Weight Enumeration for Convolutional Codes

该项目旨在进一步推进卷积码的数学理论。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测量。该项目的主要思想是研究基于权重邻接矩阵的卷积码，这是一个包含所有不同距离度量的单一参数。到目前为止，研究者已经建立了两个主要的结果：一个MacWilliams恒等定理，说明码的权值邻接矩阵完全决定对偶码的权值邻接矩阵；一个定理，说明没有非零常数码字的码和共享相同权值邻接矩阵的码是单等价的。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。除了自对偶码在theMacWilliams变换下的权重邻接矩阵是不变的明显事实外，自对偶卷积码与咬尾获得的自对偶分组码之间的紧密联系起着至关重要的作用。相反，可以预期，自对偶卷积码的积极结果也将对自对偶咬尾块码的理论产生影响。此外，研究者探讨了mcwilliams身份及其对线性分组码的最小常规和咬尾格架的影响。网格块码和卷积码的图形表示中的类比确实表明了这两类代码的并行处理。另一个子项目旨在调查在什么条件下可以声明两个卷积码在代数结构和纠错能力方面是相同的。根据上述关于一元等值码的研究结果，权重邻接矩阵也有望在本项目中发挥关键作用。这个子项目的目标是在经典编码理论的意义上分类和卷积码的比较。本项目主要研究代数理论提供的纠错码。这样的代码保护数据不受噪声的影响，并使原始数据能够从损坏的信息中重建。目前所有的数据传输和数据存储标准都内置了纠错机制。最著名的例子是光盘播放器、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。这类特殊的编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。随着对复杂信息技术的需求不断增长，为了提高这种通信设备的性能，不断需要一个全面的数学理论。

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.