Abelian Group Codes for Channel Coding and Source Coding

Abstract

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

How to compute modulo prime-power sums

“…is achievable using linear codes over the field Z 7 [2]. As is shown in [39], group codes in this example outperform linear codes. The largest achievable region using group codes is described by all rate pair (R 1 , R 2 ) such that…”

“…Proof: Note that (38) and (39) give an alternative characterization of the achievable region. Using these equations, observe that this region is convex in R 2 .…”

“…In [35], Ahlswede showed that group codes do not achieve the capacity of a general discrete memoryless channel. In [39], Sahebi et.al., unified the previously known works, and characterized the ensemble of all group codes over finite commutative groups. In addition, the authors derived the optimum asymptotic performance limits of group codes for PtP channel/source coding problems.…”

On the non-Abelian group code capacity of memoryless channels

“…Group codes allow to use non-binary, highly spectral-efficient, geometrically uniform modulations, while inheriting many of the nice structural properties enjoyed by binary linear codes. An overview of the different research lines on group codes developed during these years can be seen at [4,3,5,9,11,17,14,7,13]. One line of research is about the channel-capacity achieved by group codes.…”

“…Finally, the equality C G = C is obtained for G = Z p r transmitted through a p r -PSK-AWGN channel and an example where C G < C is given. On the other hand, the paper [17] uses the asymptotic equipartition property (AEP) to deal with ensembles of group codes, over Abelian groups, with rates achieving the group encoding capacity C G . It is not concerned with the equality C G = C.…”