A note on the Assmus–Mattson theorem for some binary 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 G₂₄.