{
  "abstract": "For an integer $q\\ge2$, a matroid is $q$-entropic if its rank function, multiplied by $\\log q$, is the joint-entropy function of random variables on a $q$-element alphabet. We prove that a matroid is $4$-entropic if and only if it is representable over $\\F_4$. The corresponding statements for alphabet sizes two and three were known. The proof combines minor closure and the excluded-minor characterization of quaternary matroids with structural properties of quasigroups of order four. Thus arbitrary four-symbol partition representations yield no matroids beyond the quaternary ones. As an application, every access structure admitting an ideal perfect scheme with a uniform four-symbol secret and four-symbol active shares also admits an ideal $\\F_4$-linear scheme.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.20553",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohammad Hossein Kalantari"
    },
    {
      "name": "Shahram Khazaei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fkc8w8y3drf4e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.20553",
  "fulltext_url": null,
  "id": "fkc8w8y3drf4e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.20553",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204144938",
  "pdf_url": "https://arxiv.org/pdf/2608.20553",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fkc8w8y3drf4e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2608.20553",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.20553"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Four-Entropic Matroids Are Quaternary",
  "title_text": "Four-Entropic Matroids Are Quaternary",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/fkc8w8y3drf4e",
  "venue": "arXiv",
  "versions": []
}