On extremal self-dual codes of length 120

TL;DR

This paper investigates the automorphism groups of hypothetical extremal self-dual binary codes of length 120, establishing prime divisors and cycle structures of automorphisms, thus constraining their possible symmetries.

Contribution

It identifies the prime divisors of automorphism group orders and characterizes automorphisms of prime order for extremal self-dual codes of length 120.

Findings

Primes dividing automorphism groups are limited to 2, 3, 5, 7, 19, 23, 29.

Automorphisms of prime order ≥ 5 have a unique cycle structure.

Constraints on automorphism groups narrow the search for such codes.

Abstract

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even [120, 60, 24] code are 2, 3, 5, 7, 19, 23 and 29. Furthermore we prove that automorphisms of prime order $p \geq 5$ have a unique cycle structure.