Hash Property and Coding Theorems for Sparse Matrices and Maximum-Likelihood Coding

Details Hash Property and Coding Theorems for Sparse Matrices and Maximum-Likelihood Coding Hash Property and Coding Theorems for Sparse Matrices and Maximum-Likelihood Coding

2008-01-25

arxiv.org/abs/0801.3878v2

IEEE Transactions on Information Theory, vol.IT-56, no.5, pp.2143-2167, May 2010

Computer Science

Information Theory

This manuscript has been submitted to IEEE Transactions on Information Theory and a part of this manuscript has been submitted

Scientific paper

The aim of this paper is to prove the achievability of several coding problems by using sparse matrices (the maximum column weight grows logarithmically in the block length) and maximal-likelihood (ML) coding. These problems are the Slepian-Wolf problem, the Gel'fand-Pinsker problem, the Wyner-Ziv problem, and the One-helps-one problem (source coding with partial side information at the decoder). To this end, the notion of a hash property for an ensemble of functions is introduced and it is proved that an ensemble of $q$-ary sparse matrices satisfies the hash property. Based on this property, it is proved that the rate of codes using sparse matrices and maximal-likelihood (ML) coding can achieve the optimal rate.

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