On s-extremal singly even self-dual [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≥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≥148. We also determine the covering radii of some extremal doubly even self-dual codes of length 80.