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Variants of Jacobi polynomials in coding theory
Designs, Codes and Cryptography ( IF 1.6 ) Pub Date : 2021-08-28 , DOI: 10.1007/s10623-021-00923-2
Himadri Shekhar Chakraborty 1, 2 , Tsuyoshi Miezaki 3
Affiliation  

In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length n over \({\mathbb {F}}_{q}\) and \({\mathbb {Z}}_{k}\). We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts of the average Jacobi polynomial and the average complete joint Jacobi polynomial over \({\mathbb {F}}_{q}\) and \({\mathbb {Z}}_{k}\). We give a representation of the average of the complete joint Jacobi polynomials of two linear codes of length n over \({\mathbb {F}}_{q}\) and \({\mathbb {Z}}_{k}\) in terms of the compositions of n and its distribution in the codes. Further we present a generalization of the representation for the average of the \((g+1)\)-fold complete joint Jacobi polynomials of codes over \({\mathbb {F}}_{q}\) and \({\mathbb {Z}}_{k}\). Finally, we give the notion of the average Jacobi intersection number of two codes.



中文翻译:

编码理论中雅可比多项式的变体

在本文中,我们介绍的长度的两个线性码完整的关节雅可比多项式的概念 Ñ超过\({\ mathbb {F}} _ {Q} \)\({\ mathbb {Z}} _ {ķ }\)。我们给出了完整的联合雅可比代码多项式的 MacWilliams 类型恒等式。我们还介绍了\({\mathbb {F}}_{q}\)\({\mathbb {Z}}_{k}\)上的平均雅可比多项式和平均完整联合雅可比多项式的概念。我们给出了两个长度为n 的线性代码 在\({\mathbb {F}}_{q}\)\({\mathbb {Z}}_{k} \)n的组合而言 及其在代码中的分布。此外,我们提出的代表性的一般化的平均值的\((G + 1)\)倍进行完全焊缝雅可比码的多项式超过 \({\ mathbb {F}} _ {Q} \)\({ \mathbb {Z}}_{k}\)。最后,我们给出了两个代码的平均 Jacobi 交集数的概念。

更新日期:2021-08-29
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