Weight enumerators, intersection enumerators and Jacobi polynomials II

TL;DR

This paper introduces Jacobi polynomials and intersection enumerators for codes over finite fields and rings, explores their interrelations, and establishes MacWilliams type identities, advancing algebraic coding theory.

Contribution

It presents new definitions of Jacobi polynomials and intersection enumerators for codes over various algebraic structures, and derives MacWilliams identities relating them.

Findings

Defined Jacobi polynomials for codes over $\\mathbb{F}_q$ and $\mathbb{Z}_k$

Established interrelations among these polynomials and enumerators

Proved MacWilliams type identities for Jacobi polynomials

Abstract

In the present paper, we introduce the concepts of Jacobi polynomials and intersection enumerators of codes over $\mathbb{F}_q$ and $\mathbb{Z}_{k}$ for arbitrary genus $g$. We also discuss the interrelation among them. Finally, we give the MacWilliams type identities for Jacobi polynomials.