{
  "abstract": "Let $V$ be an $n$-dimensional vector space over the finite field $\\mathbb F_q$, where $n\\ge 2$. It is classical that $q+1$ proper subspaces are necessary and sufficient to cover $V$. We study extremal refinements of this covering theorem. For $1\\le m\\le q+1$, we determine the maximum possible size of the union of $m$ proper subspaces of $V$, proving that $$ \\max_{W_1,\\ldots,W_m<V} \\left|W_1\\cup\\cdots\\cup W_m\\right| = q^{n-2}\\bigl(1+m(q-1)\\bigr). $$ We also classify all equality cases: for $m\\ge 2$, equality holds precisely when the subspaces are distinct hyperplanes containing a common codimension-two subspace. As consequences, we obtain a structural classification of minimum covers of $V$ by proper subspaces and a sharp defect estimate for unions of $q$ proper subspaces. We then introduce basis-blocking families, namely families of proper subspaces whose union meets every basis of $V$. We prove that the minimum size of such a family is $q$ and classify all extremal families of this size. Finally, we establish the affine analogue and give an elementary recognition criterion for extremal families. Together, these results provide a unified extremal-combinatorial description of coverings of finite vector spaces by proper subspaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07051",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohsen Aliabadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/byxcf2rj0rhwb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07051",
  "fulltext_url": null,
  "id": "byxcf2rj0rhwb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07051",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221039209",
  "pdf_url": "https://arxiv.org/pdf/2610.07051",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/byxcf2rj0rhwb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.07051",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07051"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal subspace covers in finite vector spaces",
  "title_text": "Extremal subspace covers in finite vector spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/byxcf2rj0rhwb",
  "venue": "arXiv",
  "versions": []
}