{
  "abstract": "Let $\\cS(k,q)$ be the point-line geometry whose points and lines correspond to one-dimensional and two-dimensional subcodes of $q$-ary simplex codes of dimension $k$, respectively. We determine the complete distance relation and the diameter of its collinearity graph. For two points whose representing vectors have different sets of zero coordinates there is a common neighbour. If the representing vectors have the same zero set, then a common neighbour exists precisely when $(q-1)q^{k-2}$ of their non-zero coordinate ratios can be selected so that their product is one. To prove this, we use a generalization of Hall's result for finite abelian groups. Applying the Erdős--Ginzburg--Ziv theorem, we establish that the above product condition holds for $k\\ge3$, which implies that the diameter is two. If $k=2$, then the diameter is one when $q=2$, two when $q=3$ and three when $q\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00916",
  "author_total": 1,
  "authors": [
    {
      "name": "Adam Tyc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9m6gkcqh9br0w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00916",
  "fulltext_url": null,
  "id": "9m6gkcqh9br0w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00916",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00916",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9m6gkcqh9br0w.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00916",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00916"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Distances in the collinearity graphs of geometries related to simplex codes",
  "title_text": "Distances in the collinearity graphs of geometries related to simplex codes",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9m6gkcqh9br0w",
  "venue": "arXiv",
  "versions": []
}