Deterministic and Linear Finite Field Networks

Abstract

In this chapter, the submodular cut model and the polymatroid broadcast model are connected to inner and outer bounds on the multicast capacity region for deterministic networks, in particular, the cut-set outer bound and the noisy network coding inner bound. These bounds have been proven to be tight in certain special cases, e.g., graphical and hypergraphical networks, networks of independent deterministic broadcast channels (Aref networks), and deterministic linear finite networks, i.e., deterministic networks where the channel outputs are linear functions of the channel inputs over any finite field. The submodularity of these bounds results in a weighted sum multicast rate maximization characterization of the corresponding multicast rate regions based on the submodular dual decomposition approach. This chapter also studies linear finite field networks, which include deterministic linear finite field networks as a special case. The submodular dual decomposition approach is applied to the cut-set outer and the noisy network coding inner bound on the multicast capacity region of these networks.