Isometries and MacWilliams Extension Property for Weighted Poset Metric

TL;DR

This paper characterizes isometries and the MacWilliams extension property for weighted poset metrics on modules, establishing conditions under which these properties hold and their implications for coding theory.

Contribution

It provides a comprehensive analysis of isometries and the MacWilliams extension property for weighted poset metrics, including necessary and sufficient conditions and structural decompositions.

Findings

The group of isometries for weighted poset metrics is characterized.

The MacWilliams extension property implies the unique decomposition property.

For hierarchical posets or constant weights, MEP relates to Hamming weight MEP.

Abstract

Let $\mathbf{H}$ be the cartesian product of a family of left modules over a ring $S$, indexed by a finite set $\Omega$. We are concerned with the $(\mathbf{P},\omega)$-weight on $\mathbf{H}$, where $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ is a poset and $\omega:\Omega\longrightarrow\mathbb{R}^{+}$ is a weight function. We characterize the group of $(\mathbf{P},\omega)$-weight isometries of $\mathbf{H}$, and give a canonical decomposition for semi-simple subcodes of $\mathbf{H}$ when $\mathbf{P}$ is hierarchical. We then study the MacWilliams extension property (MEP) for $(\mathbf{P},\omega)$-weight. We show that the MEP implies the unique decomposition property (UDP) of $(\mathbf{P},\omega)$, which further implies that $\mathbf{P}$ is hierarchical if $\omega$ is identically $1$. For the case that either $\mathbf{P}$ is hierarchical or $\omega$ is identically $1$, we show that the MEP for $(\mathbf{P},\omega)$-weight can be characterized in terms of the MEP for Hamming weight, and give necessary and sufficient conditions for $\mathbf{H}$ to satisfy the MEP for $(\mathbf{P},\omega)$-weight when $S$ is an Artinian simple ring (either finite or infinite). When $S$ is a finite field, in the context of $(\mathbf{P},\omega)$-weight, we compare the MEP with other coding theoretic properties including the MacWilliams identity, Fourier-reflexivity of partitions and the UDP, and show that the MEP is strictly stronger than all the rest among them.