Zeta functions for formal weight enumerators and an analogue of the Mallows-Sloane bound

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It has various properties similar to those of the zeta function of an algebraic curve. This article extends Duursma's theory to the case of the formal weight enumerators and shows that there exists a similar structure to that of the weight enumerators of type II codes. The extremal property of the formal weight enumerators is also considered and an analogue of the Mallows-Sloane bound is deduced.