{
  "abstract": "The Berger-Charpin conjecture predicts that the permutation automorphism group of a cyclic code is generally the $q$-affine group, which is generated by the shift and the Frobenius multiplier on the index set. The word {\\em generally} is not made precise in the literature, and the goal of this series of papers is to study when the Berger-Charpin conjecture or its variants hold. In this first part, we establish a natural connection between the theory of cyclic codes and the theory of cyclotomic association schemes (equivalently, Schur rings over cyclic groups). Using it, we show that the $q$-affine group is not the correct group to expect for certain code lengths. We characterize these lengths by an arithmetic condition, and show that they have density zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33050",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanni Wu"
    },
    {
      "name": "Ziqing Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7d77bp1c8e68x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33050",
  "fulltext_url": null,
  "id": "7d77bp1c8e68x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33050",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214767787",
  "pdf_url": "https://arxiv.org/pdf/2609.33050",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7d77bp1c8e68x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33050",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33050"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Permutation automorphism groups of cyclic codes I: cyclotomic association schemes",
  "title_text": "Permutation automorphism groups of cyclic codes I: cyclotomic association schemes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/7d77bp1c8e68x",
  "venue": "arXiv",
  "versions": []
}