{
  "abstract": "Let $Λ_m$ be the Barnes-Wall lattice of dimension $2^m$, normalized to have minimum squared norm $d_m:=2^{m-1}$. We determine its complete norm spectrum strictly between $d_m$ and $2d_m$, proving that, for every integer $m\\ge1$, $\\{\\|x\\|^2:x\\inΛ_m,\\ d_m<\\|x\\|^2<2d_m\\}=\\{2d_m-2^j:j\\in\\mathbb Z,\\ \\lceil m/2\\rceil\\le j\\le m-2\\}$. In particular, the norm spectrum in this interval is nonempty for $m\\ge4$. The proof uses a Gaussian-integer model in which lattice automorphisms arising from Clifford unitaries transform a vector into a pair of coordinate subvectors in a lower-dimensional lattice. Divisibility of the differences of their squared norms and a matrix Parseval identity provide the arithmetic and averaging constraints needed for induction on $m$. Every listed norm is realized explicitly by a sum of two minimal vectors of the lattice.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15288",
  "author_total": 4,
  "authors": [
    {
      "name": "Yuhang Cao"
    },
    {
      "name": "Sihuang Hu"
    },
    {
      "name": "Hexiang Huang"
    },
    {
      "name": "Mengrui Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/92bc9kyw77656.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15288",
  "fulltext_url": null,
  "id": "92bc9kyw77656",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15288",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221764230",
  "pdf_url": "https://arxiv.org/pdf/2610.15288",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/92bc9kyw77656.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15288",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15288"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum",
  "title_text": "The Norm Spectrum of Barnes-Wall Lattices Below Twice the Minimum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/92bc9kyw77656",
  "venue": "arXiv",
  "versions": []
}