A Unified Graphical Approach to Random Coding for Single-Hop Networks

Abstract

A unified graphical approach to random coding for any memoryless, single-hop, $K$ -user channel with or without common information is defined through two steps. The first step is user virtualization. Each user is divided into multiple virtual sub-users according to a chosen rate-splitting strategy. This results in an enhanced channel with a possibly larger number of users for which more coding possibilities are available and for which common messages to any subset of users can be encoded. Following user virtualization, the message of each user in the enhanced model is coded using a chosen combination of coded time-sharing, superposition coding, and joint binning. A graph is used to represent the chosen coding strategies. Nodes in the graph represent codewords, while edges represent coding operations. This graph is used to construct a graphical Markov model, which illustrates the statistical dependence among codewords that can be introduced by the superposition coding or joint binning. Using this statistical representation of the overall codebook distribution, the error probability of the code is shown to vanish through a unified analysis. The rate bounds that define the achievable rate region are obtained by linking the error analysis to the properties of the graphical Markov model. This proposed framework makes it possible to numerically obtain an achievable rate region by specifying a user virtualization strategy and describing a set of coding operations. The union of these rate regions defines the maximum achievable rate region of our unified coding strategy. The achievable rates obtained based on this unified graphical approach to random coding encompass the best random coding achievable rates for all memoryless single-hop networks known to date, including broadcast, multiple access, interference, and cognitive radio channels, as well as new results for topologies not previously studied, as we illustrate with several examples.