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

TL;DR

This work addresses the MacWilliams extension property (MEP) and constant weight codes under the $ω$-weighted Hamming metric on $F^{Ω}$ by an elementary linear-algebraic approach. It derives two key identities for $ω$-weight of subspaces via double counting and develops local-to-global equivalence criteria, yielding necessary and sufficient conditions for MEP in terms of transitivity and the unique decomposition property (UDP). It also provides generator-matrix characterizations for constant weight codes and unifies these results with the classical Hamming metric when $ω≡1$, recovering known results about MEP and repetitions of the dual of the Hamming code. The framework offers a character-theory-free, combinatorial perspective with potential applications to channels exhibiting nonuniform coordinate weights.

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 $ω:Ω\longrightarrow\mathbb{R}^{+}$ 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.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.1
• Proposition 2.2