Zeta functions for formal weight enumerators and the extremal property

Abstract

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 formal weight enumerators. It is shown that the zeta function for a formal weight enumerator has a similar structure to that of the weight enumerator of a Type II code. The notion of the extremal formal weight enumerators is introduced and an analogue of the Mallows-Sloane bound is obtained. Moreover the ternary case is considered.