Let D be the support design of the minimum weight of an extremal binary doubly even self-dual [24m,12m,4m+4] code. In this note, we consider the case when D becomes a t-design with t \geq 6.Comment: 8 page

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In this paper, we present examples of codes all of whose weight classes support 1‐designs, with duals whose classes include two that support 2‐designs. We can find these examples in the triply even binary codes of length 48, which have been classified by Betsumiya and Munemasa.

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 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 δ(C) and s(C).