Codes over rings, complex lattices and Hermitian modular forms

We introduce the finite ring S2m=Z2m+iZ2m. We develop a theory of self-dual codes over this ring and relate self-dual codes over this ring to complex unimodular lattices. We describe a theory of shadows for these codes and lattices. We construct a gray map from this ring to the ring Z2m and relate codes over these rings, giving special attention to the case when m=2. We construct various Hermitian modular forms from weight enumerators and give the correspondence between the invariant space, where the weight enumerators of codes reside, and the space of Hermitian modular forms.