The automorphism group of a self-dual [72,36,16] code is not an elementary abelian group of order 8

TL;DR

This paper investigates the automorphism group of a hypothetical extremal self-dual binary linear code of length 72, proving it cannot be an elementary abelian group of order 8, thus narrowing possible automorphism group structures.

Contribution

It demonstrates that the automorphism group of such a code is not elementary abelian of order 8, refining the understanding of its possible automorphism groups.

Findings

Aut(C) is not isomorphic to elementary abelian group of order 8

Aut(C) has order at most 5

Provides computational evidence using Magma

Abstract

The existence of an extremal self-dual binary linear code C of length 72 is a long-standing open problem. We continue the investigation of its automorphism group: looking at the combination of the subcodes fixed by different involutions and doing a computer calculation with Magma, we prove that Aut(C) is not isomorphic to the elementary abelian group of order 8. Combining this with the known results in the literature one obtains that Aut(C) has order at most 5.