Variants of Jacobi polynomials in coding theory

TL;DR

This paper introduces new polynomial invariants called complete joint Jacobi polynomials for linear codes over finite fields and rings, establishing identities and generalizations that enhance the algebraic understanding of code properties.

Contribution

It defines the complete joint Jacobi polynomial, derives a MacWilliams type identity, and generalizes the average polynomial representations for codes over different algebraic structures.

Findings

Established the MacWilliams type identity for complete joint Jacobi polynomials.

Introduced the concepts of average Jacobi polynomial and average complete joint Jacobi polynomial.

Provided a generalization for the average of higher-fold complete joint Jacobi polynomials.

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.