# Some new results on the self-dual [120,60,24] code

## TL;DR

This paper investigates the automorphism group of the long-standing open problem of the extremal self-dual binary linear code of length 120, providing new restrictions and conditions that advance understanding of its structure.

## Contribution

The paper proves that automorphisms of certain orders cannot occur and establishes new necessary conditions for the code's existence based on shadow and design theory.

## Key findings

- Automorphisms of order 30 and 57 cannot occur.
- Automorphisms of order 8 cannot occur under fixed point free involutions.
- Automorphism group order is at most 120 with additional restrictions.

## Abstract

The existence of an extremal self-dual binary linear code of length 120 is a long-standing open problem. We continue the investigation of its automorphism group, proving that automorphisms of order 30 and 57 cannot occur. Supposing the involutions acting fixed point freely, we show that also automorphisms of order 8 cannot occur and the automorphism group is of order at most 120, with further restrictions. Finally, we present some necessary conditions for the existence of the code, based on shadow and design theory.