An Elementary Approach to MacWilliams Extension Property and Constant Weight Code with Respect to Weighted Hamming Metric

TL;DR

This paper presents an elementary linear algebra-based characterization of the MacWilliams extension property and constant weight codes under weighted Hamming metrics, generalizing known results for standard Hamming metrics.

Contribution

It introduces a simple approach to analyze MEP and constant weight codes with respect to weighted Hamming metrics using elementary linear algebra and double-counting identities.

Findings

Characterizes MEP for weighted Hamming metrics

Shows extension of weight-preserving maps to isometries

Recovers classical results for Hamming metric codes

Abstract

In this paper, we characterize the MacWilliams extension property (MEP) and constant weight codes with respect to $\omega$-weight defined on $\mathbb{F}^{\Omega}$ via an elementary approach, where $\mathbb{F}$ is a finite field, $\Omega$ is a finite set, and $\omega:\Omega\longrightarrow\mathbb{R}^{+}$ is a weight function. Our approach relies solely on elementary linear algebra and two key identities for $\omega$-weight of subspaces derived from a double-counting argument. When $\omega$ is the constant $1$ map, our results recover two well-known results for Hamming metric code: (1) any Hamming weight preserving map between linear codes extends to a Hamming weight isometry of the entire ambient space; and (2) any constant weight Hamming metric code is a repetition of the dual of Hamming code.