  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.
