On $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes

TL;DR

This paper explores the relationship between s-extremal singly even self-dual codes and extremal doubly even self-dual codes, constructs new examples, and establishes non-existence results for certain code lengths meeting the Delsarte bound.

Contribution

It establishes a relationship between two classes of self-dual codes, constructs the first known s-extremal singly even self-dual [56,28,10] code, and proves non-existence of certain extremal codes for large lengths.

Findings

Constructed the first s-extremal singly even self-dual [56,28,10] code.

Proved no extremal doubly even self-dual code of length 24k+8 with covering radius meeting the Delsarte bound exists for k ≥ 137.

Proved no extremal doubly even self-dual code of length 24k+16 with covering radius meeting the Delsarte bound exists for k ≥ 148.

Abstract

A relationship between $s$-extremal singly even self-dual $[24k+8,12k+4,4k+2]$ codes and extremal doubly even self-dual $[24k+8,12k+4,4k+4]$ codes with covering radius meeting the Delsarte bound, is established. As an example of the relationship, $s$-extremal singly even self-dual $[56,28,10]$ codes are constructed for the first time. In addition, we show that there is no extremal doubly even self-dual code of length $24k+8$ with covering radius meeting the Delsarte bound for $k \ge 137$. Similarly, we show that there is no extremal doubly even self-dual code of length $24k+16$ with covering radius meeting the Delsarte bound for $k \ge 148$.