Type II Codes over

Type II 4-codes are introduced as self-dual codes over the integers modulo 4 containing the all-one vector and with Euclidean weights multiple of 8. Their weight enumerators are characterized by means of invariant theory. A notion of extremality for the Euclidean weight is introduced. Their binary images under the Gray map are formally self-dual with even weights. Extended quadratic residue 4-codes are the main example of this family of codes. They are obtained by Hensel lifting of the classical binary quadratic residue codes. Their binary images have good parameters. With every type II 4-code is associated via construction A modulo 4 an even unimodular lattice (type II lattice). In dimension 32, we construct two unimodular lattices of norm 4 with an automorphism of order 31. One of them is the Barnes–Wall lattice BW32.