The Case for Structured Random Codes in Network Capacity Theorems

Abstract: Random coding arguments are the backbone of most channel capacity achievability proofs. In this paper, we show that in their standard form, such arguments are insufficient for proving some network capacity theorems: structured coding arguments, such as random linear or lattice codes, attain higher rates. Historically, structured codes have been studied as a stepping stone to practical constructions. However, K\"{o}rner and Marton demonstrated their usefulness for capacity theorems through the derivation of the optimal rate region of a distributed functional source coding problem. Here, we use multicasting over finite field and Gaussian multiple-access networks as canonical examples to demonstrate that even if we want to send bits over a network, structured codes succeed where simple random codes fail. Beyond network coding, we also consider distributed computation over noisy channels and a special relay-type problem.