$r$-Minimal Poset Codes

2026-07-15

AI Summary

This work investigates the structure and existence of $r$-minimal codes under poset metrics, extending the classical theory of minimal codes in the Hamming metric. By introducing cut $r$-blocking maps and $(P,\omega)$-weight functions, the paper proposes a novel definition of $r$-minimal $P$-codes and establishes their connection to minimal Hamming codes. The main contributions include a generalization of the Ashikhmin–Barg criterion, revealing an equivalence between $r$-minimal $P$-codes and blocking sets; necessary and sufficient conditions for their existence; proofs of existence for both general posets and disjoint unions of chains; and a complete characterization of cut $r$-blocking sets induced by two-level hierarchical posets, thereby resolving an open problem in the field.

Abstract

In this paper, we propose and study $r$-minimal codes with respect to $\mathbf{P}$-support, where $\mathbf{P}=(Ω,\preccurlyeq_{\mathbf{P}})$ is a poset defined on the coordinate set of the ambient space $\mathbf{H}$. $r$-Minimal $\mathbf{P}$-codes are natural extensions of Hamming metric minimal codes that have been extensively studied in the literature. We characterize $r$-minimal $\mathbf{P}$-codes in terms of the notion so called cutting $r$-blocking maps, which generalizes the well-known equivalence between minimal Hamming metric codes and cutting blocking sets. We also give a necessary and sufficient condition for $r$-minimality in terms of $(\mathbf{P},ω)$-weight defined on $\mathbf{H}$, where $ω:Ω\longrightarrow\mathbb{R}^{+}$ is an arbitrary weight function. This leads to a generalization of the well-known Ashikhmin-Barg criterion for Hamming metric minimal codes. We then prove two existence results for $r$-minimal $\mathbf{P}$-codes, both for general $\mathbf{P}$ and for the special case that $\mathbf{P}$ is a disjoint union of chains. When $\mathbf{P}$ is hierarchical, we characterize $r$-minimal $\mathbf{P}$-codes in terms of $r$-minimal Hamming metric codes. Finally, we characterize cutting $r$-blocking sets induced by hierarchical posets with two levels, which further enables us to answer a question raised in Hyun, Kim, Wu and Yue \cite{28}.

Problem

• r-minimal codes
• poset codes
• cutting r-blocking sets
• P-support
• Ashikhmin-Barg criterion

Innovation

• r-minimal codes
• poset metric
• cutting r-blocking sets
• Ashikhmin-Barg criterion
• hierarchical posets