Codes over rings, complex lattices and Hermitian modular forms

Abstract

We introduce the finite ring S-2m = Z(2m) + iZ(2m). 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 Z(2m) 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. (C) 2004 Elsevier Ltd. All rights reserved.