Jacobi polynomials, invariant rings, and generalized $t$-designs

TL;DR

This paper explores the relationships between Jacobi polynomials, invariant rings, and generalized $t$-designs in coding theory, providing new formulas, generators, and identities for these mathematical objects across different genera.

Contribution

It introduces new connections between Jacobi polynomials and weight enumerators, identifies generators of invariant rings for genus one, and generalizes relations to higher genus cases.

Findings

Jacobi polynomials in genus g relate to weight enumerators via polarization.

Generators of the invariant ring for g=1 are explicitly obtained.

MacWilliams type identities for split Jacobi polynomials are established.

Abstract

In the present paper, we provide results that relate the Jacobi polynomials in genus $g$. We show that if a code is $t$-homogeneous that is, the codewords of the code for every given weight hold a $t$-design, then its Jacobi polynomial in genus $g$ with composition $T$ with $|T|\leq t$ can be obtained from its weight enumerator in genus~$g$ using the polarization operator. Using this fact, we investigate the invariant ring, which relates the homogeneous Jacobi polynomials of the binary codes in genus $g$. Specifically, the generators of the invariant ring appearing for $g=1$ are obtained. Moreover, we define the split Jacobi polynomials in genus~$g$ and obtain the MacWilliams type identity for it. A split generalization for higher genus cases of the relation between the Jacobi polynomials and weight enumerator of a $t$-homogeneous code also given.