A note on the Assmus--Mattson theorem for some binary codes

Abstract

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 $t^{\prime }$-designs for some weights with some $t^{\prime }>t$. 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 ${\mathcal{G}}_{24}$.

21 pages. arXiv admin note: text overlap with arXiv:1805.10019