Automorphism of order 2p in binary self-dual extremal codes of length a multiple of 24

TL;DR

This paper investigates automorphisms of order 2p in binary self-dual extremal codes of length multiple of 24, establishing conditions for projectivity and analyzing automorphism groups of length 120 codes.

Contribution

It introduces a module-theoretic criterion linking automorphism properties with subcodes, and applies it to classify automorphisms in length 120 extremal codes.

Findings

All involutions are fixed point free except in Golay and length 120 codes.

Provides criteria to determine if a code is projective as an F_2<g>-module.

Shows automorphism groups of length 120 codes lack elements of order 38 and 58.

Abstract

Let C be a binary self-dual code with an automorphism g of order 2p, where p is an odd prime, such that g^p is a fixed point free involution. If C is extremal of length a multiple of 24 all the involutions are fixed point free, except the Golay Code and eventually putative codes of length 120. Connecting module theoretical properties of a self-dual code C with coding theoretical ones of the subcode C(g^p) which consists of the set of fixed points of g^p, we prove that C is a projective F_2<g>-module if and only if a natural projection of C(g^p) is a self-dual code. We then discuss easy to handle criteria to decide if C is projective or not. As an application we consider in the last part extremal self-dual codes of length 120, proving that their automorphism group does not contain elements of order 38 and 58.