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

2025-11-02

AI Summary

This paper investigates the MacWilliams extension property (MEP) and constant-weight code structure for linear codes under weighted Hamming metrics. Specifically, it addresses two classical problems over finite fields endowed with a general weight function ω: (1) whether a linear map preserving ω-weight admits an extension to a global ω-isometry, and (2) whether every constant-ω-weight linear code must arise as a repetition of a dual Hamming code. Using elementary linear algebra and double counting, the authors derive two fundamental identities characterizing the ω-weight distribution of subspaces and establish a unified framework linking ω-weight with orthogonal complements. The results fully recover and generalize—under arbitrary (non-uniform) weight functions—the two landmark theorems known in the classical (uniform) Hamming metric. This work reveals how weight functions govern the structural universality of codes and provides a concise, unified, and elementary approach to generalized MacWilliams theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Mechanism Design
• Machine Learning: Other Foundations of Machine Learning

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• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
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Abstract

In this paper, we characterize the MacWilliams extension property (MEP) and constant weight codes with respect to $ω$-weight defined on $mathbb{F}^Ω$ via an elementary approach, where $mathbb{F}$ is a finite field, $Ω$ is a finite set, and $ω:Ωlongrightarrowmathbb{R}^{+}$ is a weight function. Our approach relies solely on elementary linear algebra and two key identities for $ω$-weight of subspaces derived from a double-counting argument. When $ω$ 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.

Problem

• Characterizing MacWilliams extension property for weighted Hamming metric codes
• Analyzing constant weight codes with respect to weighted Hamming metric
• Developing elementary approach using linear algebra and weight identities

Innovation

• Elementary linear algebra approach for extension property
• Double-counting argument for weighted Hamming metric
• Characterizing constant weight codes via subspace identities