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Random coding arguments are the backbone of most channel capacityachievability proofs. In this paper, we show that in their standard form, sucharguments are insufficient for proving some network capacity theorems:structured coding arguments, such as random linear or lattice codes, attainhigher rates. Historically, structured codes have been studied as a steppingstone to practical constructions. However, K\"{o}rner and Marton demonstratedtheir usefulness for capacity theorems through the derivation of the optimalrate region of a distributed functional source coding problem. Here, we usemulticasting over finite field and Gaussian multiple-access networks ascanonical examples to demonstrate that even if we want to send bits over anetwork, structured codes succeed where simple random codes fail. Beyondnetwork coding, we also consider distributed computation over noisy channelsand a special relay-type problem.

The Case for Structured Random Codes in Network Capacity Theorems