Automorphisms of doubly-even self-dual binary codes

TL;DR

This paper investigates the automorphism groups of binary doubly-even self-dual codes, establishing their containment in the alternating group and characterizing the existence of such codes with specific symmetry groups.

Contribution

It provides a complete characterization of the automorphism groups of these codes and conditions for their existence based on group and module properties.

Findings

Automorphism groups are always contained in the alternating group.

Existence of G-invariant codes depends on divisibility and module multiplicity conditions.

Characterization links code symmetry to group-theoretic and module-theoretic criteria.

Abstract

The automorphism group of a binary doubly-even self-dual code is always contained in the alternating group. On the other hand, given a permutation group $G$ of degree $n$ there exists a doubly-even self-dual $G$-invariant code if and only if $n$ is a multiple of 8, every simple self-dual $\F_2G$-module occurs with even multiplicity in $\F_2^n$, and $G$ is contained in the alternating group.