{
  "abstract": "Recent work shows that ordinary Reed--Solomon codes over prime fields admit polynomial-size lists up to list-decoding capacity on every evaluation set. We show a contrasting phenomenon over binary extension fields. For every fixed rate $0<R<1$, there exists $\\varepsilon_R>0$ such that, for all sufficiently large $h$, the full-field Reed--Solomon code over $\\mathbb{F}_{2^h}$ has a received word with $n^{Ω_R(\\log n)}$ codewords within relative distance $1-R-\\varepsilon_R$. To our knowledge, these are the first superpolynomial ordinary Reed--Solomon list-size lower bounds at fixed positive rate and a fixed positive agreement gap above the rate. Thus capacity-approaching polynomial list size for Reed--Solomon codes is not uniform over the underlying field, and the obstruction occurs at every fixed rate. As a puncturing consequence, the same holds for primitive Reed--Solomon codes; the obstruction is also robust under controlled dense puncturing. The proof begins with an exact correspondence that turns certain binary Bose--Chaudhuri--Hocquenghem weight-enumerator coefficients into sizes of single Reed--Solomon lists. A classical binary weight family gives the initial quarter-rate construction, while a quadratic-form enumeration supplies its superpolynomial multiplicity. A common-kernel quotient/descent argument and common-root padding then extend the construction to all rates. We also determine the optimized agreement envelope achieved by the construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15815",
  "author_total": 1,
  "authors": [
    {
      "name": "Minshen Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f8788mt9j9vme.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15815",
  "fulltext_url": null,
  "id": "f8788mt9j9vme",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15815",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
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    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
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      "openalex",
      "1702|Coding theory and cryptography"
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  "openalex": "W7221816052",
  "pdf_url": "https://arxiv.org/pdf/2610.15815",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f8788mt9j9vme.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15815",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15815"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Superpolynomial Lists for Reed-Solomon Codes over Binary Extension Fields",
  "title_text": "Superpolynomial Lists for Reed-Solomon Codes over Binary Extension Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/f8788mt9j9vme",
  "venue": "arXiv",
  "versions": []
}