New extremal singly even self-dual codes of lengths $64$ and $66$

TL;DR

This paper constructs new extremal singly even self-dual codes of lengths 64 and 66, and introduces 40 new inequivalent extremal doubly even self-dual codes of length 64 with optimal covering radius, expanding known code classes.

Contribution

It presents the first known extremal singly even self-dual codes for lengths 64 and 66, and 40 new extremal doubly even codes of length 64 with optimal covering radius.

Findings

New extremal singly even self-dual codes for lengths 64 and 66.

40 new inequivalent extremal doubly even self-dual codes of length 64.

Codes meet known optimal bounds such as the Delsarte bound.

Abstract

For lengths $64$ and $66$, we construct extremal singly even self-dual codes with weight enumerators for which no extremal singly even self-dual codes were previously known to exist. We also construct new $40$ inequivalent extremal doubly even self-dual $[64,32,12]$ codes with covering radius $12$ meeting the Delsarte bound.