Jacobi polynomials and design theory I

TL;DR

This paper introduces Jacobi polynomials with multiple reference vectors in coding theory, establishes a MacWilliams type identity, and explores their relation to designs through new formulas and code types.

Contribution

It presents a novel definition of Jacobi polynomials with multiple references, derives a new formula using the Aronhold polarization operator, and links these polynomials to code design theory.

Findings

Derived a MacWilliams type identity for Jacobi polynomials with multiple references.

Obtained a formula for Jacobi polynomials using the Aronhold polarization operator.

Explored the relationship between Jacobi polynomials and designs via Type III and IV codes.

Abstract

In this paper, we introduce the notion of Jacobi polynomials with multiple reference vectors of a code, and give the MacWilliams type identity for it. Moreover, we derive a formula to obtain the Jacobi polynomials using the Aronhold polarization operator. Finally, we describe some facts obtained from Type III and Type IV codes that interpret the relation between the Jacobi polynomials and designs.