Variants of Jacobi polynomials in coding theory

Abstract

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.