{
  "abstract": "Proximity gaps are central to the soundness of interactive oracle proofs of proximity (IOPPs) and polynomial commitment schemes (PCSs). An $[n,k,d]$ linear code $C\\subseteq\\mathbb F^n$ has a $δ$-proximity gap with error $ε$ if, for every $u_0,u_1\\in\\mathbb F^n$, either all points on $\\ell_{u_0,u_1}=\\{u_0+αu_1:α\\in\\mathbb F\\}$ are $δ$-close to $C$, or at most an $ε$ fraction are. Although proximity gaps for Hamming-metric codes are well understood, their rank-metric counterparts remain largely unexplored despite their applications in coding theory and cryptography. In this work, we study proximity gaps for linear rank-metric codes and their cryptographic applications. First, we show that every $[n,k,d]$ linear rank-metric code $C$ over $\\mathbb F_{q^m}$ admits a proximity gap for every $δ\\le(d-1)/(3n)$, with error at most $q^{e+1}/q^m$, where $e=\\lfloorδn\\rfloor$. For Gabidulin codes, we improve the gap to $(d-1)/(2n)$ with error $10q^{n-1}/q^m$. These two proximity gaps match those for general linear Hamming-metric codes and Reed--Solomon (RS) codes, respectively. We prove the $(d-1)/(2n)$ bound is tight by constructing an infinite family of constant-rate Gabidulin codes and affine lines $\\ell_{u_0,u_1}$ on which a $1-o(1)$ fraction of points are $d/(2n)$-close to the code, while $u_1$ is at least $3d/(4n)$-far from it. At the $d/(3n)$ gap, we also give a counterexample establishing a lower bound on $ε$. As applications, we construct an IOPP for interleaved Gabidulin codes by adapting the Ligero IOPP for interleaved RS codes. We then adapt the Ligero-based PCS for ordinary polynomials to obtain a $q$-linearized polynomial commitment scheme. To our knowledge, this is the first PCS framework based on rank-metric error-correcting codes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09838",
  "author_total": 4,
  "authors": [
    {
      "name": "Songsong Li"
    },
    {
      "name": "Chaoping Xing"
    },
    {
      "name": "Chen Yuan"
    },
    {
      "name": "Ruiqi Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7sbpkw3ae4172.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09838",
  "fulltext_url": null,
  "id": "7sbpkw3ae4172",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09838",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7212157576",
  "pdf_url": "https://arxiv.org/pdf/2609.09838",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7sbpkw3ae4172.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09838",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09838"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Proximity Gaps for Gabidulin Codes and Applications",
  "title_text": "Proximity Gaps for Gabidulin Codes and Applications",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7sbpkw3ae4172",
  "venue": "arXiv",
  "versions": []
}