Type II Codes over $$\mathbb{Z}/2k\mathbb{Z}$$ , Invariant Rings and Theta Series

Abstract

We consider type II codes over finite rings $$\mathbb{Z}/2k\mathbb{Z} $$ . It is well-known that their gth complete weight enumerator polynomials are invariant under the action of a certain finite subgroup of $$GL((2k)^g,\mathbb{C})$$ , which we denote Hk,g. We show that the invariant ring with respect to Hk,g is generated by such polynomials. This is carried out by using some closely related results concerning theta series and Siegel modular forms with respect to $$Sp(g,\mathbb{Z})$$ .