Codes and Invariant Theory
Abstract
The main theorem in this paper is a far-reaching generalization of Gleason's theorem on the weight enumerators of codes which applies to arbitrary-genus weight enumerators of self-dual codes defined over a large class of finite rings and modules. The proof of the theorem uses a categorical approach, and will be the subject of a forthcoming book. However, the theorem can be stated and applied without using category theory, and we illustrate it here by applying it to generalized doubly-even codes over fields of characteristic 2, doubly-even codes over the integers modulo a power of 2, and self-dual codes over the noncommutative ring $\F_q + \F_q u$, where $u^2 = 0$..
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- November 2003
- DOI:
- 10.48550/arXiv.math/0311046
- arXiv:
- arXiv:math/0311046
- Bibcode:
- 2003math.....11046N
- Keywords:
-
- Mathematics - Number Theory;
- Computer Science - Information Theory;
- 94B05;
- 13A50;
- 94B60