A coding theoretic approach to extending designs

Abstract We introduce the study of designs in a coset of a binary code which can be held by vectors of a fixed weight. If C is a binary [2 n , n , d ] code with n odd and the words of weights n - 1 and n + 1 hold complementary t -designs, then we show that the vectors of weight n in a coset of weight 1 also hold a t -design. We also show how to “extend” these designs. We then consider designs in cosets of type I self-dual codes, in particular in the shadow. If the vectors of a fixed weight in the code hold t -designs then so do the vectors of a fixed weight in the shadow. For [24 k - 2, 12 k - 1, 2 + 4 k ] type I codes, these designs extend to designs in the type II parent code.