Pseudo-Codeword Analysis and Design of Quasi-Cyclic and Convolutional Codes

提案：DMS -0708033 PI： Smarandache，Roxana机构：San Diego State University圣地亚哥州立大学准循环和卷积码的伪码字分析和设计摘要本项目集中于现代数字通信系统的数学方面-特别是有限域上某些向量空间的分析和设计，称为低密度奇偶校验（LDPC）码，它具有一组生成对偶空间的“稀疏”向量，其中，准循环和卷积LDPC码。 结合迭代解码算法的基础上通过概率估计沿着自然相关的代码的图形的边缘，这些代码显示出很大的希望，为当前和未来的通信系统。分析包括相关的代码图，基本锥（与线性编程解码和迭代解码的影响），伪码字集（罪魁祸首，防止迭代解码的收敛）和伪权重（迭代解码下的性能的措施）的分析。目标是：发展有限长度LDPC码的理论并提供具有相关代数结构的不同码之间的比较;通过研究每个对其他的影响来统一相关概念，如近码字、伪码字、围长、最小距离、最小伪权重;基于这些参数及其影响来推导LDPC块和卷积码的性能的上下界;并最终为设计在迭代解码下具有可预测的良好性能的有限长度LDPC码提供指导。将LDPC分组码纳入数字视频广播、以太网和第三代（3G）移动电话网络等新兴标准中，将使3G电话的无线接入速度比老式拨号上网快十倍以上，最近激发了人们对LDPC卷积码的兴趣。对LDPC卷积码的初步研究使PI相信，与LDPC块码相比，这些码具有几个潜在的实际优势。 这个项目的目标是分析这些代码，并构建具有可预测的良好性能的新代码。这项研究将自然地整合抽象理论和现实世界的应用，为跨领域研究提供令人兴奋的机会。PI与来自学术界和工业界的领先工程专家的持续合作，使该项目的理论研究成果能够过渡到实际的通信系统。预计该项目将对未来的信息基础设施产生影响，并有助于普及信息。这种影响将扩展到数据压缩和通信协议和安全领域，因为它们的基础理论与编码理论密切相关。

Proposal: DMS - 0708033PI: Smarandache, RoxanaInstitution: San Diego State UniversityTitle: Pseudo-Codeword Analysis and Design of Quasi-Cyclic and Convolutional CodesAbstractThis project focuses on the mathematical aspects of modern digital communication systems --- in particular on the analysis and design of certain vector spaces over finite fields, called low-density parity-check (LDPC) codes, that have a set of "sparse'' vectors generating the dual space, and among these, on quasi-cyclic and convolutional LDPC codes. In conjunction with an iterative decoding algorithm based on passing probability estimates along the edges of a graph naturally associated to the code, these codes show great promise for current and future communication systems. The analysis consists of an analysis of the associated code graphs, fundamental cones (with implications in linear programming decoding and iterative decoding), pseudo-codeword sets (the culprits that prevent the convergence of iterative decoding) and pseudo-weights (a measure of performance under iterative decoding). The goals are: to develop a theory of finite-length LDPC codes and provide a comparison between different codes having related algebraic structures; to unify relevant notions like near-codewords, pseudo-codewords, girth, minimum distance, minimum pseudo-weight by studying the effect that each has on the others; to derive lower and upper bounds on the performance of LDPC block and convolutional codes based on these parameters and their influences; and to ultimately provide guidelines for designing finite-length LDPC codes that have predictably good performance under iterative decoding.The inclusion of LDPC block codes in emerging standards such as digital video broadcasting, Ethernet and third-generation (3G) mobile-telephone networks, which will allow wireless access from 3G phones to be over ten times faster than from an old-fashioned dial-up, has recently spurred interest in LDPC convolutional codes. Initial research on LDPC convolutional codes leads the PI to the belief that these codes have several potential practical advantages compared to LDPC block codes. The goal of this project is to analyze these codes and construct new ones with predictably good performance. This research will naturally integrate abstract theory and real-world applications, providing exciting opportunities for cross-cutting research. The PI's continuing collaboration with leading engineering experts from academia and industry enables a transition from the theoretical research findings in this project to practical communication systems. It is anticipated that the project will have an impact on the future information infrastructure and contribute to universal accessibility. This impact will extend to the areas of data compression and communication protocols and security, since their underlying theories are closely related to coding theory.