# Binary extremal self-dual codes of length $60$ and related codes

## TL;DR

This paper classifies specific self-dual codes of length 60, constructs new extremal codes with previously unknown weight enumerators, and explores restrictions on weight enumerators of related codes.

## Contribution

It provides a classification of four-circulant singly even self-dual codes of length 60 and constructs new extremal codes with novel weight enumerators.

## Key findings

- New extremal singly even self-dual [60,30,12] codes constructed.
- Existence of extremal codes with previously unknown weight enumerators.
- Restrictions on weight enumerators of certain self-dual codes with shadow of minimum weight 1.

## Abstract

We give a classification of four-circulant singly even self-dual $[60,30,d]$ codes for $d=10$ and $12$. These codes are used to construct extremal singly even self-dual $[60,30,12]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual $[60,30,12]$ codes, we also construct extremal singly even self-dual $[58,29,10]$ codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight $1$.