  Let $C$ be an extremal Type III or IV code and $D_{w}$ be the support design
of $C$ for a weight $w$. We introduce the two numbers $\delta(C)$ and $s(C)$:
$\delta(C)$ is the largest integer $t$ such that, for all weights, $D_{w}$ is a
$t$-design; $s(C)$ denotes the largest integer $t$ such that there exists a $w$
such that $D_{w}$ is a $t$-design. In the present paper, we consider the
possible values of $\delta(C)$ and $s(C)$.
