The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6

TL;DR

This paper proves that an extremal [72,36,16] code cannot have automorphisms of order 6 by analyzing its module structure and conducting an exhaustive search, thus resolving a long-standing open problem.

Contribution

It demonstrates that the automorphism group of an extremal [72,36,16] code lacks elements of order 6, using module decomposition and exhaustive computational methods.

Findings

No extremal [72,36,16] code has automorphisms of order 6.

The module structure under an order 6 automorphism is highly restrictive.

Exhaustive search confirms the non-existence of such automorphisms.

Abstract

The existence of an extremal code of length 72 is a long-standing open problem. Let C be a putative extremal code of length 72 and suppose that C has an automorphism g of order 6. We show that C, as an F_2<g>-module, is the direct sum of two modules, one easily determinable and the other one which has a very restrictive structure. We use this fact to do an exhaustive search and we do not find any code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.