{
  "abstract": "A Hadamard matrix is a square matrix with entries $\\pm 1$ whose rows are mutually orthogonal. Hadamard matrices are conjectured to exist at every order divisible by $4$; for approximately the last twenty years, the smallest open case of this conjecture was order 668. In August 2026, Alpöge, Voinov and Reynolds-Haertle announced Hadamard matrices at the twelve orders below $2000$ at which none was known. In this paper we verify and analyse the claimed matrices. We survey results on the existence of Hadamard matrices and list the $95$ orders up to $10\\,000$ at which no Hadamard matrix is currently known. In pursuing this work we constructed Hadamard matrices at sixteen orders that were open in the published literature. We conclude with a list of open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27175",
  "author_total": 4,
  "authors": [
    {
      "name": "Ronan Egan"
    },
    {
      "name": "Ilias Kotsireas"
    },
    {
      "name": "Stephen London"
    },
    {
      "name": "Pádraig Ó Catháin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0jq7zjygd75s7.bib",
  "cited_by": 1,
  "doi": "10.48550/arxiv.2610.27175",
  "fulltext_url": null,
  "id": "0jq7zjygd75s7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27175",
  "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": "W7223155683",
  "pdf_url": "https://arxiv.org/pdf/2610.27175",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0jq7zjygd75s7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27175",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27175"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Recent progress in the construction of Hadamard matrices",
  "title_text": "Recent progress in the construction of Hadamard matrices",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:28Z",
  "url": "https://research.draconitic.com/paper/0jq7zjygd75s7",
  "venue": "arXiv",
  "versions": []
}