Singly even self-dual codes of length $24k+10$ and minimum weight $4k+2$

TL;DR

This paper investigates the properties and construction methods of singly even self-dual codes of length 24k+10, providing new restrictions, a construction example, and exploring code neighbors with previously unknown weight enumerators.

Contribution

It introduces restrictions on weight enumerators, a method for constructing codes with minimal shadow, and presents the first known example of a singly even self-dual [82,41,14] code with minimal shadow.

Findings

Restrictions on weight enumerators for certain codes

First construction of a singly even self-dual [82,41,14] code with minimal shadow

New codes with previously unknown weight enumerators

Abstract

Currently, the existence of an extremal singly even self-dual code of length $24k+10$ is unknown for all nonnegative integers $k$. In this note, we study singly even self-dual $[24k+10,12k+5,4k+2]$ codes. We give some restrictions on the possible weight enumerators of singly even self-dual $[24k+10,12k+5,4k+2]$ codes with shadows of minimum weight at least $5$ for $k=2,3,4,5$. We discuss a method for constructing singly even self-dual codes with minimal shadow. As an example, a singly even self-dual $[82,41,14]$ code with minimal shadow is constructed for the first time. In addition, as neighbors of the code, we construct singly even self-dual $[82,41,14]$ codes with weight enumerator for which no singly even self-dual code was previously known to exist.