{
  "abstract": "Cooperative MSR codes are a kind of storage codes which enable optimal-bandwidth repair of any $h\\geq2$ node erasures in a cooperative way, while retaining the minimum storage as an $[n,k]$ MDS code. Each code coordinate (node) is assumed to store an array of $\\ell$ symbols, where $\\ell$ is termed as sub-packetization. To address the disk IO (input/output) capability, a cooperative MSR code is said to have optimal-access property, if during node repair, the amount of data accessed at each helper node meets a lower bound on this quantity. In this paper, we focus on reducing the sub-packetization level of optimal-access cooperative MSR codes. We propose new constructions of optimal-access cooperative MSR codes through two methods. At first, we propose a direct explicit construction by designing its parity-check matrix. Such parity-check matrix is built by repeatedly employing two crucial parity-check matrices as building blocks. Secondly, we propose a generic transformation framework. Starting from an arbitrary $[n+d-k,d]$ MDS scalar code, one can derive a final cooperative MSR code by systematically applying two basic transformations. Both approaches yield $(n,k,\\ell=δ^m)$ optimal-access cooperative MSR codes with $δ=d-k+h$ and $m=\\binom{n}{h}-\\lfloor\\frac{n}δ\\rfloor(\\binomδ{h}-1)$. Compared with the state of the art (with $\\ell=δ^{\\binom{n}{h}}$), the derived codes can reduce the sub-packetization $\\ell$ by a fraction of $1/δ^{\\lfloor\\frac{n}δ\\rfloor(\\binomδ{h}-1)}$, where $δ=d-k+h$. Moreover, we also show that some previous code structures of optimal-access cooperative MSR codes and optimal-access MSR codes with $h=1$ are included as special cases of our transformation construction. At last, we note that all of the constructions are built over a finite field of linear size $\\geq n+d-k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06372",
  "author_total": 3,
  "authors": [
    {
      "name": "Yaqian Zhang"
    },
    {
      "name": "Jingke Xu"
    },
    {
      "name": "Ya-Feng Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7vfy1z709zrj8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06372",
  "fulltext_url": null,
  "id": "7vfy1z709zrj8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06372",
  "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",
      "1705|Advanced Data Storage Technologies"
    ]
  ],
  "openalex": "W7212073181",
  "pdf_url": "https://arxiv.org/pdf/2609.06372",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7vfy1z709zrj8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06372",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06372"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation",
  "title_text": "Optimal-Access Cooperative MSR Codes: Parity-Check Matrix Construction And a Unified Transformation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/7vfy1z709zrj8",
  "venue": "arXiv",
  "versions": []
}