{
  "abstract": "In this paper, we study the minimum subpacketization of $(n,k)$ Reed-Solomon (RS) codes over $\\mathbb{F}_{q^\\ell}$ admitting cut-set-optimal exact $\\mathbb{F}_q$-linear repair of every coordinate from any $d$ helpers. For $3\\le k<d\\le n-1$, write $s=d+1-k$ and denote this minimum by $\\ell_{min}(n,k,d;q)$. Using divisibility and relative-degree constraints on cross-ratio fields of evaluation sets, we establish $\\ell_{min}(n,k,d;q)\\ge sG_s(d+1)$, where $G_s(m)$ is the minimum product of $m$ pairwise coprime integers, each at least $s$. Let $Δ_s(n,d)$ be the minimum of $lcm(a_1,\\ldots,a_n)$ over positive integers $a_1,\\ldots,a_n$ satisfying $lcm(a_j:j\\in J\\cup\\{i\\})\\ge s~lcm(a_j:j\\in J)$ for every $d$-subset $J\\subseteq[n]$ and every $i\\in[n]\\setminus J$. We construct RS codes with subpacketization $sΔ_s(n,d)$ and further prove that $\\ell_{min}(n,k,d;q)=sΔ_s(n,d)=sG_s(n)$ whenever $d(d+2)\\geq 4(n+1)$, for every prime power $q$. In particular, this determines the minimum subpacketization in the full-helper case $d=n-1$, resolving the corresponding exact linear-repair case of Open Problem 20 posed by Ramkumar et al. in their survey on codes for distributed storage. We further give a finite formula for $G_s(n)$ and determine the exact threshold for its closed-form expression in terms of the first $n$ primes. For fixed $k$ and $d$, we prove $\\log\\ell_{min}(n,k,d;q)= Θ_{k,d}(\\log n\\log\\log n)$ as $n\\to\\infty$, using divisor-function estimates and cover-free families. An affine-plane construction shows that $Δ_s(n,d)<G_s(n)$ for some parameters, so $sG_s(n)$ is not a universal formula. We conjecture that $\\ell_{min}(n,k,d;q)=sΔ_s(n,d)$ for every prime power $q$ and all $3\\le k<d\\le n-1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.37522",
  "author_total": 4,
  "authors": [
    {
      "name": "Weijun Fang"
    },
    {
      "name": "Jing Qiu"
    },
    {
      "name": "Shu-Tao Xia"
    },
    {
      "name": "Fang-Wei Fu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afvg8vep451dt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.37522",
  "fulltext_url": null,
  "id": "afvg8vep451dt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.37522",
  "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",
      "1705|Advanced Data Storage Technologies"
    ]
  ],
  "openalex": "W7224135243",
  "pdf_url": "https://arxiv.org/pdf/2610.37522",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afvg8vep451dt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.37522",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.37522"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Minimum Subpacketization of Reed-Solomon Codes with Linear Repair Achieving the Cut-Set Bound",
  "title_text": "The Minimum Subpacketization of Reed-Solomon Codes with Linear Repair Achieving the Cut-Set Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/afvg8vep451dt",
  "venue": "arXiv",
  "versions": []
}