How to Compute Modulo Prime-Power Sums ?

TL;DR

This paper introduces Quasi Group Codes (QGC), a new class of structured codes that generalize group codes, and demonstrates their effectiveness in achieving optimal rates for various information-theoretic problems, including modulo prime-power sums.

Contribution

The paper proposes Quasi Group Codes (QGC), analyzes their performance, and applies them to solve fundamental multi-terminal problems with improved coding strategies over existing methods.

Findings

QGCs can be tailored to achieve any size between |mathcal{C}| and |mathcal{C}|^2.

QGC-based coding strategies outperform unstructured, linear, and group codes in key problems.

QGCs achieve channel capacity and rate-distortion bounds in the studied scenarios.

Abstract

A new class of structured codes called Quasi Group Codes (QGC) is introduced. A QGC is a subset of a group code. In contrast with group codes, QGCs are not closed under group addition. The parameters of the QGC can be chosen such that the size of $\mathcal{C}+\mathcal{C}$ is equal to any number between $|\mathcal{C}|$ and $|\mathcal{C}|^2$ . We analyze the performance of a specific class of QGCs. This class of QGCs is constructed by assigning single-letter distributions to the indices of the codewords in a group code. Then, the QGC is defined as the set of codewords whose index is in the typical set corresponding to these single-letter distributions. The asymptotic performance limits of this class of QGCs is characterized using single-letter information quantities. Corresponding covering and packing bounds are derived. It is shown that the point-to-point channel capacity and optimal rate-distortion function are achievable using QGCs. Coding strategies based on QGCs are introduced for three fundamental multi-terminal problems: the K\"orner-Marton problem for modulo prime-power sums, computation over the multiple access channel (MAC), and MAC with distributed states. For each problem a single-letter achievable rate-region is derived. It is shown, through examples, that the coding strategies improve upon the previous strategies based on unstructured codes, linear codes and group codes.