Hash property and coding theorems for sparse matrices and maximum-likelihood coding

The aim of this paper is to prove the achievability of rate regions for several coding problems by using sparse matrices (with logarithmic column degree) and maximum-likelihood (ML) coding. These problems are 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 ML coding can achieve the optimal rate.