{
  "abstract": "Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the $Z$-logical quotient $K/S$ over $\\mathbb F_2[x,y]/(x^\\ell-1,y^m-1)$ and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity $d_Z=\\min(d_{\\mathcal A},d_{\\mathcal C})$. Using the Frobenius structure of the finite group algebra, we prove $r_{\\mathcal A}=r_{\\mathcal C}=k/2$ for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances $4,6,10,10,12,18$ of the six standard BB codes of lengths $18$ to $288$ and enumerates every minimum-weight logical operator. The resulting census shows that $[[108,8,10]]$ is the only one of the six whose distance is attained in a single component, with $d_{\\mathcal C}=10$ and $d_{\\mathcal A}=12$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39395",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohammad Rowshan"
    },
    {
      "name": "Simon Devitt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29nhtm2889sf1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39395",
  "fulltext_url": null,
  "id": "29nhtm2889sf1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39395",
  "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.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7215080573",
  "pdf_url": "https://arxiv.org/pdf/2609.39395",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29nhtm2889sf1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2609.39395",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39395"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes",
  "title_text": "Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/29nhtm2889sf1",
  "venue": "arXiv",
  "versions": []
}