The project aims at furthering the mathematical theory of convolutional codes. The error-correcting quality of these codes can be measured by a variety of distance parameters that have been introduced in the engineering-oriented literature. The main idea of the project is the investigation of convolutional codes based on the weight adjacency matrix, a single parameter that comprises, among other things, all those different distance measures. So far two major results have been established by the investigator: a MacWilliams Identity Theorem stating that the weight adjacency matrix of a code fully determines that of the dual code, and a theorem stating that codes without non-zero constant codewords and sharing the same weight adjacency matrix are monomially equivalent. These results open up new directions in convolutional coding theory. Firstly, self-dual codes can be studied theoretically. Besides the obvious fact that the weight adjacency matrix of a self-dual code is invariant under the MacWilliams transformation, the close link between self-dual convolutional codes and self-dual block codes obtained by tail-biting plays a crucial role. It can be expected that, conversely, positive results for self-dual convolutional codes will also have an impact on the theory of self-dual tail-biting block codes. Furthermore, the investigator explores the MacWilliams Identity and its consequences for minimal conventional and tail-biting trellises for linear block codes. The analogy in the graphical representation of trellis block codes and convolutional codes indeed suggests a parallel treatment of these two classes of codes. Another subproject aims at investigating under what conditions one may declare two convolutional codes identical with respect to their algebraic structure and error-correcting capabilities. In light of the result about monomially equivalent codes mentioned above, the weight adjacency matrix is expected to play a crucial role in this project as well. The goal of this subproject is a classification and a comparison of convolutional codes in the sense of classical coding theory.
This project focuses on the algebraic theory of error-correcting codes. Such codes protect data against alteration through noise and enable the reconstruction of the original data from the corrupted information. All current standards of data transmission and data storage have built-in error correcting mechanisms. The most famous examples are the compact disk player, cell phones, the internet, satellite and deep space communication systems. The project aims at deepening the mathematical theory of convolutional codes. This specific class of codes forms one of the major players in many communication schemes, mainly in deep space communication and other wireless data transmission schemes. With the ever-growing demand of sophisticated information technology there is a continuing need of a thorough mathematical theory in order to improve the performance of such communication devices.
