{
  "abstract": "The Permutation Equivalence Problem (PEP) for linear complementary dual (LCD) codes reduces to Graph Isomorphism (GI) through the orthogonal projector onto the code, but this requires a trivial hull. We ask whether the reduction extends when the standard inner product is replaced by a nondegenerate symmetric bilinear form \\(M\\). We make precise what it means for \\(M\\) to yield a valid reduction to GI and prove that, for length \\(n \\geq 3\\) and every field size, this holds if and only if \\(M\\) is permutation-invariant. The reduction then reaches, beyond LCD codes, a class of codes with one-dimensional hull: exactly those whose hull is spanned by a vector with nonzero coordinate sum. We count these codes exactly; for odd \\(q\\), asymptotically the method covers a fraction \\(\\prod_{i \\geq 2}(1+q^{-i})^{-1}\\) of all codes, \\(1 + q^{-1}\\) times the fraction of LCD codes. A polynomial-time reduction algorithm is given.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.23120",
  "author_total": 1,
  "authors": [
    {
      "name": "Keita Ishizuka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bmx7jhwnm8e8f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.23120",
  "fulltext_url": null,
  "id": "bmx7jhwnm8e8f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.23120",
  "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.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7162313823",
  "pdf_url": "https://arxiv.org/pdf/2605.23120",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bmx7jhwnm8e8f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2605.23120",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.23120"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Closure of LCD-to-GI Reductions via Generalized Inner Products",
  "title_text": "The Closure of LCD-to-GI Reductions via Generalized Inner Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/bmx7jhwnm8e8f",
  "venue": "arXiv",
  "versions": []
}