The case for structured random codes in network communication theorems
In the Shannon-theoretic analysis of joint source-channel coding problems, achievability is usually established via a two-stage approach: The sources are compressed into bits' and these bits are reliably communicated across the noisy channels. Random coding arguments are the backbone of both stages of the proof. This strategy not only establishes the optimal performance for stationary ergodic point-to-point problems, but also for a number of simple network situations, such as independent sources that are communicated with respect to separate fidelity criteria across a multiple-access channel. Beyond such simple cases, for general networks, unstructured random coding arguments are not sufficient. This was first realized for source coding by Korner and Marton, who showed that for a distributed source coding problem where one only needs to recover a function of the sources random linear codes are necessary. The goal of this note is to extend this insight to pure channel coding as well as to joint source-channel coding problems, such as the problem of reliable computation over a multiple-access channel and a multi-access network with relays.