{
  "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}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13520",
  "author_total": 5,
  "authors": [
    {
      "name": "Jianhua Zheng"
    },
    {
      "name": "Yang Xu"
    },
    {
      "name": "Yidan Chen"
    },
    {
      "name": "Haibin Kan"
    },
    {
      "name": "Guangyue Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/54cdsh5r08hy4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13520",
  "fulltext_url": null,
  "id": "54cdsh5r08hy4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13520",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7168743507",
  "pdf_url": "https://arxiv.org/pdf/2607.13520",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/54cdsh5r08hy4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2607.13520",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13520"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "$r$-Minimal Poset Codes",
  "title_text": "r-Minimal Poset Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/54cdsh5r08hy4",
  "venue": "arXiv",
  "versions": []
}