{
  "abstract": "In their work on sphere packing and the modular bootstrap, Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini conjectured the exact high-dimensional exponent of the Cohn-Elkies sphere-packing linear program. OpenAI's Chapter 1 subsequently proved their conjecture by establishing that both Fourier sign-uncertainty radii are $(1/π+o(1))\\sqrt d$. We prove the binary coding analogue: \\[ R_D\\!\\left(\\frac12-\\frac1π\\right)=\\frac12. \\] We also formulate the two Krawtchouk sign-uncertainty problems and determine both of their asymptotics. If $A^{\\mathrm K}_{\\pm}(n)$ denotes the smallest radius $r$ for which a nonzero Krawtchouk $(\\pm1)$-eigenfunction $f$ exists with $f(0) = 0$ and $f(x) \\ge 0$ for all $|x| \\ge r$, then \\[ \\frac{A^{\\mathrm K}_{\\pm}(n)}n\\longrightarrow \\frac12-\\frac1π. \\] The lower bound proves a mass-concentration principle similar to OpenAI's Chapter 1 for Hamming space. The upper bound, on the other hand, follows the approach of the spherical-code construction in OpenAI's Chapter 2. Gay, Jeronimo, and Liu formulated a hierarchy for binary codes analogous to the spherical-code construction and improved the best known binary coding rate bounds by evaluating the first level of the corresponding hierarchy. We prove this hierarchy bounds the Delsarte program and give a construction at arbitrarily deep levels of the hierarchy, attaining the upper bound in the limit. The construction uses an $N$-qubit generalization of the pure-state channel of Alrabiah and Guruswami.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03736",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew Salmon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8tbwyjdd4sz3h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03736",
  "fulltext_url": null,
  "id": "8tbwyjdd4sz3h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03736",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7208786058",
  "pdf_url": "https://arxiv.org/pdf/2609.03736",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8tbwyjdd4sz3h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.03736",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03736"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The half-rate linear programming bound for binary codes is $\\frac12-\\frac1π$",
  "title_text": "The half-rate linear programming bound for binary codes is 1/2−1/π",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/8tbwyjdd4sz3h",
  "venue": "arXiv",
  "versions": []
}