[Paper Review] Abelian Group Codes for Source Coding and Channel Coding

Aria G. Sahebi, S. Sandeep Pradhan|arXiv (Cornell University)|May 7, 2013

Cooperative Communication and Network Coding30 references3 citations

TL;DR

This paper introduces a single-letter information-theoretic characterization of achievable rate-distortion and channel coding rates using Abelian group codes for arbitrary discrete memoryless sources and channels. It shows that when the group is a field, the results reduce to symmetric rate-distortion and symmetric capacity, and demonstrates superior performance over linear codes in non-symmetric settings, such as Z4-based codes for correlated quaternary sources.

ABSTRACT

In this paper, we study the asymptotic performance of Abelian group codes for the lossy source coding problem for arbitrary discrete (finite alphabet) memoryless sources as well as the channel coding problem for arbitrary discrete (finite alphabet) memoryless channels. For the source coding problem, we derive an achievable rate-distortion function that is characterized in a single-letter information-theoretic form using the ensemble of Abelian group codes. When the underlying group is a field, it simplifies to the symmetric rate-distortion function. Similarly, for the channel coding problem, we find an achievable rate characterized in a single-letter information-theoretic form using group codes. This simplifies to the symmetric capacity of the channel when the underlying group is a field. We compute the rate-distortion function and the achievable rate for several examples of sources and channels. Due to the non-symmetric nature of the sources and channels considered, our analysis uses a synergy of information theoretic and group-theoretic tools.

Motivation & Objective

• To derive a single-letter achievable rate-distortion function for lossy source coding using Abelian group codes over arbitrary finite Abelian groups.
• To establish a single-letter achievable rate for channel coding using Abelian group codes over arbitrary finite Abelian groups.
• To demonstrate that group codes can outperform standard random coding ensembles and linear codes in non-symmetric source and channel models.
• To unify information-theoretic and group-theoretic tools for analyzing structured code performance in multi-terminal and single-terminal communication problems.

Proposed method

• The authors use ensemble analysis of Abelian group codes to derive a single-letter rate-distortion function based on group structure and entropy of quotient groups.
• They apply group-theoretic tools, including subgroup decomposition and coset analysis, to characterize the size of typical sets under group actions.
• The method involves defining equivalence classes via quotient groups and analyzing the conditional typicality of sequences under group cosets.
• The analysis leverages the structure of p-adic representations in Z_{p^r} to handle non-field groups and derive solution sets for linear equations over rings.
• The authors prove that the number of solutions to ax = b in Z_{p^r} is independent of representative choices in the p-adic decomposition, ensuring consistency in group code constructions.
• They establish a bound on the size of the intersection of a group coset with a typical set, showing concentration around the entropy of the conditional distribution.

Experimental results

Research questions

• RQ1 Can Abelian group codes achieve better asymptotic performance than random coding ensembles in non-symmetric source and channel models?
• RQ2 How does the choice of group structure—especially non-field groups like Z_{p^r}—affect the achievable rate-distortion and channel coding rates?
• RQ3 Under what conditions does the use of Abelian group codes lead to a strict performance gain over linear codes over Galois fields?
• RQ4 How can group-theoretic tools be combined with information-theoretic typicality to analyze structured code ensembles?
• RQ5 What is the precise single-letter characterization of the rate-distortion function for Abelian group codes over arbitrary finite Abelian groups?

Key findings

• The achievable rate-distortion function for Abelian group codes is characterized in a single-letter form using group entropy and quotient group structure, generalizing the symmetric rate-distortion function.
• When the underlying group is a field, the derived rate-distortion function reduces to the symmetric rate-distortion function, confirming consistency with known results.
• For channel coding, the achievable rate using Abelian group codes is characterized in a single-letter form, which reduces to the symmetric channel capacity when the group is a field.
• In non-symmetric settings, such as correlated quaternary sources over Z4, Abelian group codes outperform linear codes over GF(4), demonstrating a performance advantage beyond complexity.
• The size of the intersection between a group coset and a typical set is tightly bounded by the conditional entropy of the source given the group quotient, with exponential concentration.
• The solution set to ax = b in Z_{p^r} is independent of the choice of representatives in the p-adic decomposition, ensuring robustness in code construction.