Type II codes, even unimodular lattices, and invariant rings

In this paper, we study self-dual codes over the ring Z(2k) Of the integers module 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z(2k) which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over Z(6) of length 24 is constructed, which gives a new construction of the Leech lattice, The complete and symmetrized weight enumerators in genus g of codes over Z(2k) are introduced, and the MacWilliams identities for these weight enumerators are given, We investigate the groups which fix these weight enumerators of Type II codes over Z(2k) and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z(2k) are also introduced.