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Codes and Invariant Theory

Gabriele Nebe·E. M. Rains·N. J. A. Sloane
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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 $u^2 = 0$..

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Gabriele Nebe, E. M. Rains, N. J. A. Sloane. 2003-11-04. Codes and Invariant Theory. https://arxiv.org/abs/math/0311046

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