Abelian Group Codes for Source Coding and Channel Coding

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.