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 C + C is equal to any number between |C| and |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 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 unstructured codes, linear codes and group codes. code ensemble, and characterize the asymptotic performance using single-letter information quantities. By choosing the single-letter distribution on the indices one can operate anywhere in the spectrum between the two extremes: group codes and unstructured codes.The contributions of this work are as follows. A new class of codes over groups called Quasi Group Codes (QGC) is introduced. These codes are constructed by taking subsets of group codes. This work considers QGCs over cyclic groups Z p r . One can use the fundamental theorem of finitely generated Abelian groups to generalize the results of this paper to QGCs over non-cyclic finite Abelian groups.Information-theoretic characterizations for the asymptotic performance limits and properties of QGCs for source coding and channel coding problems are derived in terms of single-letter information quantities.Covering and packing bounds are derived for an ensemble of QGCs. Next, a binning technique for the QGCs is developed by constructing nested QGCs. As a result of these bounds, the PtP channel capacity and optimal rate-distortion function of sources are shown to be achievable using nested QGCs. The applications of QGCs in some multi-terminal communications problems are considered. More specifically our study includes the following problems:Distributed Source Coding: A more general version of Körner-Marton problem is considered. In this problem, there are two distributed sources taking values from Z p r . The sources are to be compressed in a distributed fashion. The decoder wishes to compute the modulo p r -addition of the sources losslessly.Computation over MAC: In this problem, two transmitters wish to communicate independent i...
A new structured coding scheme based on transver sal group codes is proposed. We investigate the information theoretic performance limits for this strategy in multi-terminal communications. Achievability results are derived for lossless reconstruction of sum of two sources. In addition, a new rate region is presented for the problem of computation over multiple access channel . We show that the application of the new coding strategy, results in strict gains in terms of achievable rates in both settings.
A new ensemble of structured codes is introduced. These codes are called Quasi Linear Codes (QLC). The QLC's are constructed by taking subsets of linear codes. They have a looser structure compared to linear codes and are not closed under addition. We argue that these codes provide gains in terms of achievable Rate-Distortions (RD) in different multi-terminal source coding problems. We derive the necessary covering bounds for analyzing the performance of QLC's. We then consider the Multiple-Descriptions (MD) problem, and prove through an example that the application of QLC's gives an improved achievable RD region for this problem. Finally, we derive an inner bound to the achievable RD region for the general MD problem which strictly contains all of the previous known achievable regions.
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