不满足黎曼假设的可除形式权枚举数和极多项式的构造
Construction of divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis
摘要：
形式权枚举数是由M.关它们形成了一个不变多项式环，类似于II型码的重量计数器。后来，发现了线性码的zeta函数，并发展了它们的理论。杜尔斯玛本文作者将它推广到某些不变多项式，包括Ozeki的形式权计数器.极值权计数器是否满足黎曼假设是一个著名而重要的问题。在本文中，我们首先制定的概念，可分的形式重量枚举，并提出了一个算法的有效搜索的形式重量枚举可被二分。主要工具是二项式矩。它导致发现了几个新的家庭的正式重量枚举。然后，作为一个结果，我们发现的极值形式的重量枚举不满足黎曼假设的例子。
英文摘要：
The formal weight enumerators were first introduced by M. Ozeki. They form a ring of invariant polynomials which is similar to that of the weight enumerators of Type II codes. Later, the zeta functions for linear codes were discovered and their theory was developed by I. Duursma. It was generalized to certain invariant polynomials including Ozeki's formal weight enumerators by the present author. One of the famous and important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this paper, first we formulate the notion of divisible formal weight enumerators and propose an algorithm for the efficient search of the formal weight enumerators divisible by two. The main tools are the binomial moments. It leads to the discovery of several new families of formal weight enumerators. Then, as a result, we find examples of extremal formal weight enumerators which do not satisfy the Riemann hypothesis.
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