathon is essentially a competition, it is necessary to look at how the scores are calculated, a more \ufb01ne-grained way of evaluation may change the \ufb01nal results depending on the models used by the teams. As mentioned before, the hackathon \fT1 F1 Score T1 Precision T1 Recall T1 Task Score 1 hrishikeshrt 97.478 (1) 97.580 (1) 97.439 (1) 97.439 (1) 2 Ours 96.189 (2) 96.439 (2) 96.061 (2) 96.061 (2) 3 jivnesh 95.436 (3) 95.793 (3) 95.237 (3) 95.237 (3) 4 maheshak 82.752 (4) 82.223 (4) 83.646 (4) 83.646 (4) 5 Asha 51.470 (5) 52.177 (5) 51.525 (5) 51.525 (5) (a) Results for T1 T2 F1-Score T2 Precision T2 Recall T2 Task Score 1 jivnesh 69.327 (1) 60.410 (1) 85.467 (1) 91.307 (1) 2 Ours 69.180 (2) 60.259 (2) 85.317 (2) 90.023 (2) 3 hrishikeshrt 42.644 (3) 52.513 (3) 39.234 (5) 48.429 (6) 4 maheshak 42.644 (3) 39.234 (4) 52.513 (3) 67.123 (3) 5 Asha 38.190 (4) 33.187 (5) 47.310 (4) 62.654 (4) 6 Manas_P_P 27.950 (5) 24.017 (6) 35.384 (6) 58.265 (5) (b) Results for T2 T3 F1 Score T3 Precision T3 Recall T3 Task Score 1 Ours 80.018 (1) 80.252 (1) 79.906 (1) 88.404 (1) 2 jivnesh 78.762 (2) 79.065 (2) 78.591 (2) 87.440 (2) 3 hrishikeshrt 63.657 (3) 63.603 (3) 63.781 (3) 81.585 (3) 4 maheshak 24.026 (4) 16.579 (4) 46.403 (4) 69.606 (4) (c) Results for T3 uses a different way to compare strings. Instead of checking for string equality, they used a counterbased method to count the characters that occurred in both the ground-truth and the model prediction. Although this measure seems more lenient, it is not very accurate and can create more confusion. To illustrate the point, consider the made-up example of \u201copt. [1] ac. sg. 3\u201d and \u201copt. [3] ac. sg. 1\u201d, according to their scoring script, this prediction is considered fully correct, when the third person is predicted as \ufb01rst person15. On the contrary, \u201cca. opt. ac. sg. 3\u201d and \u201copt. [10] ac. sg. 3\u201d are considered different when they are actually more similar to each other than the previous pair. For nouns, consider \u201cf. du. abl.\u201d and \u201cm. pl. dat.\u201d, none of the subtags are correctly predicted, but 8 out of 11 characters are counted as the same16, merely because there happen to be overlapping characters in different morphological categories. This would imply that locative (loc.) and vocative (voc.) are considered more similar than nominative (nom.)and vocative (voc.), while the truth is the exact opposite. In this regard, this way of evaluation is not very 15the conjugate class is also wrong, but is of lesser importance 16including dots and spaces informative. When models achieve similar high scores, it is also interesting to know on what they fail and whether model performance is consistent over the \"dif\ufb01cult\" cases. To better understand individual model performance, we propose to subdivide the test dataset into multiple sets, each sharing some linguistically-oriented properties. For instance, an expert in Sanskrit can compose lists of hard-toanalyse forms categorised into forms with reduplication, ambiguous forms, idiosyncratic forms, forms with nontransparent vowel gradation etc. Subsets of the test set can be automatically created from there on. It would also be useful to have a set testing the models on out-of-vocabulary (OOV) forms of known roots and unseen roots. In doing so, we have a more \ufb01ne-grained qualitative evaluation of the models. The results would be more informative especially when model structures are not speci\ufb01ed in the competition. 7 Discussion and"
+                }
+            ],
+            "Qadeer Khan": [
+                {
+                    "url": "http://arxiv.org/abs/2103.11204v1",
+                    "title": "Self-Supervised Steering Angle Prediction for Vehicle Control Using Visual Odometry",
+                    "abstract": "Vision-based learning methods for self-driving cars have primarily used\nsupervised approaches that require a large number of labels for training.\nHowever, those labels are usually difficult and expensive to obtain. In this\npaper, we demonstrate how a model can be trained to control a vehicle's\ntrajectory using camera poses estimated through visual odometry methods in an\nentirely self-supervised fashion. We propose a scalable framework that\nleverages trajectory information from several different runs using a camera\nsetup placed at the front of a car. Experimental results on the CARLA simulator\ndemonstrate that our proposed approach performs at par with the model trained\nwith supervision.",
+                    "authors": "Qadeer Khan, Patrick Wenzel, Daniel Cremers",
+                    "published": "2021-03-20",
+                    "updated": "2021-03-20",
+                    "primary_cat": "cs.CV",
+                    "cats": [
+                        "cs.CV"
+                    ],
