Reflexivity of Partitions Induced by Weighted Poset Metric and Combinatorial Metric

TL;DR

This paper investigates the reflexivity of partitions induced by weighted poset and combinatorial metrics on finite abelian groups, establishing conditions for reflexivity, duality, and properties like MacWilliams identity, with applications to counterexamples of existing conjectures.

Contribution

It provides new sufficient and necessary conditions for reflexivity of partitions induced by weighted poset metrics and explores properties like MacWilliams identity and extension, including counterexamples to prior conjectures.

Findings

Reflexivity conditions for partitions induced by weighted poset metrics.

Existence of non-reflexive partitions induced by combinatorial metrics.

Partitions satisfying MacWilliams identity are mutually dual and reflexive.

Abstract

Let $\mathbf{H}$ be the Cartesian product of a family of finite abelian groups. Via a polynomial approach, we give sufficient conditions for a partition of $\mathbf{H}$ induced by weighted poset metric to be reflexive, which also become necessary for some special cases. Moreover, by examining the roots of the Krawtchouk polynomials, we establish non-reflexive partitions of $\mathbf{H}$ induced by combinatorial metric. When $\mathbf{H}$ is a vector space over a finite field $\mathbb{F}$, we consider the property of admitting MacWilliams identity (PAMI) and the MacWilliams extension property (MEP) for partitions of $\mathbf{H}$. With some invariance assumptions, we show that two partitions of $\mathbf{H}$ admit MacWilliams identity if and only if they are mutually dual and reflexive, and any partition of $\mathbf{H}$ satisfying the MEP is in fact an orbit partition induced by some subgroup of $\Aut_{\mathbb{F}}(\mathbf{H})$, which is necessarily reflexive. As an application of the aforementioned results, we establish partitions of $\mathbf{H}$ induced by combinatorial metric that do not satisfy the MEP, which further enable us to provide counter-examples to a conjecture proposed by Pinheiro, Machado and Firer in \cite{39}.