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By the Assmus and Mattson theorem, the codewords of each nontrivial weight inan extremal doubly even self-dual code of length 24m form a self-orthogonal5-design. In this paper, we study the codes constructed from self-orthogonal5-designs with the same parameters as the above 5-designs. We give someparameters of a self-orthogonal 5-design whose existence is equivalent to thatof 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 $(m,k) \ne (6,18)$, thenit is shown that an extremal doubly even self-dual code of length 24m isgenerated by codewords of weight 4k.

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