CIF: Small: An Algebraic Approach to Distributed Source Coding

存在需要将相关信源从远程信源传输到一组终端(例如，大规模传感器网络和分布式数据存储系统)的各种情况。分布式信源编码领域考虑编码策略，该编码策略以分布式方式利用源之间的相关性，同时允许以尽可能低的比特率将信息传送到终端。本研究针对一般的分布式信源编码问题，研究了一种实用的编码方案，该领域以往的绝大多数工作都考虑了两个二进制源或具有小字母表和受限相关结构的数目较多的信源的情况。研究了基于里德-所罗门(RS)码的分布式源代码的新构造。非二进制源的连续实现被视为有限域上多项式的系数。每个信源编码器在来自现场的某些点上传输该多项式的评估。在终端的信源恢复是通过使用精心选择的根重数的多变量多项式内插，然后是因式分解来执行的。多重性矩阵使我们能够在译码过程中捕获不同信源序列的联合似然，同时确保译码是容易处理的。该方案是针对多个非二进制源设计的。期望以这种方式获得具有任意相关性的源的速率区域的角点。对于一般税率点，该方法将扩展到一大类相关结构，概括出可由当前最先进的技术处理的结构集。研究被整合到研究生/本科生的课程中，并在适当的水平上纳入高级设计。

There are various instances of problems where correlated sources need to be transmitted from remote sources to a set of terminals, e.g., large scale sensor networks and distributed data storage systems. The field of distributed source coding considers coding strategies that leverage the correlation between the sources in a distributed manner while allowing the information to be communicated at a bitrate as low as possible to the terminals. This research studies practical coding schemes for the general distributed source coding problem.The vast majority of prior work in this area considers the case of two binary sources, or a higher number of sources with small alphabets and restrictive correlation structures. This research investigates novel constructions of distributed source codes based on Reed Solomon (RS) codes. Consecutive realizations of the nonbinary sources are viewed as coefficients of polynomials over a finite field. Each source encoder transmits evaluations of this polynomial at certain points from the field. Source recovery at the terminal is performed by multivariate polynomial interpolation with carefully chosen root multiplicities, followed by factorization. The multiplicity matrices allow us to capture the joint likelihood of the different source sequences in the decoding process, while ensuring that the decoding is tractable. The scheme is designed for multiple nonbinary sources. Corner points of the rate region for sources with arbitrary correlation are expected to be achieved in this manner. For general rate points, the approach will be extended for a large class of correlation structures that generalizes the set of structures that can be handled by current state of the art techniques.The research is integrated into the graduate/undergraduate curriculum and into senior-design at appropriate levels.