A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

中文翻译：

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形式权重枚举器和切比雪夫多项式

形式权重枚举器是两个变量中的齐次多项式，其行为类似于自对偶线性码的汉明权重枚举器，不同之处在于系数不一定是非负整数。形式权枚举器的概念首先由 Ozeki 在与模形式相关的情况下引入，并且 Chien 已结合 zeta 函数和线性代码的黎曼假设对形式权枚举器进行了系统的研究。在本文中，我们建立了形式权重枚举器和切比雪夫多项式之间的关系。具体而言，具有指定参数 $$(n,\varepsilon ,q)$$ 的形式权重枚举器的存在条件是根据切比雪夫多项式给出的。根据 n 的奇偶性和符号 $$\varepsilon$$ ，四种切比雪夫多项式出现在结果的陈述中。此外，我们使用 Dickson 多项式获得了在 n 为奇数或 $$\varepsilon =-1$$ 的情况下形式权重枚举数的显式表达式，该多项式概括了切比雪夫多项式。我们还从二项式矩的角度陈述了一个猜想，并看到本文的结果部分支持了该猜想。