A note on Assmus–Mattson type theorems

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 24 m with minimum weight 4 m provides a combinatorial 1-design and an even unimodular lattice of rank 24 m with minimum norm 2 m 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 24 m with minimum weight 4 m . Ternary and quaternary analogues are also given.