The Automorphism Group of a Self Dual Binary [72,36,16] Code Does Not Contain Z4

TL;DR

This paper proves that extremal self-dual binary [72,36,16] codes do not have automorphisms of order 4, combining theoretical analysis with extensive computational verification.

Contribution

It provides a complete parametrization of self-dual binary codes with automorphisms of order 4 and shows such automorphisms do not exist in extremal codes.

Findings

Extremal binary [72,36,16] codes lack automorphisms of order 4.

A parametrization of codes with automorphisms of order 4 is established.

Computational methods confirm the non-existence of order 4 automorphisms in extremal codes.

Abstract

It has been proven in a series of works that the order of the automorphism group of a binary [72,36,16] code does not exceed five. We obtain a parametrization of all self-dual binary codes of length 72 with automorphism of order 4 which can be extremal. We use extensive computations in MAGMA and on a supercomputer to show that an extremal binary code of length 72 does not have an element of order 4.