+                    "main_content": "Introduction The success of current learning-based approaches on various tasks can be attributed to the availability of a large amount of labeled data. In most circumstances, data is being generated at a rate that is far superior than it is humanly possible to annotate it. Even if possible, it would be expensive, time-consuming, and prone to errors. Vision-based sensorimotor control tackles the following problem: Given an image captured by a camera, what should be the corresponding steering command? Controlling a car using supervised steering labels has shown fairly good performance [Bojarski et al., 2016, Chen et al., 2015, Codevilla et al., 2018b, Muller et al., 2006]. However, this performance is mostly limited to the domain in which the data was collected [Wenzel et al., 2018]. In order to generalize across a variety of conditions, a car \ufb01tted with driveProceedings of the 24th International Conference on Arti\ufb01cial Intelligence and Statistics (AISTATS) 2021, San Diego, California, USA. PMLR: Volume 130. Copyright 2021 by the author(s). by-wire needs to be driven by an expert driver for all such possible conditions. This is an in-feasible, inef\ufb01cient, and expensive solution for the following reasons: \u2022 Dynamic conditions such as lighting and weather are beyond human control. Hence, data collection for a speci\ufb01c scenario cannot be pre-planned. \u2022 The trained model may over\ufb01t to the speci\ufb01c position at which the camera is placed. Any deviation, in the position of the camera at test time, may lead to deteriorated performance. \u2022 Collection of supervised data requires drive by wire cars. Since many vehicles driven on the street by drivers do not have this option, we lose out on bootstrapping this huge treasure of valuable data. It may be argued that modern cars have the CAN bus which may be used to extract out important information including steering commands which can then be used for supervision. However, common people are not expected to modify their vehicle, if even allowed by the law of the country they reside in. Moreover, self-modi\ufb01cations also have implications on the warranty/insurance premiums. Therefore, we have referred to an expert as a dedicated driver who has the necessary license and permits from the relevant traf\ufb01c and regulatory authorities to drive a modi\ufb01ed vehicle on public roads. Traditional people are not expected to hold these permits. In contrast to supervised approaches, an alternative would be to deploy reinforcement learning (RL) based techniques, wherein instead of collecting expert supervised labels, a reward function is de\ufb01ned [Abbeel et al., 2007, Abbeel and Ng, 2004, Lillicrap et al., 2015]. The model learns a policy by a hit and trial method using the reward function as an indicator for updating the model\u2019s weights. However, this is an extremely data-inef\ufb01cient solution as it requires far more sessions to train a model when compared with supervised techniques. Moreover, car manufacturers would be unwilling to use an unstable learning-based arXiv:2103.11204v1  [cs.CV]  20 Mar 2021 \fSelf-Supervised Steering Angle Prediction for Vehicle Control Using Visual Odometry method for their autonomous functions. Recently, the authors of [Kendall et al., 2019] demonstrated an example of using deep RL for training a policy for maintaining the car on the road in a controlled environment. However, this approach requires an expert driver to intervene for any correction whenever the car attempts to deviate away from the road. In this paper, we propose a self-supervised approach which that does not suffer from the limitations of the supervised and RL-based approaches as described earlier. We use a camera setup, that can be placed at the front of any car model. Ideally, we would like to retrieve the precise ground truth poses of the cameras as the car moves along its trajectory. This could be done using precise GNSS positions or through localization against landmarks. However, acquiring accurate camera poses using these methods may not always be possible such as when a car enters a tunnel or drives through an urban canyon due to the unavailability of GNSS signals or landmarks [Weiss et al., 2011]. Therefore, as an alternative, we demonstrate that using visual odometry (VO) methods [Engel et al., 2017, Mur-Artal et al., 2015, Wang et al., 2017] to estimate the camera poses in a self-supervised manner can yield similar performance as with supervised labels. Note that, the VO methods can themselves be integrated with GNSS [Schleicher et al., 2009, Carlson, 2010], or inertial measurement units (IMUs) [Qin et al., 2017, von Stumberg et al., 2018] to improve the precision. However, to disambiguate the contribution of our method from additional sensor information, we only consider visual information, i.e. camera images. The overall framework works as follows. Given an image, we train a model which predicts the lateral movement of the relative translation vector between 2 camera frames. The relative translation vector predicted by our model