Strengthening of the Assmus--Mattson theorem for some dual codes

We previously proposed the first nontrivial examples of a code having support $t$-designs for all weights obtained from the Assmus--Mattson theorem and having support $t'$-designs for some weights with some $t'>t$. This suggests the possibility of generalizing the Assmus--Mattson theorem, which is very important in design and coding theory. In the present paper, we generalize this example as a strengthening of the Assmus--Mattson theorem along this direction. As a corollary, we provide a new characterization of the extended Golay code $G_24$.