{
  "abstract": "Fix a prime power q. We prove that deciding whether a received word y in Fnqm lies at rank distance at most n-2 from the one-dimensional Gabidulin code {{λ.g : λ in Fqm } is NP-complete. Thus maximum-likelihood decoding of Gabidulin codes is NP-hard even for k = 1, an analogue of the Guruswami-Vardy theorem for Reed-Solomon codes. In the Hamming metric, by contrast, hardness at this radius requires the dimension to grow with the length, as in the Guruswami-Vardy theorem. The reduction is deterministic; consequently Rank Syndrome Decoding (RSD) and Rank Minimum Distance (RMD) for Fqm -linear codes are NP-complete under deterministic polynomial reductions, whereas the previously known reduction was randomized. The proof goes through a pivot problem that we introduce, Twisted Square Membership (TSM): given Fq-subspaces U, W in FqN , decide whether some non-zero a in U satisfies a^(q+1) in W . Quadratic systems in d unknowns are encoded into TSM, then TSM into Gabidulin decoding. The last step views the received word as an Fq-linear map and decoding at radius n - 2 as the search for an Fq-plane on which it acts as a scalar. With support U + τ W in a cubic extension and map a + τ b to a^(q^2) + τ bq^q , two Frobenius twists force solution planes to encode the relation a^(q+1) in W . All constructions are explicit and polynomial, with N = O(qd^2 ), n less than N + d, and m = 3N .",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34404",
  "author_total": 1,
  "authors": [
    {
      "name": "Philippe Gaborit"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ez3smmb440c32.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34404",
  "fulltext_url": null,
  "id": "ez3smmb440c32",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34404",
  "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.CR"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223917855",
  "pdf_url": "https://arxiv.org/pdf/2610.34404",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ez3smmb440c32.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34404",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34404"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Maximum-Likelihood Decoding of Gabidulin Codes Is NP-Hard Even in Dimension One",
  "title_text": "Maximum-Likelihood Decoding of Gabidulin Codes Is NP-Hard Even in Dimension One",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/ez3smmb440c32",
  "venue": "arXiv",
  "versions": []
}