The Mathematics of Pseudocodewords

这个项目专注于分析有限域上的某些向量空间，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。虽然这些码在当前和未来的通信系统中显示出巨大的前景，但它们的性能受到所谓伪码字的存在的限制。本研究的主要目标是通过结合编码和信息论、线性代数、组合学和图论的工具，开发一个全面的这些对象的数学理论。它是通过分析与相关代码图的封面图相对应的码字（称为基于原型的代码）和通过将这些码字投影到代码图上获得的伪码字来实现的，分析基本锥体中的伪码字（具有线性规划解码和迭代解码的含义），并分析了对代码性能的影响的伪权重（迭代解码下的性能度量）。尽管在过去几十年中取得了重大进展，但当前通信系统的性能在许多情况下仍然不令人满意，例如当延迟很重要，功率很关键或带宽受限时。该项目研究了当前编码技术相对于理论极限表现不佳的原因，并开发了一个数学理论来解释它们的缺点，并帮助设计更好的代码。研究者与电气工程系的联合任命，以及她与学术界和工业界的工程专家的合作，确保了该项目具有超越理论领域的影响；分析结果被用于提高当前和未来通信系统的可靠性。由于可靠的通信是无处不在的数据访问和交换的核心，国家的信息技术基础设施将从该项目中受益。然而，它的影响并不局限于信道编码，而是扩展到计算机科学、网络和压缩感知等领域。

Smarandache1313221 This project focuses on the analysis 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. While these codes show great promise in current and future communication systems, their performance has been limited by the existence of so-called pseudocodewords. The main goal of this research is to develop a comprehensive mathematical theory of these objects by combining tools from coding and information theory, linear algebra, combinatorics, and graph theory. It is achieved by an analysis of the codewords corresponding to the cover graphs of the associated code graphs -- called protograph-based codes -- and the pseudocodewords obtained by projecting these codewords onto the code graphs, an analysis of the pseudocodewords in the fundamental cones (with implications in linear programming decoding and iterative decoding), and an analysis of the pseudo-weights (the measure of performance under iterative decoding) with implications in code performance. Despite significant progress in the last decades, the performance of current communication systems is still not satisfactory in many situations, such as when delay is important, power is critical, or bandwidth is constrained. This project studies the reasons why current coding techniques underperform relative to theoretical limits and develops a mathematical theory that explains their shortcomings and helps design better codes. The investigator's joint appointment with the department of electrical engineering and her collaboration with engineering experts from academia and industry ensure that the project has an impact beyond the theoretical realm; the analytical findings are exploited to improve the reliability of current and future communication systems. Because reliable communication is at the heart of ubiquitous access to and exchange of data, the nation's information technology infrastructure benefits from the project. Its impact is, however, not restricted to channel coding, but extends to areas such as computer science, networking, and compressed sensing.