at inference time can be considered as a virtual target indicating to the vehicle where to traverse next. Therefore, our framework can be seen as a waypoint-based control approach. However, the main difference to other approaches [Kaufmann et al., 2018, Agarwal et al., 2019] is that we do not require physical waypoints nor accurate GNSS reception at any given time. The authors of [Bansal et al., 2019] also use virtual waypoints to navigate a robot indoors, however, they assume that the odometry is perfectly known. Figure 1 shows the high-level framework of the proposed self-supervised vehicle control method. Our emphasis is on retrieving the relative translation vector which would be the same irrespective of which car the module is placed on. Hence, our approach of placing the camera module at the front of any vehicle can be scaled up to as many numbers of running cars on the road as possible without drive-by-wire limitations or regulatory restrictions. To this end, we make the following contributions: Figure 1: This \ufb01gure shows the high-level overview for vehicle control. An image is fed to a deep neural network which is trained in a self-supervised manner using camera poses obtained by visual odometry. The model predicts the relative lateral translation vector for the next camera position, hence indicating the direction of where the car ought to move next. This vector value is then passed on to the post-processing module to produce an appropriate steering command. 1. Ability to control a car in a self-supervised fashion without the need for an expert driver. This is achieved by using camera poses obtained through visual odometry methods. 2. We show how a combination of trajectory information from multiple runs leads to improved performance. 3. Using our proposed scalable solution, we demonstrate how our model can generalize even to variations in translation and rotation of the camera at inference time. Note that under dynamic weather and lighting conditions, VO methods may fail to furnish the correct camera poses. Therefore, we additionally show how our approach can be extended to such circumstances where VO may fail. Although there are many publicly available data sets [Geiger et al., 2012, Cordts et al., 2016, Maddern et al., 2017, Sun et al., 2020, Huang et al., 2018, Wenzel et al., 2020] focused on driving applications, they do not offer the possibility to test the online performance of self-driving cars necessary to evaluate our approach. As discussed in Section 3, this is because the online performance of embodied agents does not necessarily correlate with the of\ufb02ine error metrics [Anderson et al., 2018, Codevilla et al., 2018a, Ross et al., 2011]. To circumvent this problem, we report results tested on the CARLA simulator [Dosovitskiy et al., 2017]. It offers the facility to conduct an online evaluation of our method. \fQadeer Khan, Patrick Wenzel, Daniel Cremers 2 Method In this section, we \ufb01rst describe the self-supervised control framework we use. Our approach focuses on low-level lateral control for vehicles. The problem we want to tackle is the following: given an image from a monocular camera at any instance in time, we would like to determine the immediate steering command where the car should maneuver next. This can be done by predicting a translation vector indicating the direction from where the car currently is to where it ought to be. But how do we get this translation vector without explicit supervision? 2.1 Camera Pose Estimation The only requirement for our approach is to have camera poses estimated by visual odometry (VO). This can be done by placing a camera setup at the front of a car recording the images as the driver controls the car. A VO method run on a sequence of these images yields the corresponding camera poses. The 6DoF pose can be estimated using a state-of-the-art VO method such as DSO [Engel et al., 2017] or ORBSLAM [Mur-Artal and Tard\u00b4 os, 2017]. The camera poses are represented as transformation matrices Ti \u2208SE(3), which are transforming a point from the world frame into the camera frame. To get the relative pose between two cameras at time i and i+1, we can extract out the rotation matrix R and the translation vector t as follows: \u0014 R t 0 1 \u0015 = TiT\u22121 i+1, (1) where the rotation matrix R performs a rotation in Euclidean space which can be represented by yaw, pitch, and roll angles. The translation vector t denotes a translation around the x,y,z axes. Since the camera is rigidly attached to the car, the relative pose between any 2 camera frames also gives the relative information between the car positions at these 2 instances. We further assume that the car has a planar motion with its state determined by x,y, and orientation angle in a global frame of reference. Throughout this paper, we will assume that we do not observe motion along the z direction of the car, as well as a constant roll and pitch. The translation vector is de\ufb01ned