Higher and extended Jacobi polynomials for codes

TL;DR

This paper introduces Jacobi polynomial generalizations of classical coding invariants, establishing new identities and methods to compute these polynomials, which enhance the understanding of code structures over finite fields.

Contribution

It develops Jacobi polynomial analogues for weight enumerators, proves a Jacobi version of MacWilliams identity, and provides methods to compute these polynomials from harmonic weight enumerators.

Findings

Jacobi polynomials generalize classical invariants in coding theory.

The Jacobi analogue of MacWilliams identity is established.

Higher Jacobi polynomials can be computed from harmonic higher weight enumerators.

Abstract

In this paper, we introduce Jacobi polynomial generalizations of several classical invariants in coding theory over finite fields, specifically, the higher and extended weight enumerators, and we establish explicit correspondences between the resulting Jacobi polynomials. Moreover, we present the Jacobi analogue of MacWilliams identity for both higher and extended weight enumerators. We also present that the higher Jacobi polynomials for linear codes whose subcode supports form $t$-designs can be uniquely determined from the higher weight enumerators of the codes via polarization technique. Finally, we demonstrate how higher Jacobi polynomials can be computed from harmonic higher weight enumerators with the help of Hahn polynomials.