Grants and Contracts Details
Description
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.
Status | Finished |
---|---|
Effective start/end date | 10/1/09 → 9/30/13 |
Funding
- National Science Foundation: $183,418.00
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