{
  "abstract": "Let $Γ^s(2,q)$ be the graph induced in the Grassmann graph by the $q$-ary simplex codes of dimension 2. For $q=4$, this graph is known to have diameter 3. We prove that the same diameter occurs for every prime power $q\\geq5$. Together with the elementary cases $q=2,3$, this gives diam$Γ^s(2,q)=0,2,3$ for $q=2$, $q=3$, and $q\\geq4$, respectively. The upper bound is obtained from a consequence of a theorem of Marshall Hall on finite abelian groups. For $q\\geq5$ an explicit diagonal pair of simplex lines gives the matching lower bound; we give two proofs, one using moments and one using products. For $q\\geq7 we also retain an independent counting proof. The case $q=4$ is handled separately.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13574",
  "author_total": 1,
  "authors": [
    {
      "name": "Artur Siemaszko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3f8gwzcafzkp3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13574",
  "fulltext_url": null,
  "id": "3f8gwzcafzkp3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13574",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7213297975",
  "pdf_url": "https://arxiv.org/pdf/2609.13574",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3f8gwzcafzkp3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.13574",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13574"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A note on the diameter of graphs of two-dimensional simplex codes",
  "title_text": "A note on the diameter of graphs of two-dimensional simplex codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/3f8gwzcafzkp3",
  "venue": "arXiv",
  "versions": []
}