TYPE-II CODES OVER Z(4)

Type II Z(4)-codes are introduced as self-dual codes over the integers module 4 containing the all-one vector and with Euclidean weights mul tiple of 8. Their weight enumerators are characterized by means of inv ariant theory, A notion of extremality for the Euclidean weight is int roduced, Their binary images under the Gray map are formally self-dual with even weights, Extended quadratic residue Z(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 h ave good parameters, With every type II Z(4)-code is associated via co nstruction A module 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.