Harmonic Tutte polynomials of matroids II

TL;DR

This paper introduces harmonic generalizations of matroid and code polynomials over finite Frobenius rings, establishing identities and invariants that extend classical results in coding theory and matroid theory.

Contribution

It develops harmonic versions of Tutte and coboundary polynomials for demi-matroids and codes, proving new identities and invariance properties.

Findings

Harmonic $m$-tuple weight enumerators satisfy a MacWilliams-type identity.

Harmonic Tutte and coboundary polynomials are linked via Greene-type identities.

Structure of invariant spaces for harmonic weight enumerators of self-dual codes is characterized.

Abstract

In this work, we introduce the harmonic generalization of the $m$-tuple weight enumerators of codes over finite Frobenius rings. A harmonic version of the MacWilliams-type identity for $m$-tuple weight enumerators of codes over finite Frobenius ring is also given. Moreover, we define the demi-matroid analogue of well-known polynomials from matroid theory, namely Tutte polynomials and coboundary polynomials, and associate them with a harmonic function. We also prove the Greene-type identity relating these polynomials to the harmonic $m$-tuple weight enumerators of codes over finite Frobenius rings. As an application of this Greene-type identity, we provide a simple combinatorial proof of the MacWilliams-type identity for harmonic $m$-tuple weight enumerators over finite Frobenius rings. Finally, we provide the structure of the relative invariant spaces containing the harmonic $m$-tuple weight enumerators of self-dual codes over finite fields.