On the achievable rates of sources having a group alphabet in a distributed source coding setting

We consider the problem of compression via homomorphic encoding of a source having a group alphabet. This is motivated by the problem of distributed function computation, where it is known that if one is only interested in computing a function of several sources, then one can at times improve upon the compression rate required by the Slepian-Wolf bound. The functions of interest are those which could be represented by the binary operation in the group. We first consider the case when the source alphabet is the cyclic Abelian group, Zpr. In this scenario, we show that the set of achievable rates provided by Krithivasan and Pradhan [1], is indeed the best possible. In addition to that, we provide a simpler proof of their achievability result. In the case of a general Abelian group, an improved achievable rate region is presented than what was obtained by Krithivasan and Pradhan. We then consider the case when the source alphabet is a non-Abelian group. We show that if all the source symbols have non-zero probability and the center of the group is trivial, then it is impossible to compress such a source if one employs a homomorphic encoder. Finally, we present certain non-homomorphic encoders, which also are suitable in the context of function computation over non-Abelian group sources and provide rate regions achieved by these encoders.