as the relative vector between 2 camera poses. Note that in the global frame of reference, trajectories corresponding to the same steering commands may undesirably have different translation vectors. Therefore, to circumvent this issue, we rede\ufb01ne the frame of reference to a local coordinate system such that the forward motion of the car at any instance in time is always in the x\u2032 direction, whereas any lateral motion is always in the y\u2032 direction. The coordinates of the relative position vector in the local frame of reference are labeled as dx and dy. Details of which can be found in the supplementary. Note that if the rate at which frames are received or the speed at which the ego-vehicle is moving is variable, then calculating the translation vector between 2 consecutive frames will yield inconsistent results. To overcome this, we select only those pair of frames that are a \ufb01xed distance dx apart up to a certain tolerance. Hence, dx can be considered to be a constant. 2.2 Bicycle Model for a Car We model the dynamics of a 4 wheel front-wheel drive car using the bicycle model [Wang and Q, 2001] depicted in Figure 2. The model assumes planar motion with the spatial coordinates and orientation describing the state of the car. The 2 rear wheels of the car are represented by a single rear wheel (point A). Likewise, the 2 front wheels are represented by a single front wheel (point B). The lateral motion of the bicycle can be controlled by maneuvering the steering angle (\u03b4) of the front wheel, whereas the longitudinal/forward motion of the vehicle is controlled by applying the throttle. Figure 2: Depicts the bicycle model for a forward drive 4-wheeled car. L is the distance between the rear (point A) and front (point B) wheels. With a steering angle of \u03b4, the car would traverse a curvature whose radius is given by R and the rear wheel would move from point A to point A\u2019 by covering a distance S. The longitudinal and lateral motion of the car in the car\u2019s local frame of reference is given by dx and dy respectively. O is the instantaneous center of rotation at the intersection of line segments drawn perpendicular to the front and rear wheels of the car. The dotted line represents the curvature of the trajectory. During turns, the front and rear wheels will be oriented in different directions. Attempting to make sharp turns at high speeds would exert excessive lateral forces, causing the car to slip. An important assumption we make is that the vehicle does not experience any slip. This is true when both wheels are oriented in the same direction irrespective of the \fSelf-Supervised Steering Angle Prediction for Vehicle Control Using Visual Odometry speed. Whereas, while executing turns, this holds at moderately low speeds (5 m/s or \u224820 km/h) [Rajamani, 2012]. We can geometrically express the steering angle, \u03b4 in terms of the lateral translation vector determined by the method described in Section 2.1 as follows: \u2220OBA+90+\u03b4 = 180 (2) \u2220OBA = 90\u2212\u03b4 (3) \u2220AOB+90+\u2220OBA = 180 (4) \u2220AOB = \u03b4 (5) tan(\u2220AOB) = tan(\u03b4) = L R (6) R = L tan(\u03b4) (7) If the frame rate at which the information of successive frames is high coupled with the assumption of the vehicle moving at moderately low speeds while executing the turn would yield the following implications: 1. The change in the orientation of the vehicle would be small giving \u2206\u03c6 \u2248\u2220A\u2019AE \u22480. tan(\u2220A\u2032AE) \u2248tan(\u2206\u03c6) \u2248\u2206\u03c6 \u2248dy dx. (8) 2. The distance S, traversed by the vehicle along the curvature is \u2248AA\u2019. Moreover, \u2206\u03c6 \u22480 implies AA\u2019 \u2248 dx. 3. The angular velocity of the car is given by d\u03c6 dt \u22481 R \u00b7 dS dt . A high frame rate implies: \u2206\u03c6 \u2248S R \u2248AA\u2032 R \u2248dx R (9) Substituting Equation (7) in (9) results in \u2206\u03c6 = dx L tan(\u03b4) (10) Equating Equations (8) and (10) produces: dy dx = dx L tan(\u03b4) (11) \u03b4 = tan\u22121( dy\u00b7L dx\u00b7dx) (12) As described in Section 2.1, the image pairs are selected such that they are approximately a distance of dx apart. Hence, denominator of Equation (12) can be taken as constant. Since L is the length between the front and rear wheels, it is \ufb01xed, hence the above equation can be rewritten as: \u03b4 = tan\u22121(dy\u00b7\u03b1) (13) Where \u03b1 is a constant that can be tuned at run-time depending on the car. Therefore, we can determine the steering angle of the vehicle from the lateral component of the relative translation vector which in turn is determined from the method described in Section 2.1. For this, we train a neural network model which takes in an image as input and outputs the corresponding value of dy by minimizing the L1 loss between the value predicted by the model and value of dy. In the supplementary material, we demonstrate the implications on the performance of the trained models upon deviating from the assumptions given in this section. 