On a 5-design related to a putative extremal doubly even self-dual code of length a multiple of 24

By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs with the same parameters as the above 5-designs. We give some parameters of a self-orthogonal 5-design whose existence is equivalent to that of an extremal doubly even self-dual code of length 24m for m = 3, 4, 5, 6. If m ∈ {1, . . . , 6}, k ∈ {m + 1, . . . , 5m − 1} and (m,k) 6= (6, 18), then it is shown that an extremal doubly even self-dual code of length 24m is generated by codewords of weight 4k.