Harmonic higher and extended weight enumerators

TL;DR

This paper introduces harmonic generalizations of classical weight enumerators for codes over finite fields, extending Greene's Theorem and deriving a MacWilliams-type identity, which leads to a new proof of the Assmus-Mattson Theorem.

Contribution

It develops harmonic versions of higher and extended weight enumerators, generalizes Greene's Theorem, and establishes a MacWilliams-type identity for these new enumerators.

Findings

Harmonic higher weight enumerators generalize classical polynomials.

A new MacWilliams-type identity for harmonic weight enumerators is derived.

The approach provides a novel proof of the Assmus-Mattson Theorem.

Abstract

In this paper, we present the harmonic generalizations of well-known polynomials of codes over finite fields, namely the higher weight enumerators and the extended weight enumerators, and we derive the correspondences between these weight enumerators. Moreover, we present the harmonic generalization of Greene's Theorem for the higher (resp. extended) weight enumerators. As an application of this Greene's-type theorem, we provide the MacWilliams-type identity for harmonic higher weight enumerators of codes over finite fields. Finally, we use this new identity to give a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields using harmonic higher weight enumerators.