{
  "abstract": "Let $RS_q(k)$ denote the full-length Reed--Solomon code over $\\mathbb{F}_q$. Every word $u=(u_x)_{x\\in \\mathbb{F}_q} \\in \\mathbb{F}_q^q$ has a unique interpolation polynomial $U(X)\\in \\mathbb{F}_q[X]$ of degree less than $q$, satisfying $U(x)=u_x$ for all $x\\in \\mathbb{F}_q$. It is well known that every word with $°U=k$ is a deep hole. The Cheng--Murray conjecture asserts that these are the only deep holes for $3\\le k\\le q-2$, except when $q$ is even and $k=q-3$. Previous results established the conjecture for $k\\le p-1$ and for $k\\ge\\lfloor(q-1)/2\\rfloor$, where $p$ is the characteristic of $\\mathbb{F}_q$. In this paper, we prove the Cheng--Murray conjecture in the remaining range $3\\le k\\le\\lfloor q/2\\rfloor$, thereby resolving it completely. Our proof combines two complementary approaches. The tangent envelope of a marked arc yields differential constraints that establish the conjecture for $3\\le k\\le q/2$ when $q$ is even, and for $3\\le k\\le\\lceil q/3\\rceil$ when $q$ is odd. For the remaining cases with $q$ odd, we combine local syndrome compatibility with combinatorial arguments to rule out additional deep holes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31531",
  "author_total": 1,
  "authors": [
    {
      "name": "Weijun Fang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/chtzah88mys7q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31531",
  "fulltext_url": null,
  "id": "chtzah88mys7q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31531",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223333330",
  "pdf_url": "https://arxiv.org/pdf/2610.31531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/chtzah88mys7q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31531"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Cheng-Murray Conjecture on Deep Holes of Full-Length Reed-Solomon Codes",
  "title_text": "The Cheng-Murray Conjecture on Deep Holes of Full-Length Reed-Solomon Codes",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/chtzah88mys7q",
  "venue": "arXiv",
  "versions": []
}