An upper bound of the value of t of the support t-designs of extremal Type III and IV codes

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 wight, 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).