An extremal [72,36,16] binary code has no automorphism group containing Z2xZ4, Q_8, or Z_{10}

TL;DR

This paper investigates the automorphism group of an extremal self-dual binary code of length 72, proving it cannot contain certain subgroups of order 8 or an element of order 10, thus narrowing its possible symmetries.

Contribution

It demonstrates that the automorphism group of the code excludes specific groups like Z2xZ4, Q_8, and Z_{10}, providing new constraints on its structure.

Findings

Automorphism group does not contain Z2xZ4, Q_8, or Z_{10}

Code is a free module over certain automorphisms

Excluded subgroups of order 8 in automorphism group

Abstract

Let $C$ be an extremal self-dual binary code of length 72 and $g\in \Aut(C) $ be an automorphism of order 2. We show that $C$ is a free $\F_2<g>$ module and use this to exclude certain subgroups of order 8 of $\Aut (C)$. We also show that $\Aut(C)$ does not contain an element of order 10.