  The recent development of deep learning methods provides a new approach to
optimize the belief propagation (BP) decoding of linear codes. However, the
limitation of existing works is that the scale of neural networks increases
rapidly with the codelength, thus they can only support short to moderate
codelengths. From the point view of practicality, we propose a high-performance
neural min-sum (MS) decoding method that makes full use of the lifting
structure of protograph low-density parity-check (LDPC) codes. By this means,
the size of the parameter array of each layer in the neural decoder only equals
the number of edge-types for arbitrary codelengths. In particular, for
protograph LDPC codes, the proposed neural MS decoder is constructed in a
special way such that identical parameters are shared by a bundle of edges
derived from the same edge-type. To reduce the complexity and overcome the
vanishing gradient problem in training the proposed neural MS decoder, an
iteration-by-iteration (i.e., layer-by-layer in neural networks) greedy
training method is proposed. With this, the proposed neural MS decoder tends to
be optimized with faster convergence, which is aligned with the early
termination mechanism widely used in practice. To further enhance the
generalization ability of the proposed neural MS decoder, a codelength/rate
compatible training method is proposed, which randomly selects samples from a
set of codes lifted from the same base code. As a theoretical performance
evaluation tool, a trajectory-based extrinsic information transfer (T-EXIT)
chart is developed for various decoders. Both T-EXIT and simulation results
show that the optimized MS decoding can provide faster convergence and up to
1dB gain compared with the plain MS decoding and its variants with only
slightly increased complexity. In addition, it can even outperform the
sum-product algorithm for some short codes.