2.3 Multi-Car Trajectories One issue with training a self-supervised vehicle control model with only a single car trajectory is that there may not be many variations in the collected data. Hence, at inference time, the car may slightly deviate from the reference trajectory (being the only trajectory on which it was trained) and come across a scene of possibly a different trajectory which it had never seen in the training set. Hence, the model may not be able to \ufb01gure out the correct action to take against the next sequence of images. This is depicted in Figure 3. To solve for this, note that drivers have a unique driving preference. Some tend to drive in the center, while others close to the lane or sidewalks. In fact, the same driver might demonstrate different trajectories while driving along the same route multiple times. Note that these behaviors lead to a diverse set of trajectories along the same section of a road. Hence, a slight deviation for one driver may be the usual trajectory for a different driver. But how can we scale our training to also incorporate this vast amount of information being generated by multiple drivers/runs across the same road section? This is also depicted in Figure 3 by the orange curve showing the trajectory generated by a different run along the same route. For any image along this new trajectory, the model aims to predict the relative position vector to the next closest image pose of the original purple trajectory. The frames in this new trajectory can be relocalized against the reference using direct visual localization approaches such as GN-Net [von Stumberg et al., 2020] or by running the PnP algorithm in a RANSAC scheme using robust visual image descriptors, e.g. R2D2 [Revaud et al., 2019]. Reference Trajectory: For any section of the road having multiple trajectories, we de\ufb01ne the reference trajectory as the one to which the relative vector from any of the many trajectories is mapped to. In Figure 3, the purple line is taken as the reference trajectory. For the case of multiple \fQadeer Khan, Patrick Wenzel, Daniel Cremers Figure 3: The purple line shows a section of the trajectory for Car1 used to train the self-supervised model. At test time, the model might slightly deviate away from this learned trajectory and come across images it has never seen in the training set. Hence, it would not be able to take the correct action to return to the original path after this slight deviation. Eventually, this deviation would accumulate with the car possibly crashing into an obstacle. The orange line shows the trajectory of another different car (Car2). During training, images from this trajectory can also be used to map the relative vector from Car2 trajectory to the next frame of Car1. Hence, at test time, even if the car deviates from the original Car1 trajectory, it will still be able to make the correct maneuver to return to its original course. trajectories, then for each, we calculate the lateral motion vector to every other trajectory and sum it up. The centermost will have the lowest sum and is taken as the reference. 2.4 Control Under Dynamic Weather/Lighting Conditions In dynamic weather conditions, VO methods may fail to generate an accurate representation of the trajectory. Hence, it may not be possible to obtain self-supervised training data for controlling a car in such a scenario. This is depicted in Figure 4, where under the CloudyNoon condition the trajectory generated by VO matches closely with the ground truth trajectory. This is in contrast to the HardRainNoon condition, where visual odometry fails for the same sequence. To overcome this issue, we adopt the teacher-student approach proposed by [Wenzel et al., 2018]. Note that our model can be divided into a feature extraction (FEM) and Steering Angle Prediction (SAP) module. The FEM extracts out features with an RGB image as input. These features are in turn fed to the SAP which predicts the coordinate for the lateral motion of the car. This model is trained with the visual odometry trajectories obtained under the CloudyNoon condition and is referred to as the teacher. This model will fail if tested on HardRainNoon condition. (a) CloudyNoon Trajectory. (b) HardRainNoon Trajectory. Figure 4: This \ufb01gure shows the in\ufb02uence of inclement weather on a small sample stretch of a trajectory produced by VO on the CARLA simulator. The curve in blue is the ground truth trajectory, and the curve in orange is the one obtained with VO. VO fails on the HardRainNoon condition. Top: Camera snapshots from the respective locations on the sequence indicated by circles in the ground truth trajectory. For this, we train another student model whose SAP module has exactly the same weights as that of the teacher model. Meanwhile, its FEM module is trained by feeding an image translated from CloudyNoon to HardRainNoon using an image-to-image translation (GAN) network [Anoosheh et al., 2018] trained in an unsupervised manner. The generator of the GAN takes in an image from domain A, IA (CloudyNoon) and translates it to an image IB belonging to domain B (HardRainNoon). The weights are updated by minimizing the mean square error between the features of the 2 models. The architecture is depicted in Figure 5. 