In the present paper, we give Assmus–Mattson type theorems for codes and lattices. We show that a binary doubly even self-dual code of length 24m with minimum weight 4m provides a combinatorial 1-design and an even unimodular lattice of rank 24m with minimum norm 2m provides a spherical 3-design. We remark that some of such codes and lattices give t-designs for higher t. As a corollary, we give some restrictions on the weight enumerators of binary doubly even self-dual codes of length 24m with minimum weight 4m. Ternary and quaternary analogues are also given.

中文翻译：

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关于Assmus-Mattson型定理的注释

在本文中，我们给出了代码和格的Assmus-Mattson型定理。我们表明，长度为24 m且最小权重为4 m的二进制双偶自对偶代码提供了组合1设计，而秩为24 m且最小范数为2 m的均匀单模晶格提供了球形3设计。我们注意到一些这样的代码和格给出牛逼-designs更高的牛逼。作为推论，我们对长度为24 m且最小权重为4 m的二进制双偶甚至对偶代码的加权枚举数施加一些限制。还给出了三元和四元类似物。