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The formal weight enumerators were first introduced by M. Ozeki. They form aring of invariant polynomials which is similar to that of the weightenumerators of Type II codes. Later, the zeta functions for linear codes werediscovered and their theory was developed by I. Duursma. It was generalized tocertain invariant polynomials including Ozeki's formal weight enumerators bythe present author. One of the famous and important problems is whetherextremal weight enumerators satisfy the Riemann hypothesis. In this paper,first we formulate the notion of divisible formal weight enumerators andpropose an algorithm for the efficient search of the formal weight enumeratorsdivisible by two. The main tools are the binomial moments. It leads to thediscovery of several new families of formal weight enumerators. Then, as aresult, we find examples of extremal formal weight enumerators which do notsatisfy the Riemann hypothesis.

Construction of divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis