The Universality of Generalized Hamming Code for Multiple Sources

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.