{
  "abstract": "We study the minimum extension degree required for a Reed--Solomon code to admit exact $\\mathbb{F}_q$-linear repair at the cut-set bound from every prescribed set of $d$ helpers. Let $s=d-k+1$, and let $Q_s(n)$ be the minimum product of $n$ pairwise coprime integers, each at least $s$. For every prime power $q$ and all $3\\le k<d<n\\le2d-2$, we prove that the minimum degree over $\\mathbb{F}_q$ is exactly $sQ_s(n)$. When all surviving nodes are helpers, we also characterize the admissible extension degrees and bound the number of coordinates that can attain the cut-set bound. The required achievability follows from existing construction methods; we give a short existence proof using splitting subspaces. For a family of parameters with $n>2d-2$, we obtain strictly smaller extension degrees, showing that the range restriction cannot be omitted altogether. Finally, we prove that a nonlinear repair scheme attaining the cut-set bound induces an $\\mathbb{F}_p$-linear repair scheme on the same code, where $p$ is the characteristic. Consequently, the extension-degree lower bounds over the prime field remain valid without a linearity assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34377",
  "author_total": 1,
  "authors": [
    {
      "name": "Yuhan Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/epn9h6pd14f4p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34377",
  "fulltext_url": null,
  "id": "epn9h6pd14f4p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34377",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.34377",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/epn9h6pd14f4p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34377",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34377"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Exact Minimum Extension Degrees for Reed Solomon Codes under Linear and Nonlinear Repair at the Cut Set Bound",
  "title_text": "Exact Minimum Extension Degrees for Reed Solomon Codes under Linear and Nonlinear Repair at the Cut Set Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/epn9h6pd14f4p",
  "venue": "arXiv",
  "versions": []
}