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 $δ(C)$ and $s(C)$: $δ(C)$ 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 $δ(C)$ and $s(C)$.