{
  "abstract": "Motivated by higher-dimensional generalizations of Sudoku, we study exact block-structured decompositions, algebraic characterizations, and orthogonality for Sudoku hypercubes. Let $n=\\prod_{i=1}^d a_i$, let $b_i=n/a_i$, and consider the $λ$-fold complete $d$-uniform $d$-partite hypergraph with $d$ vertex classes of size $n$, where the $i$th class is partitioned into $a_i$ groups of size $b_i$. Given positive integers $m_1,\\dots,m_k$ with $\\sum_{j=1}^k m_j=λn^d$, we partition the edges into color classes of sizes $m_1,\\dots,m_k$ so that, in color $j$, vertex degrees and block counts are each either $\\lfloor m_j/n\\rfloor$ or $\\lceil m_j/n\\rceil$, while the multiplicity of an underlying edge is either $\\lfloor m_j/n^d\\rfloor$ or $\\lceil m_j/n^d\\rceil$. When $m_j=nr_j$, the vertex and block balances are exact, yielding block factorizations and higher-dimensional Sudoku analogues of Baranyai's theorem. Within the same block framework, we give a Delsarte characterization of the Sudoku condition using association schemes and study mutually orthogonal Sudoku hypercubes of order $q^3$ for prime powers $q$. For block sizes $(q^3,q^2,q)$ and $(q^3,q^3,1)$, the resulting families attain a general upper bound and are best possible. For block size $(q^2,q^2,q^2)$, we construct $q^2(q^2-1)(q^2-q)$ mutually orthogonal hypercubes; this construction is asymptotically best possible as $q\\to\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23975",
  "author_total": 2,
  "authors": [
    {
      "name": "Amin Bahmanian"
    },
    {
      "name": "Sho Suda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6kmhtfh302teq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23975",
  "fulltext_url": null,
  "id": "6kmhtfh302teq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23975",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.23975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6kmhtfh302teq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sudoku Analogues of Baranyai's Theorem",
  "title_text": "Sudoku Analogues of Baranyai's Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/6kmhtfh302teq",
  "venue": "arXiv",
  "versions": []
}