{
  "abstract": "We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical $T/T^\\dagger$ gates and whose distance is at least 3. These codes start with 8-divisible $X$-stabilizers defined by projective geometry, which guarantees $d_Z \\geq 3$. A SAT solver then finds three $X$-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from $n=48$ to $n=496$, with $d_Z=3$ or 4, and illustrate the method with the [[48,3,3]] code $Q_{48}$. Additionally, we prove that no CSS code with three logical qubits, $Z$-distance at least 3, and a quasi-transversal CCZ gate (transversal $T$ with a diagonal Clifford correction) exists below $n=39$. The [[47,3,3]] code of Jacinto et al. attains the minimum length in this class, as established by the classification of Wills, Jain, and Singh.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09341",
  "author_total": 2,
  "authors": [
    {
      "name": "Bohan Lu"
    },
    {
      "name": "Kenneth R. Brown"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f00maaz0rygv4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09341",
  "fulltext_url": null,
  "id": "f00maaz0rygv4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09341",
  "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|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7221030017",
  "pdf_url": "https://arxiv.org/pdf/2610.09341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f00maaz0rygv4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:32Z",
      "native_id": "2610.09341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Building codes with transversal CCZ using projective geometry and SAT solvers",
  "title_text": "Building codes with transversal CCZ using projective geometry and SAT solvers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/f00maaz0rygv4",
  "venue": "arXiv",
  "versions": []
}