3 Experiments Coming up with good metrics for evaluating the task of visual navigation is rather challenging. In particular, for autonomous driving, it is not obvious how to measure the performance of the driving policy. The authors of [Anderson et al., 2018] present different problem scenarios for the analysis of embodied agent navigation and suggest some evaluation metrics for standard scenarios. Moreover, [Codevilla et al., 2018a] conduct further studies on online and of\ufb02ine metrics, especially for vision-based sensorimotor control. Speci\ufb01cally, in this work, the difference between metrics on static images compared to evaluating online in the simulation is analyzed. Their result showed that driving models can have similar of\ufb02ine metrics but drastically different driving performance. They concluded from their experiments that of\ufb02ine metrics are weakly correlated with actual driving quality. Therefore, it is not trivial to link from of\ufb02ine to online performance due to a low correlation between them. Due to the aforementioned reasons, we interpret our con\fSelf-Supervised Steering Angle Prediction for Vehicle Control Using Visual Odometry Figure 5: This \ufb01gure shows the teacher-student architecture used to produce the steering commands independently of the image input domain. The generator is pre-trained using samples from both domains. The solid arrows represent forward propagation into the network, whereas the dotted line depicts back-propagation to update the weights of the student model. clusions in an online setting as this is directly correlated to the actual driving performance. For this, we use the stable version v0.8.2 of CARLA [Dosovitskiy et al., 2017] simulator for training and evaluation of our models. CARLA provides 15 different weather conditions. We train our models only on the CloudyNoon condition but test our results on all other conditions. We compare the performance of our method with that of [Khan et al., 2019] which trains a supervised model that is robust across all the weather conditions. The results were reported for executing turns around corners at \ufb01xed speeds. The metric to measure the performance of the models was done by calculating the ratio of time the car remains within its own driving lane. We adapt the same for comparison in our experiments. 3.1 Models For the purpose of evaluation, we evaluate 4 different models whose descriptions are given below. While in principle, our approach can be deployed in conjunction with any VO based method, we have used [Wang et al., 2017] for our experiments: 1. Supervised model with ground truth steering angle: This is the method of [Khan et al., 2019] trained in a supervised manner using the ground truth steering angles directly obtained from the CARLA simulator. Figure 6: This \ufb01gure shows the reference trajectory obtained by running the VO algorithm. 4 additional trajectories in relation to the reference are also depicted. The network is capable of handling images from all 15 weather conditions. We may use this model trained with ground truth supervised steering angle labels as the Oracle to compare against the other models not trained with any ground truth data. 2. Visual Odometry, with Multiple Trajectories: Here we use the estimated camera poses obtained through VO to implicitly map to the steering angle of the car. Figure 6 shows the reference VO trajectory along with 4 other trajectories in relation to the reference. 3. Visual Odometry, with Single Trajectory: Similar to the previous model. The images used for training this model are only taken from the reference trajectory. 4. Visual Odometry, with Multiple Trajectories, Robust: Same as the model described previously using VO on multiple trajectories except this model is also made robust to all the remaining weather conditions in a completely unsupervised manner by the method described in Section 2.4. However, to cater for multiple weathers not only the generator for conversion between CloudyNoon to HardRainNoon is trained but also the generators for HardRainSunset, WetCloudyNoon, CloudySunset, MidRainSunset, ClearSunset, ClearNoon, WetSunset, WetCloudySunet, WetNoon. Images produced by all these generators are used for training the student model. Figure 7 is a plot showing the performance of the aforementioned 4 models against all the 15 weather conditions. 4 Discussion In the following, we will elaborate on important key points regarding the performance of the 4 models and conduct additional studies to further investigate the \ufb01ndings. \fQadeer Khan, Patrick Wenzel, Daniel Cremers Figure 7: The plot exhibits the ratio of time the car remains within its own driving track for the 4 models across all the 15 different weather conditions. Higher is better. 