  The capacity of multiuser networks has been a long-standing problem in
information theory. Recently, Avestimehr et al. have proposed a deterministic
network model to approximate multiuser wireless networks. This model, known as
the ADT network model, takes into account the broadcast nature of wireless
medium and interference.
  We show that the ADT network model can be described within the algebraic
network coding framework introduced by Koetter and Medard. We prove that the
ADT network problem can be captured by a single matrix, and show that the
min-cut of an ADT network is the rank of this matrix; thus, eliminating the
need to optimize over exponential number of cuts between two nodes to compute
the min-cut of an ADT network. We extend the capacity characterization for ADT
networks to a more general set of connections, including single
unicast/multicast connection and non-multicast connections such as multiple
multicast, disjoint multicast, and two-level multicast. We also provide
sufficiency conditions for achievability in ADT networks for any general
connection set. In addition, we show that random linear network coding, a
randomized distributed algorithm for network code construction, achieves the
capacity for the connections listed above. Furthermore, we extend the ADT
networks to those with random erasures and cycles (thus, allowing
bi-directional links).
  In addition, we propose an efficient linear code construction for the
deterministic wireless multicast relay network model. Avestimehr et al.'s
proposed code construction is not guaranteed to be efficient and may
potentially involve an infinite block length. Unlike several previous coding
schemes, we do not attempt to find flows in the network. Instead, for a layered
network, we maintain an invariant where it is required that at each stage of
the code construction, certain sets of codewords are linearly independent.
