In this paper, we use linear codes to study zero-error Slepian–Wolf coding of a set of sources with deviation symmetry, where the sources are generalization of the Hamming sources over an arbitrary field. We extend our previous codes, generalized Hamming codes for multiple sources, to matrix partition codes and use the latter to efficiently compress the target sources. We further show that every perfect or linear-optimal code is a matrix partition code. We also present some conditions when matrix partition codes are perfect and/or linear-optimal. Detail discussions of matrix partition codes on Hamming sources are given at last as examples.

We consider zero-error Slepian-Wolf coding for a special kind of correlated sources known as Hamming sources. Moreover, we focus on the design of codes with minimum redundancy (i.e., perfect codes). As shown in a prior work by Koulgi et al., the design of a perfect code for a general source is very difficult and in fact is NP-hard. In our recent work, we introduce a subset of perfect codes for Hamming sources known as Hamming Codes for Multiple Sources (HCMSs). In this work, we extend HCMSs to generalized HCMSs, which can be proved to include all perfect codes for Hamming sources. To prove our main result, we first show that any perfect code for a Hamming source with two terminals is equivalent to a Hamming code for asymmetric Slepian Wolf coding (c.f. Lemma 2). We then show that any multi-terminal (of more than two terminals) perfect code can be transformed to a perfect code for two terminals (c.f. Lemma 3) and to a perfect code with an asymmetric form (c.f. Lemma 4). Equipped with these results, we prove that every perfect Slepian-Wolf code for Hamming sources is equivalent to a generalized HCMS.

We consider Slepian-Wolf (SW) coding of multiple sources and extend the packing bound and the notion of perfect code from conventional channel coding to SW coding with more than two sources. Moreover, we show that there does not exist perfect SW code of length-5 for three sources.

We introduce Hamming Codes for Multiple Sources (HCMSs) as a potential solution of perfect Slepian-Wolf (SW) coding for arbitrary number of terminals. Moreover, we study the case with three sources in detail. We present the necessary conditions of a perfect SW code and show that there exists infinite number of HCMSs. Moreover, we show that for perfect SW code with sufficiently long code length, the compression rates of different sources can be trade-off flexibly.