{
  "abstract": "Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width $f$, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance $d$ and check weight $w$. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations. First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by $1$ guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an $[[1088,128,22]]$ LP code with $f=22$ and an $[[1024,256,24]]$ PP code with $f \\leq 27$, both at $w=10$. At $w=11$, our PP constructions attain either $d=25$ with $f \\leq 27$ or $d=24$ with the proved optimum $f=25$, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06820",
  "author_total": 1,
  "authors": [
    {
      "name": "Jong Yeon Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6v5q24kg2t237.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06820",
  "fulltext_url": null,
  "id": "6v5q24kg2t237",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06820",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220700470",
  "pdf_url": "https://arxiv.org/pdf/2610.06820",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6v5q24kg2t237.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:05Z",
      "native_id": "2610.06820",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06820"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Designing Group-Valued Codes with Full Regular Low-Weight Bases",
  "title_text": "Designing Group-Valued Codes with Full Regular Low-Weight Bases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:05Z",
  "url": "https://research.draconitic.com/paper/6v5q24kg2t237",
  "venue": "arXiv",
  "versions": []
}