Gleason's Theorem on Self-Dual Codes and Its Generalizations

TL;DR

This paper discusses Gleason's theorem on self-dual codes, presents a comprehensive generalization that encompasses previous versions, and highlights its significance in coding theory and invariant theory.

Contribution

It introduces a broad generalization of Gleason's theorem that unifies all earlier versions and applications in a single framework.

Findings

Unified theorems for self-dual codes

Extended applicability to various code types

Connections to invariant theory

Abstract

One of the most remarkable theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes. In the past 36 years there have been hundreds of papers written about generalizations and applications of this theorem to different types of codes, always on a case-by-case basis. In this paper I state the theorem and then describe the far-reaching generalization that Gabriele Nebe, Eric Rains and I have developed which includes all the earlier generalizations at once. The full proof has just appeared in our book "Self-Dual Codes and Invariant Theory" (Springer, 2006). This paper is based on my talk at the conference on Algebraic Combinatorics in honor of Eiichi Bannai, held in Sendai, Japan, June 26-30, 2006.