{
  "abstract": "A finite set of unit vectors is $k$-irreducible if every signed sum of between two and $k$ distinct elements has norm greater than one. Let $\\mathcal{R}_k$ be the maximal asymptotic rate of such sets, and let $κ(α)$ be the maximal asymptotic rate of spherical codes with pairwise inner products at most $α$. For $k \\ge 2$ we show: \\begin{align} \\mathcal{R}_k \\le \\min_{1 \\le r \\le \\lfloor k/2 \\rfloor} \\frac{1}{r} \\, κ\\!\\left(1 - \\frac{1}{2r}\\right) \\, . \\end{align} Combining this with standard sphere packing bounds, for large $k$ we obtain an almost-tight asymptotic comparison with the known lower bounds: \\begin{align} \\left(\\tfrac{1}{2}-o(1)\\right) \\, \\frac{\\log_2 k}{k} \\le \\mathcal{R}_k \\le (1 + o(1)) \\, \\frac{\\log_2 k}{k} \\, . \\end{align}",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08190",
  "author_total": 1,
  "authors": [
    {
      "name": "Thijs Laarhoven"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2n5436ewxz676.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08190",
  "fulltext_url": null,
  "id": "2n5436ewxz676",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08190",
  "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",
      "cs.CR"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212077052",
  "pdf_url": "https://arxiv.org/pdf/2609.08190",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2n5436ewxz676.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08190",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08190"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Note on Sphere Packing Bounds for Tuple Lattice Sieving",
  "title_text": "A Note on Sphere Packing Bounds for Tuple Lattice Sieving",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/2n5436ewxz676",
  "venue": "arXiv",
  "versions": []
}