  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 $k \ge 148$.