1. Single Trajectory Performance: All models demonstrate good performance on the weather condition on which they were trained with the model trained on 1 trajectory being an exception. This is because, during the online evaluation, slight deviations from the reference trajectory might lead to a collapse of the model since it has no way of coming back to its original course. This is due to the fact that the model had never seen such scenes from different trajectories during training. Another reason that would explain this poor performance is that if the poses estimated by visual odometry used to train this model are by themselves not perfect, then the model is being trained to the imperfections of this imprecise dataset. Although these predictions may be marginally incorrect locally, however, in an online setting they tend to accumulate over the long run and thereby causing a collision with obstacles. 2. Multi Trajectory Performance: In contrast, the 2 models trained with VO on multiple trajectories exhibit superior online performance alluding to the utility of combining information from multiple trajectories. In fact, the performance of these 2 models is comparable with the Oracle that was trained in a supervised manner using ground truth labels. 3. Method Scalability: A plausible explanation for this is depicted in Figure 8 which demonstrates the scaling capability of our method. The blue curve shows that increasing the number of training trajectories improves the online performance. In fact, after a certain number of trajectories, the performance tends to saturate since any addition is already captured by the previous. Moreover, the \ufb01gure also shows that our method is robust to changes in camera parameters at Figure 8: This \ufb01gure shows the ratio of the time the car remains within its own driving track for different number of trajectories and changes in camera parameters from the default. The yaw and pitch values report the degree change in these parameters from the default. Whereas the relative movement of the camera along the width of the car (y) with respect to the default position is given in meters. Higher is better. inference time. Hence the position/orientation of the camera can be adjusted to yield the best performance at inference. 4. Performance under Dynamic Weather Conditions: Note that both the models trained on multiple trajectories give good performance on the weathers that closely resemble the CloudyNoon condition. However, on all other conditions, only our model trained for domain adaptation in a completely unsupervised manner by the method described in Section 2.4 maintains equivalent performance. It is interesting to note that our approach even works for the HardRainNoon condition, where, VO had failed to produce the correct trajectory of the car as depicted in Figure 4. 5. Robustness to Perturbations: Figure 9 shows the implication of inducing varying degrees of perturbations to the \ufb01nal steering command. As the perturbations are progressively increased, the performance for the model trained with only one trajectory, tends to correspondingly deteriorate. The models trained with multiple trajectories are less in\ufb02uenced by these perturbations. This can be observed in Figure 10 wherein, not only the mean accuracy tends to improve but the deviation in performance across the perturbations also shows a diminishing trend with the increasing number of trajectories. 6. In\ufb02uence of Scalability on Perturbations: It is interesting to note from Figure 8 that models which were trained with 5 or more trajectories maintain a performance ratio of remaining in the driving lane of greater \fSelf-Supervised Steering Angle Prediction for Vehicle Control Using Visual Odometry Figure 9: This \ufb01gure shows the in\ufb02uence on the performance of the models trained with different number of trajectories as varying levels of perturbations (shown as a percentage) are introduced into the \ufb01nal steering command. For a speci\ufb01c percentage of perturbation, each line in the \ufb01gure represents the mean ratio of the time the car remains within its own driving track across all the camera positions described in Figure 8. than 0.9. However, the model trained with 5 trajectories is inferior in performance when perturbations are introduced as depicted in Figure 10. Here, a minimum of 8 trajectories was needed to achieve similar performance. This alludes that scaling our method across more trajectories tends to lead to better robustness to perturbations. The online performance of the models along with a short summary of the proposed method can be found at this link: https://youtu.be/_22gF9GuwTg. 5"
+                }
+            ]
+        },
+        "edge_feat": {}
+    }
+}
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