{
  "abstract": "We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most $(J+1)!$, where $J$ is the column weight. For every fixed pair of column and row weights $2\\le J<L$, we show that this bound is attained for all sufficiently large integer lift sizes. First, we give one integer exponent matrix independent of the lift size $P$. Second, we show that independent uniform exponent choices attain the bound with probability $1-O_{J,L}(P^{-1})$. Both proofs use cycle conditions required by low-weight codewords and a lower bound on the number of terms in vectors satisfying polynomial check equations. Neither construction requires $P$ to be prime. We also give small-lift arrays attaining the bound 24 for $J=3$, $L=4,\\ldots,8$, and arrays with distance at least 28 for $J=4$, $L=5,\\ldots,8$, together with computational distance verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16836",
  "author_total": 1,
  "authors": [
    {
      "name": "Kenta Kasai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dg1gb6m7h7610.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16836",
  "fulltext_url": null,
  "id": "dg1gb6m7h7610",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16836",
  "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"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7213414170",
  "pdf_url": "https://arxiv.org/pdf/2609.16836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dg1gb6m7h7610.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "CPM-LDPC Codes Attaining the Minimum-Distance Bound",
  "title_text": "CPM-LDPC Codes Attaining the Minimum-Distance Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/dg1gb6m7h7610",
  "venue": "arXiv",
  "versions": []
}