On the Automorphism Group of a Binary Self-dual [120, 60, 24] Code

TL;DR

This paper investigates the automorphism group of a hypothetical binary self-dual [120, 60, 24] code, proving restrictions on automorphisms of order 3 and describing possible group structures and composition factors.

Contribution

It provides new constraints on the automorphism group of the code, including fixed point properties and divisibility conditions, which were previously unknown.

Findings

Automorphisms of order 3 have no fixed points.

The automorphism group's order divides 2^a·3·5·7·19·23·29.

The automorphism group is solvable if it contains a prime order p≥7.

Abstract

We prove that an automorphism of order 3 of a putative binary self-dual [120, 60, 24] code C has no fixed points. Moreover, the order of the automorphism group of C divides 2^a.3.5.7.19.23.29 where a is a nonegative integer. Automorphisms of odd composite order r may occur only for r=15, 57 or r=115 with corresponding cycle structures 15-(0,0,8;0), 57-(2,0,2;0) or 115-(1,0,1;0), respectively. In case that all involutions act fixed point freely we have |Aut(C)|<=920, and Aut(C) is solvable if it contains an element of prime order p>=7. Moreover, the alternating group A_5 is the only non-abelian composition factor which may occur.