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

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 $k \ge 148$.Comment: 15 pages, minor revisio

Singly even self-dual codes of length 24k+1024k+10 and minimum weight 4k+24k+2

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.Comment: 16 page

On the existence of ss-extremal singly even self-dual codes

We construct new $s$-extremal singly even self-dual codes of minimum weights $8,10,12$ and $14$. We also give tables for the currently known results on the existence of $s$-extremal singly even self-dual codes of minimum weights $8,10,12$ and $14$.Comment: 16 page