Structured Random Codes and Sensor Network Coding Theorems

TL;DR: This note generalizes the Shannon-theoretic analysis of joint source-channel coding problems by employing a lattice code, and finds that the underlying linear structure of the code is crucial to its success.

Abstract: 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 "separation" 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, separation-based coding is suboptimal. For instance, for a simple Gaussian sensor network, uncoded transmission is exactly optimal and performs exponentially better than a separation-based solution. In this note, we generalize this sensor network strategy by employing a lattice code. The underlying linear structure of our code is crucial to its success.