{
  "abstract": "Let $q$ be a power of an odd prime $p$. We prove the MDS conjecture over $\\mathbb F_q$ in every dimension $k$ satisfying \\[ 2\\leqslant k\\leqslant B(p,q) \\quad\\text{or}\\quad q+2-B(p,q)\\leqslant k\\leqslant q, \\qquad B(p,q)=\\left\\lfloor\\frac{(p-2)q+6p-10}{2p-3}\\right\\rfloor. \\] For fixed $p$, the first interval is linear in $q$; together with duality, the theorem covers an asymptotic proportion $1-1/(2p-3)$ of all dimensions. The proof rests on a vanishing theorem for determinant relations over an arbitrary field of characteristic $p>0$. It yields full row rank for Chowdhury's matrices over a larger range of arc sizes. In particular, at $|G|=2k-3+n$ it proves Chowdhury's full-row-rank conjecture without the $q$-dependent restriction. Specialization to $\\mathbb F_q$, together with the Ball--Lavrauw construction, gives the stated MDS range. We also prove that, for every odd prime power $q$, every normal rational curve in $\\mathrm{PG}(N,q)$ is complete for $2\\leqslant N\\leqslant q-2$, and every projective Reed--Solomon code of length $q+1$ and dimension $2\\leqslant k\\leqslant q-2$ has covering radius $q-k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03740",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiang Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2z0p5ttcr9mfn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03740",
  "fulltext_url": null,
  "id": "2z0p5ttcr9mfn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03740",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
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      "openalex",
      "1702|Coding theory and cryptography"
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  "openalex": "W7220685889",
  "pdf_url": "https://arxiv.org/pdf/2610.03740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2z0p5ttcr9mfn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.03740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A linear-in-$q$ range of dimensions for the MDS conjecture over $\\mathbb F_q$ in odd characteristic",
  "title_text": "A linear-in-q range of dimensions for the MDS conjecture over 𝔽_q in odd characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/2z0p5ttcr9mfn",
  "venue": "arXiv",
  "versions": []
}