编码理论中的雅可比多项式变体
Variants of Jacobi polynomials in coding theory
摘要：
本文引入了$\mathbb{F}_q$和$\mathbb{Z}_k$上长度为$n$的两个线性码的完全联合雅可比多项式的概念。给出了码的完备联合雅可比多项式的MacWilliams型恒等式.我们还引入了$\mathbb{F}_q$和$\mathbb{Z}_k$上的平均雅可比多项式和平均完全联合雅可比多项式的概念。利用$n$的合成及其在码中的分布，给出了长度为$n$的两个线性码在$\mathbb{F}_q$和$\mathbb{Z}_k$上的完全联合雅可比多项式的平均表示.进一步给出了$\mathbb{F}_{q}$和$\mathbb{Z}_{k}$上码的$(g+1)$ -折叠完全联合雅可比多项式的平均值的一个推广.最后，我们给出了两个码的平均Jacobi相交数的概念。
英文摘要：
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.
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