Hamming Code for Multiple Sources

TL;DR

This paper extends the concept of Hamming codes to multiple sources in Slepian-Wolf coding, proposing Hamming Codes for Multiple Sources (HCMSs) as a potential perfect coding solution for arbitrary numbers of sources.

Contribution

It introduces HCMSs for perfect SW coding with multiple sources, analyzes conditions for perfect codes, and generalizes the construction method, establishing equivalence with all perfect codes for Hamming sources.

Findings

Existence of infinite HCMSs.

Flexible rate trade-offs for long codes.

Equivalence of perfect SW codes to generalized HCMS.

Abstract

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. We then introduce Hamming Codes for Multiple Sources (HCMSs) as a potential solution of perfect 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 a perfect SW code with sufficiently long code length, the compression rates of different sources can be trade-off flexibly. Finally, we relax the construction procedure of HCMS and call the resulting code generalized HCMS. We prove that every perfect SW code for Hamming sources is equivalent to a generalized HCMS.