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

Abstract: 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,...,6. If $m \in {1,\ldots,6}$, $k \in {m+1,\ldots,5m-1}$ and $(m,k) \ne (6,18)$, then it is shown that an extremal doubly even self-dual code of length 24m is generated by codewords of weight 4k.