Codes and Invariant Theory
<|reference_start|>Codes and Invariant Theory: 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 $u^2 = 0$..<|reference_end|>
@article{nebe2003codes,
    title={Codes and Invariant Theory},
    author={Gabriele Nebe, E.M. Rains, N.J.A. Sloane},
    journal={arXiv preprint arXiv:math/0311046},
    year={2003},
    archivePrefix={arXiv},
    eprint={math/0311046},
    primaryClass={math.NT cs.IT math.IT}
}