{
  "abstract": "A Legendre pair of length 333 would yield a Hadamard matrix of order 668, the smallest order presently unresolved by the Hadamard conjecture. We study the structured case in which both sequences are fixed by a common subgroup $H\\leq(\\mathbb Z/333\\mathbb Z)^\\times$ acting by coordinate multiplication. We prove that such a pair can exist only when $|H|\\leq 6$. After a mod-3 compression reduces the problem to an order-108 kernel, there are exactly 30 subgroups. We exclude 21 of them, including all 19 subgroups of order at least 9. The final order-9 subgroup is eliminated analytically: its orbit structure restricts the 9-compressed entries to $\\{\\pm1,\\pm17,\\pm19,\\pm35,\\pm37\\}$; the Legendre equations force a $+17,-17$ pair in one compressed sequence, and a single-shift autocorrelation bound then contradicts the required compressed correlation. The remaining exclusions use full-image compression, a row-sum congruence, exact meet-in-the-middle enumeration, and proof-carrying pseudo-Boolean encodings. The solver-assisted cases are accompanied by independently checked DRAT proofs or direct arithmetic certificates. The result constrains fixed common-multiplier symmetry only; the unrestricted existence problems remain open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.20765",
  "author_total": 3,
  "authors": [
    {
      "name": "Arthur F. Ramos"
    },
    {
      "name": "David B. Hulak"
    },
    {
      "name": "Ruy J. G. B. de Queiroz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6v0x40pjjmn44.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.20765",
  "fulltext_url": null,
  "id": "6v0x40pjjmn44",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.20765",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7170513268",
  "pdf_url": "https://arxiv.org/pdf/2607.20765",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6v0x40pjjmn44.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2607.20765",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.20765"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Multiplier obstructions for Legendre pairs of length 333",
  "title_text": "Multiplier obstructions for Legendre pairs of length 333",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/6v0x40pjjmn44",
  "venue": "arXiv",
  "versions": []
}