Quasi Structured Codes for Multi-Terminal Communications

Abstract

A new class of structured codes called quasi group codes (QGCs) is introduced. A QGC is a subset of a group code. In contrast with the group codes, QGCs are not closed under group addition. The parameters of the QGC can be chosen, such that the size of <inline-formula> <tex-math notation= LaTeX >$\\mathcal {C}+\\mathcal {C}$ </tex-math></inline-formula> is equal to any number between <inline-formula> <tex-math notation= LaTeX >$|\\mathcal {C}|$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$|\\mathcal {C}|^{2}$ </tex-math></inline-formula>. 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 are 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 Körner-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 the unstructured codes, linear codes, and group codes.