Nonexistence of certain singly even self-dual codes with minimal shadow

TL;DR

This paper investigates the existence and properties of certain singly even self-dual codes with minimal shadow, establishing nonexistence results for specific parameters and characterizing their weight enumerators.

Contribution

It proves nonexistence of certain codes with minimal shadow for large parameters and determines the weight enumerator for some classes, advancing understanding of code structure.

Findings

No such codes exist for large m in some parameter sets.

Weight enumerator is uniquely determined for certain code parameters.

The weight enumerator is not unique for other specific parameters.

Abstract

It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even self-dual codes with minimal shadow having minimum weight $d-2$ for these $(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow is uniquely determined and we also show that there is no singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ and $m \ge 160$, respectively. We demonstrate that the weight enumerator of a singly even self-dual code with minimal shadow is not uniquely determined for parameters $[24m+6,12m+3,4m+2]$ and $[24m+22,12m+11,4m+4]$.