{
  "abstract": "A quaternary Legendre pair of length $\\ell$ is a pair of sequences over the fourth roots of unity whose periodic autocorrelations sum to $-2$ at every nonzero shift. Each such pair yields a quaternary Hadamard matrix of order $2\\ell+2$ and a binary Hadamard matrix of order $4\\ell+4$. Fourteen even lengths up to 100 remained open in the 2025 tables of Kotsireas, Koutschan and Winterhof and of Jedwab and Pender. We settle seven of them --- the four smallest open lengths, 42, 46, 52, and 58, together with 66, 72, and 80 --- and print every pair in full. Checking one takes a few lines of code and exact arithmetic over the Gaussian integers, so the main results can be confirmed without running any of our software. Each search writes a pair as four binary sequences under a Gray map, restricts to reflection-symmetric rows, imposes an exact spectral constraint at the midpoint frequency, and matches candidates meet-in-the-middle on their autocorrelation vectors; from length 52 onward, exact compression along divisor chains shrinks the search before any candidate is lifted. Everything is deterministic: no seed, heuristic stopping rule, or floating-point value enters an acceptance decision. At lengths 64, 70 and 76 the same machinery ran to exhaustion and found nothing: the reversible four-block queue built at each length contains no pair. These are exhaustions of a specific recorded queue, not nonexistence results, and the certificates do not by themselves establish that the queue captures every reversible object of its length. A closing section sorts every claim in the paper by what a reader must accept in order to believe it, and states exactly how far the accompanying artifacts support each one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04589",
  "author_total": 1,
  "authors": [
    {
      "name": "Nikita Lebedev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5xzpszb55k3p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04589",
  "fulltext_url": null,
  "id": "e5xzpszb55k3p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04589",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7211867043",
  "pdf_url": "https://arxiv.org/pdf/2609.04589",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5xzpszb55k3p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.04589",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04589"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Quaternary Legendre Pairs of Lengths 42, 46, 52, 58, 66, 72, and 80",
  "title_text": "Quaternary Legendre Pairs of Lengths 42, 46, 52, 58, 66, 72, and 80",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/e5xzpszb55k3p",
  "venue": "arXiv",
  "versions": []
}