{
  "abstract": "Let $u$ be a received word for the full-length Reed--Solomon code $\\operatorname{RS}_q(k)$, with interpolation polynomial $f_u$ of degree $k+m$. For each $m\\in\\{3,4,5,6,7\\}$, we classify all such deep holes over every finite field, without a field-size hypothesis. None occur for odd $m$ in this range. At $m=4$ and $m=6$, the only exceptions are, respectively, the Segre sextic and Frobenius-power families; these occur only in dimension two over fields of characteristic two. For arbitrary $m\\geqslant3$, we also establish two uniform bounds. If $\\sqrt q\\geqslant12(m+1)\\log(2m+2)$, no word of interpolation degree $k+m$ is a deep hole when $q$ is odd, $k\\geqslant3$, or $m$ is odd. If $q>16(m+2)^4$, we obtain the corresponding complete classification at every admissible $k$, including the explicit dimension-two exceptions. Our proofs use an agreement criterion expressed in complete homogeneous symmetric functions, agreement-count estimates, finite geometry, permutation-polynomial restrictions, and subset constructions for solving the resulting symmetric-function equations. The remaining finite cases are settled by finite computations with supplied certificates and deterministic verification programs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31460",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiang Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9zv8c1q4kvs6b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31460",
  "fulltext_url": null,
  "id": "9zv8c1q4kvs6b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31460",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223318598",
  "pdf_url": "https://arxiv.org/pdf/2610.31460",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9zv8c1q4kvs6b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31460",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31460"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Deep holes of full-length Reed--Solomon codes: uniform bounds and exact classifications",
  "title_text": "Deep holes of full-length Reed--Solomon codes: uniform bounds and exact classifications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/9zv8c1q4kvs6b",
  "venue": "arXiv",
  "versions": []
}