From weight enumerators to zeta functions
From weight enumerators to zeta functions
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从权重枚举器到 zeta 函数
DOI：
10.1016/s0166-218x(00)00344-9
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发表时间：
2001
期刊：
影响因子：
--
通讯作者：
I. Duursma
中科院分区：
文献类型：
--
作者：
I. Duursma
在中，我们引入了任意线性码的Zeta函数。该定义是由代数曲线的性质以及由这些曲线构造的码的性质引起的。在这篇文章中，我们给出了作用于线性码的删截和缩短算子的另一个等价定义。对于某些无限族的可除码，我们计算了Zeta函数。利用Zeta函数的概念，可以用类似的Riemann假设来表示代码。证明了这种黎曼假设与线性码参数的上界之间的关系。黎曼假设类比的证明是开放的，上界是猜测的。
In , we introduced, for an arbitrary linear code, its zeta function. The definition is motivated by properties of algebraic curves and of codes constructed with these curves. In this paper, we give an alternative but equivalent definition in terms of the puncturing and shortening operators acting on a linear code. For certain infinite families of divisible codes, we compute the zeta functions. With the notion of a zeta function, an analogue of the Riemann hypothesis can be formulated for codes. We show the relation between such a Riemann hypothesis and upper bounds on the parameters of linear codes. The proof of the Riemann hypothesis analogue is open and the upper bounds are conjectural.