{
  "abstract": "We give a unified hidden-derivative framework for list decoding and mutual correlated agreement of ordinary Reed--Solomon codes over prime fields, on arbitrary prescribed evaluation sets. For every fixed slack $γ>0$, every sufficiently large block length $n$, every prime $q\\ge n$, and every dimension $1\\le k\\le(1-γ)n$, a deterministic algorithm finds all codewords within relative distance $1-k/n-γ$ in $q^{O_γ(1)}$ time. The final list has size $n^{O_γ(1)}$, independently of $q$. Both statements extend to bounded-input-list recovery, with constants depending additionally on the input-list bound. For every fixed curve degree $\\ell$, at most $n^{O_{γ,\\ell}(1)}$ parameters on a curve $f_0+zf_1+\\cdots+z^\\ell f_\\ell$ admit a nearby codeword whose exact agreement support is not a maximal jointly explained support of the coefficient words. For lines this gives MCA error $n^{O_γ(1)}/q$, with no proximity loss. The interpolation stage reparameterizes and optimizes the hidden-derivative construction of Brakensiek, Chen, Putterman, Zhang, and Zheng; differential root enumeration uses Kopparty's algorithm. We then prove that a specialization-safe differential equation has a cover by constant-dimensional varieties of polynomial cumulative degree, outside polynomially many parameter values. Intersecting these varieties with equations from the full agreement support yields both the field-size-independent list bound and exact-support MCA.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05870",
  "author_total": 1,
  "authors": [
    {
      "name": "Fernando Granha Jeronimo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/60t6w81h6benr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05870",
  "fulltext_url": null,
  "id": "60t6w81h6benr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05870",
  "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": "W7212063356",
  "pdf_url": "https://arxiv.org/pdf/2609.05870",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/60t6w81h6benr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05870",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05870"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed-Solomon Codes",
  "title_text": "Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed-Solomon Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/60t6w81h6benr",
  "venue": "arXiv",
  "versions": []
}