{
  "abstract": "A complete morphism of Butson classes replaces each entry of a matrix in $\\mathrm{BH}(n,k)$ by an $m \\times m$ block, uniformly, to produce a matrix in $\\mathrm{BH}(mn,t)$. We show that every complete morphism is described by a pair $(U,Z)$, where $Z \\in \\mathrm{BH}(m,t)$ and $U$ is a unitary matrix annihilated by the $k^{\\textrm{th}}$ cyclotomic polynomial. We deduce from a theorem of Lam and Leung that complete morphisms onto real Hadamard matrices exist precisely when $k$ is a power of $2$. The exponent-reducing morphisms of Östergård--Paavola and Ó Catháin--Swartz achieve the smallest block size the degree bound allows, and we show that there are no other complete morphisms for which $U$ is monomial. We construct new complete morphisms where $U$ is not monomial: a complete morphism $\\mathrm{BH}(n,7) \\rightarrow \\mathrm{BH}(6n,6)$ is given, the first to eliminate a prime from the alphabet.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28394",
  "author_total": 2,
  "authors": [
    {
      "name": "Ronan Egan"
    },
    {
      "name": "Pádraig Ó Catháin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6azrp7eancd95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28394",
  "fulltext_url": null,
  "id": "6azrp7eancd95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28394",
  "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": "W7223142952",
  "pdf_url": "https://arxiv.org/pdf/2610.28394",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6azrp7eancd95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28394",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28394"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On complete morphisms of Butson Hadamard matrices",
  "title_text": "On complete morphisms of Butson Hadamard matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/6azrp7eancd95",
  "venue": "arXiv",
  "versions": []
}