Linear source coding over rings and applications

Abstract

This paper studies linear coding (LC) techniques in the setting of computing functions of correlated memoryless sources. Instead of linear mappings over finite fields, we consider using linear mappings over finite rings as encoders. It is shown that generally the region c×R, where c ≥ 1 is a constant and R is the Slepian-Wolf (SW) region, is achievable with LC over ring (LCoR) when the function to compute is the identity function. c = 1 if the ring used is a field. Hence, LCoR could be suboptimal in terms of achieving the best coding rates (the SW region) for computing the identity function. In spite of that, the ring version shows several advantages. It is demonstrated that there exists a function that is neither linear nor can be linearized over any finite field. Thus, LC over field (LCoF) does not apply directly for computing such a function unless the polynomial approach [1], [2] is used. On the contrary, such a function is linear over some ring. Using LCoR, an achievable region containing the SW region can be obtained for computing this function. In addition, the alphabet sizes of the encoders are strictly smaller than using LCoF. More interestingly, LCoF is not useful if some special requirement is imposed.