{
  "abstract": "We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For $r=g=3$, every integer $\\ell\\geq4$, and each channel of capacity greater than $R=r/\\ell$, there is a sequence of code realizations whose actual transmitted rate tends to $R$ and whose average sum-product bit error tends to zero. For $r=g=2$ and each $\\ell\\in\\{3,4,5\\}$, an interval of binary symmetric channels has capacity greater than $R$ but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for $\\ell\\geq33$ with exact interval certificates for $4\\leq\\ell\\leq32$, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15526",
  "author_total": 1,
  "authors": [
    {
      "name": "Kenta Kasai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cqx6rm6m8qpwm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15526",
  "fulltext_url": null,
  "id": "cqx6rm6m8qpwm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15526",
  "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",
      "1705|Error Correcting Code Techniques"
    ]
  ],
  "openalex": "W7213260367",
  "pdf_url": "https://arxiv.org/pdf/2609.15526",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cqx6rm6m8qpwm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15526",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15526"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees",
  "title_text": "Spatially Coupled MacKay-Neal Codes Achieve Capacity on BMS Channels at Fixed Degrees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/cqx6rm6m8qpwm",
  "venue": "arXiv",
  "versions": []
}