{
  "abstract": "For every prime power $q>19683$, we settle the Cheng--Murray deep-hole problem for full-length Reed--Solomon codes in every dimension $2\\le k\\le q-2$, with the characteristic-two cases stated explicitly. For odd $q$, the deep holes are precisely the words of degree $k$. For even $q$ and $k\\ge3$, the same conclusion holds except for an additional family when $k=q-3$. In even characteristic and dimension two, every deep hole has a unique expression $a h(X)+bX+c$, where $a\\ne0$ and $h$ is a normalized $o$-polynomial. The proof combines quadratic quotient algebras, graph-theoretic weighted independent-set inequalities, and planar arc completion. We also obtain normal rational curve rigidity in the corresponding nonexceptional dimensions, the projective Reed--Solomon covering radii, and distinct-coordinate solvability and remainder criteria.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29429",
  "author_total": 2,
  "authors": [
    {
      "name": "Bocong Chen"
    },
    {
      "name": "Jing Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51s754hsa8yp3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29429",
  "fulltext_url": null,
  "id": "51s754hsa8yp3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29429",
  "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.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223226826",
  "pdf_url": "https://arxiv.org/pdf/2610.29429",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51s754hsa8yp3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29429",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29429"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Deep holes of full-length Reed--Solomon codes",
  "title_text": "Deep holes of full-length Reed--Solomon codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/51s754hsa8yp3",
  "venue": "arXiv",
  "versions": []
}