A distributed source coding problem with a joint distortion criterion that depends on the sources and the reconstruction is considered in this work. While the prevalent trend in information theory has been to prove achievability results using Shannon's random coding arguments, using structured random codes offer rate gains over unstructured random codes for many problems. Motivated by this, a new achievable rate-distortion region (an inner bound to the performance limit) is presented for this problem for discrete memoryless sources based on “good” structured random nested codes built over abelian groups. For certain sources and distortion functions, the new rate region is shown to be strictly bigger than the Berger-Tung rate region, which has been the best known achievable rate region for this problem till now. This is done using numerical plots. Achievable rates for single-user source coding using abelian group codes are also obtained as a corollary of the main coding theorem. It is shown that nested linear codes achieve the Shannon rate-distortion function in the arbitrary discrete memoryless case.

Let X and Y be finite alphabets and PXYa joint distribution over them, with PXand PYrepresenting the marginals. For any ϵ > 0, the set of n-length sequences xnand ynthat are jointly typical according to PXYcan be represented on a bipartite graph. We present a formal definition of such a graph, known as a typicality graph, and study some of its properties. These properties arise in the study of several multiuser communication problems.

Consider a pair of correlated Gaussian sources(X1,X2). Two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder. The decoder is interested in reconstructing a linear combination ofX1andX2to within a mean-square distortion of D. We obtain an inner bound to the optimal rate-distortion region for this problem. A portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual componentsX1andX2first. This results in a better rate region for certain parameter values. Our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory. We then consider the case of linear reconstruction of K sources and provide an inner bound to the optimal rate-distortion region. Some parts of the inner bound are achieved using the following coding structure: lattice vector quantization followed by ldquocorrelatedrdquo lattice-structured binning.

A new rate region is presented for a general framework of distributed source coding where the decoder is interested in lossy reconstruction of an arbitrary function of the sources. The coding scheme involves vector quantization of the sources followed by "correlated" binning. Nested linear codes are used for quantization and binning as against the more prevalent random codes. Our rate region recovers many known rate regions in distributed source coding while unifying them under the same general framework.