{
  "abstract": "Let $\\mathcal{C}$ be an $[n,k,d]$ linear code over $\\F_{q^2}$ with parity-check matrix $H$. The rank $r=\\Rank(HH^{\\dagger})$ measures how far the Hermitian dual $\\mathcal{C}^{\\perp_h}$ is from being Hermitian self-orthogonal, and it is the least number of coordinates that must be adjoined to $\\mathcal{C}^{\\perp_h}$ to make it Hermitian self-orthogonal; equivalently, the Hermitian hull $\\mathcal{C}\\cap\\mathcal{C}^{\\perp_h}$ has dimension $(n-k)-r$. In this paper, we develop a structure theory of the Hermitian hull of a cyclic code and apply it to the construction of quantum stabilizer codes. For a cyclic code with defining set $T\\subseteq\\mathbb{Z}_n$, we show that $r$ is additive over the orbits of the involution $τ(ν)=-qν$ on the $q^2$-cyclotomic cosets and satisfies a parity constraint. A parameter dictionary exhibits $r$ as a gauge trading one-for-one against the code length, and a complementation duality shows that the complement of $T$ realises the logical-qubit count of the quantum code. The extreme values of $r$ are settled by closed subset-sum criteria, and the $τ$-invariant defining sets are shown to be the unique locus on which the Hermitian dual is again cyclic. On the distance side, we determine the Hermitian dual distance of an $r$-fold extension of an arbitrary linear code exactly, and we prove that the admissible extension vectors lie in the image of $HH^{\\dagger}$, which reduces the search space from $q^{2(n-k)}$ to $q^{2r}$. Applied to cyclic, quasi-cyclic and BCH codes, the construction improves $23$ best-known parameters in Grassl's code tables, one of them provably optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.12242",
  "author_total": 5,
  "authors": [
    {
      "name": "Liangdong Lu"
    },
    {
      "name": "Ruipan Yang"
    },
    {
      "name": "Yang Liu"
    },
    {
      "name": "Qiang Fu"
    },
    {
      "name": "Guanmin Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/42qah90pq9jqe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.12242",
  "fulltext_url": null,
  "id": "42qah90pq9jqe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.12242",
  "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"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7168393492",
  "pdf_url": "https://arxiv.org/pdf/2607.12242",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/42qah90pq9jqe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:31Z",
      "native_id": "2607.12242",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.12242"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes",
  "title_text": "Hermitian Hulls of Cyclic Codes and Exact Dual Distances of Extended Linear Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/42qah90pq9jqe",
  "venue": "arXiv",
  "versions": []
}