Algebraic Structures for Multi-Terminal Communication Systems.

We study a distributed source coding problem with multiple encoders, a central decoder and a joint distortion criterion. The encoders do not communicate with each other. The encoders observe correlated sources which they quantize and communicate noiselessly to a central decoder which is interested in minimizing a joint distortion criterion that depends on the sources and the reconstruction. We are interested in characterizing an inner bound to the optimal rate-distortion region. We first consider a special case where the sources are jointly Gaussian and the decoder wants to reconstruct a linear function of the sources under mean square error distortion. We demonstrate a coding scheme involving nested lattice codes that reconstructs the linear function by encoding in such a fashion that the decoder is able to reconstruct the function directly. For certain source distributions, this approach yields a larger rate-distortion region compared to when the decoder reconstructs lossy versions of the sources first and then estimates the function from them. We then extend this approach to the case of reconstructing a linear function of an arbitrary number of jointly Gaussian sources. Next, we consider the general distributed source coding problem with discrete sources. This formulation includes as a special case many famous distributed source coding problems. We present a new achievable rate-distortion region for this problem based on “good” structured nested random codes built over abelian groups. We demonstrate rate gains for this problem over traditional coding schemes using unstructured random codes. For certain sources and distortion functions, the new rate region is strictly bigger than the Berger-Tung rate region, which has been the best known achievable rate region for the problem till now. Further, there is no known way of achieving these rate gains without exploiting the structure of the coding scheme. Achievable performance limits for single-user source coding using abelian group codes are also obtained as corollaries of the main coding theorem. Our results also imply that nested linear codes achieve the Shannon rate-distortion bound in the single-user setting. Finally, we conclude by outlining some future research directions.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/75917/1/dineshk_1.pd

How to Compute Modulo Prime-Power Sums

The problem of computing modulo prime-power sums is investigated in distributed source coding as well as computation over Multiple-Access Channel (MAC). We build upon group codes and present a new class of codes called Quasi Group Codes (QGC). A QGC is a subset of a group code. These codes are not closed under the group addition. We investigate some properties of QGC's, and provide a packing and a covering bound. Next, we use these bounds to derived achievable rates for distributed source coding as well as computation over MAC. We show that strict improvements over the previously known schemes can be obtained using QGC's