Towards an Algebraic Network Information Theory: Simultaneous Joint Typicality Decoding

TL;DR

This paper develops bounds for simultaneous joint typicality decoding with nested linear codebooks, enhancing achievable rate regions for compute-forward in multi-user MACs by addressing statistical dependencies in codeword decoding.

Contribution

It introduces bounds for joint typicality decoding with linear codes, improving the achievable region for compute-forward in multi-user channels.

Findings

Larger achievable region for compute-forward over K-user MACs

Bounds account for dependencies introduced by linear codebooks

Enhanced decoding techniques for multi-user information theory

Abstract

Consider a receiver in a multi-user network that wishes to decode several messages. Simultaneous joint typicality decoding is one of the most powerful techniques for determining the fundamental limits at which reliable decoding is possible. This technique has historically been used in conjunction with random i.i.d. codebooks to establish achievable rate regions for networks. Recently, it has been shown that, in certain scenarios, nested linear codebooks in conjunction with "single-user" or sequential decoding can yield better achievable rates. For instance, the compute-forward problem examines the scenario of recovering $L \le K$ linear combinations of transmitted codewords over a $K$-user multiple-access channel (MAC), and it is well established that linear codebooks can yield higher rates. Here, we develop bounds for simultaneous joint typicality decoding used in conjunction with nested linear codebooks, and apply them to obtain a larger achievable region for compute-forward over a $K$-user discrete memoryless MAC. The key technical challenge is that competing codeword tuples that are linearly dependent on the true codeword tuple introduce statistical dependencies, which requires careful partitioning of the associated error events.