Weight Enumeration for Convolutional Codes

Intellectual Merit Statement: The proposed research aims at deepethng the mathematical theory of convolutional codes. This class of codes forms one of the major players in many communication schemes, mainly in deep space communication and other wireless data transmission schemes. This leads to a continuing need of a thorough mathematical theory in order to design codes with excellent performance. The error-correcting quality of a code can be measured by a variety of distance parameters that have been introduced in the engineering-oriented literature. The main idea of the proposed research is the investigation of convolutional codes based on the weight adjacency matrix, a single parameter that carries, among other things, all those different distance measures. So far two major theorems have been established by the P1: 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. For instance, only now self- dual codes can be studied theoretically. This and other consequences of the MacWilliams Identity will be explored in the first subproject of the proposed research. So far, only few individual examples of self-dual convolutional codes are known and no algebraic approach has been undertaken. The obvious fact that the weight adjacency matrix of a self-dual code is invariant under the MacWilliams transformation combined with the PT's detailed knowledge about that matrix will be crucial, if not indispensable, for pursuing this project. In addition, the close link between self-dual convolutional codes and self-dual block codes obtained by tail-biting will play a crucial role. It can be expected that, conversely, positive results will also have an impact on the theory of sell-dual tail-biting block codes. Furthermore, as yet another application of the MacWilliams Identity it is planned to explore the methods developed so far for minimal tail-biting trellises for block codes. It can be expected that positive results will also have an impact on minimal trellises for convolutional codes.