Classification of Binary Self-Dual [48,24,10] Codes with an Automorphism of Odd Prime Order

TL;DR

This paper completes the classification of binary self-dual [48,24,10] codes with odd prime automorphisms, establishing that only automorphisms of order 3 exist and identifying exactly 264 such codes.

Contribution

It proves that only automorphisms of order 3 are possible for these codes and precisely counts the number of inequivalent codes with such automorphisms.

Findings

Only automorphisms of order 3 are possible.

Exactly 264 inequivalent codes have an automorphism of order 3.

The classification is complete for these codes with odd prime automorphisms.

Abstract

The purpose of this paper is to complete the classification of binary self-dual [48,24,10] codes with an automorphism of odd prime order. We prove that if there is a self-dual [48, 24, 10] code with an automorphism of type p-(c,f) with p being an odd prime, then p=3, c=16, f=0. By considering only an automorphism of type 3-(16,0), we prove that there are exactly 264 inequivalent self-dual [48, 24, 10] codes with an automorphism of odd prime order, equivalently, there are exactly 264 inequivalent cubic self-dual [48, 24, 10] codes.