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 notclosed under group addition. The parameters of the QGC can be chosen such thatthe size of $\mathcal{C}+\mathcal{C}$ is equal to any number between$|\mathcal{C}|$ and $|\mathcal{C}|^2$ . We analyze the performance of aspecific class of QGCs. This class of QGCs is constructed by assigningsingle-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 typicalset corresponding to these single-letter distributions. The asymptoticperformance limits of this class of QGCs is characterized using single-letterinformation quantities. Corresponding covering and packing bounds are derived.It is shown that the point-to-point channel capacity and optimalrate-distortion function are achievable using QGCs. Coding strategies based onQGCs are introduced for three fundamental multi-terminal problems: theK\"orner-Marton problem for modulo prime-power sums, computation over themultiple access channel (MAC), and MAC with distributed states. For eachproblem a single-letter achievable rate-region is derived. It is shown, throughexamples, that the coding strategies improve upon the previous strategies basedon unstructured codes, linear codes and group codes.
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