Weight Enumeration for Convolutional Codes

卷积码的权重枚举

• 批准号: 0908379
• 负责人: Heide Gluesing-Luerssen
• 金额: $ 18.34万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2013-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908379&HistoricalAwards=false
• 关键词: Weight; Enumeration; Convolutional; Codes

Gluesing-LuerssenDMS-0908379 The project aims at furthering the mathematical theory ofconvolutional codes. The error-correcting quality of these codescan be measured by a variety of distance parameters that havebeen introduced in the engineering-oriented literature. The mainidea of the project is the investigation of convolutional codesbased on the weight adjacency matrix, a single parameter thatcomprises, among other things, all those different distancemeasures. So far two major results have been established by theinvestigator: a MacWilliams Identity Theorem stating that theweight adjacency matrix of a code fully determines that of thedual code, and a theorem stating that codes without non-zeroconstant codewords and sharing the same weight adjacency matrixare monomially equivalent. These results open up new directionsin convolutional coding theory. Firstly, self-dual codes can bestudied theoretically. Besides the obvious fact that the weightadjacency matrix of a self-dual code is invariant under theMacWilliams transformation, the close link between self-dualconvolutional codes and self-dual block codes obtained bytail-biting plays a crucial role. It can be expected that,conversely, positive results for self-dual convolutional codeswill also have an impact on the theory of self-dual tail-bitingblock codes. Furthermore, the investigator explores theMacWilliams Identity and its consequences for minimalconventional and tail-biting trellises for linear block codes. The analogy in the graphical representation of trellis blockcodes and convolutional codes indeed suggests a paralleltreatment of these two classes of codes. Another subproject aimsat investigating under what conditions one may declare twoconvolutional codes identical with respect to their algebraicstructure and error-correcting capabilities. In light of theresult about monomially equivalent codes mentioned above, theweight adjacency matrix is expected to play a crucial role inthis project as well. The goal of this subproject is aclassification and a comparison of convolutional codes in thesense of classical coding theory. This project focuses on the algebraic theory oferror-correcting codes. Such codes protect data againstalteration through noise and enable the reconstruction of theoriginal data from the corrupted information. All currentstandards of data transmission and data storage have built-inerror correcting mechanisms. The most famous examples are thecompact disk player, cell phones, the internet, satellite anddeep space communication systems. The project aims at deepeningthe mathematical theory of convolutional codes. This specificclass of codes forms one of the major players in manycommunication schemes, mainly in deep space communication andother wireless data transmission schemes. With the ever-growingdemand of sophisticated information technology there is acontinuing need of a thorough mathematical theory in order toimprove the performance of such communication devices.

Gluesing-LuerssenDMS-0908379该项目旨在进一步发展卷积码的数学理论。这些码的纠错质量可以通过工程文献中介绍的各种距离参数来衡量。该项目的主要思想是研究基于权重邻接矩阵的卷积码，该矩阵是一个单一参数，其中包括所有这些不同的距离测量。到目前为止，作者已经建立了两个主要结果：一个是麦克威廉姆斯恒等式定理，它说明码的重量邻接矩阵完全决定了对偶码的重量邻接矩阵；另一个定理是，没有非常零常数码字的码和具有相同重量邻接矩阵的码是单项等价的。这些结果为卷积编码理论开辟了新的方向。首先，可以从理论上研究自对偶码。除了自对偶码的加权邻接矩阵在MacWilliams变换下是不变的这一明显事实外，自对偶卷积码与咬尾得到的自对偶分组码之间的密切联系也起着至关重要的作用。可以预期，反过来，自对偶卷积码的积极结果也将对自对偶尾比特分组码的理论产生影响。此外，研究人员还探讨了麦克威廉姆斯恒等式及其在线性分组码的最小传统格子结构和咬尾格子结构上的结果。网格分组码和卷积码的图形表示中的类比确实暗示了这两类码的并行处理。另一个子项目是研究在什么条件下可以宣布两个卷积码在代数结构和纠错能力方面是相同的。鉴于上述关于单项等价码的结果，预计权重邻接矩阵也将在该项目中发挥关键作用。这个子项目的目标是对经典编码理论意义上的卷积码进行分类和比较。本课题主要研究纠错码的代数理论。这样的编码保护数据不受噪声的干扰，并能够从损坏的信息中重建原始数据。目前所有的数据传输和数据存储标准都有内置的纠错机制。最著名的例子是紧凑型光盘播放机、手机、互联网、卫星和深空通信系统。该项目旨在深化卷积码的数学理论。这类特定的码构成了许多通信方案的主要参与者之一，主要是在深空通信和其他无线数据传输方案中。随着对尖端信息技术的需求不断增长，为了提高这类通信设备的性能，不断需要一个完善的数学理论。