r-Minimal Poset Codes
Abstract
In this paper, we propose and study r-minimal codes with respect to P-support, where P=(Ω,preccurlyeqP) is a poset defined on the coordinate set of the ambient space H. r-Minimal P-codes are natural extensions of Hamming metric minimal codes that have been extensively studied in the literature. We characterize r-minimal 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 (P,ω)-weight defined on H, where ω:Ω⟶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 P-codes, both for general P and for the special case that P is a disjoint union of chains. When P is hierarchical, we characterize r-minimal 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}.
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BibTeX
@misc{zheng2026rminimal,
title = {{$r$-Minimal Poset Codes}},
author = {Jianhua Zheng and Yang Xu and Yidan Chen and Haibin Kan and Guangyue Han},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.13520},
eprint = {2607.13520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13520},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - r-Minimal Poset Codes
AU - Jianhua Zheng
AU - Yang Xu
AU - Yidan Chen
AU - Haibin Kan
AU - Guangyue Han
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.13520
UR - https://arxiv.org/abs/2607.13520
AB - In this paper, we propose and study r-minimal codes with respect to P-support, where P=(Ω,preccurlyeq_P) is a poset defined on the coordinate set of the ambient space H. r-Minimal P-codes are natural extensions of Hamming metric minimal codes that have been extensively studied in the literature. We characterize r-minimal 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 (P,ω)-weight defined on H, where ω:Ω⟶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 P-codes, both for general P and for the special case that P is a disjoint union of chains. When P is hierarchical, we characterize r-minimal 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}.
ER -