On the Capacity of Abelian Group Codes Over Discrete Memoryless Channels

TL;DR

This paper investigates the capacity of Abelian group codes over discrete memoryless channels, providing bounds and exploring how relaxing linearity constraints can improve code design for various input alphabets.

Contribution

It introduces capacity bounds for Abelian group codes, extending the understanding of algebraic code structures beyond linear codes for discrete memoryless channels.

Findings

Lower and upper bounds on Abelian group code capacity

Existence of Abelian group structures for any input alphabet size

Linearity restrictions limit code optimality for many channels

Abstract

For most discrete memoryless channels, there does not exist a linear code for the channel which uses all of the channel's input symbols. Therefore, linearity of the code for such channels is a very restrictive condition and there should be a loosening of the algebraic structure of the code to a degree that the code can admit any channel input alphabet. For any channel input alphabet size, there always exists an Abelian group structure defined on the alphabet. We investigate the capacity of Abelian group codes over discrete memoryless channels and provide lower and upper bounds on the capacity.