{
  "abstract": "We prove that a nonempty subset of the binary vector space $\\mathbb{F}_2^7$ is spectral if and only if it tiles $\\mathbb{F}_2^7$ by translations. This is the smallest binary dimension left open by Ferguson and Sothanaphan. For the implication from tiling to spectrality we show that every eight-point subset of $\\mathbb{F}_2^d$ admits $2^{d-3}$ frequencies whose pairwise differences avoid the zeros of its Fourier transform; consequently every tile of $\\mathbb{F}_2^d$ with at least $2^{d-3}$ elements is spectral. For the converse implication, we use the Smith normal form of the Walsh matrix to exclude every spectral subset of $\\mathbb{F}_2^7$, and of $\\mathbb{F}_2^8$, whose size is not a power of two. The sizes $16$ and $32$ are reduced, by a projection argument, to the tiling of point sets attached to binary log-Hadamard matrices of small rank. Using the published classifications of Hadamard matrices of orders $16$ and $32$, we find that exactly sixteen order-$32$ classes up to transposition have dephased binary rank at most seven, and we give explicit tilings for both orientations of each.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24022",
  "author_total": 1,
  "authors": [
    {
      "name": "Atsushi Kadowaki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/707bzz6vztrfn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24022",
  "fulltext_url": null,
  "id": "707bzz6vztrfn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24022",
  "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.CA"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7223069933",
  "pdf_url": "https://arxiv.org/pdf/2610.24022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/707bzz6vztrfn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fuglede's conjecture holds in $\\mathbb{F}_2^7$",
  "title_text": "Fuglede's conjecture holds in 𝔽_2^7",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/707bzz6vztrfn",
  "venue": "arXiv",
  "versions": []
}