{
  "abstract": "We prove two arithmetic nonexistence conditions for tight spherical $5$-designs in dimension $(2m+1)^2-2$. Let $m$ be a positive even integer satisfying $ν_2(m(m+1))\\leq5$, where $ν_p(a)$ denotes the exponent of the prime $p$ in the positive integer $a$. We show that such a design does not exist if either (i) $m\\equiv1\\pmod3$ and $ν_p(m(m+1))\\leq1$ for every prime $p\\geq7$ with $p\\equiv3\\pmod4$, or (ii) $m\\equiv0\\pmod3$, $ν_3(m(m+1))=1$, and $ν_p(m(m+1))\\leq1$ for every prime $p\\geq5$ with $p\\not\\equiv1\\pmod{12}$. Together, these conditions recover the corresponding nonexistence results of Bannai, Munemasa, and Venkov and of Nebe and Venkov, and exclude parameters not covered by either earlier result, including $m=16,40,88,96,100$. The proofs use lattice theory to constrain the discriminant groups of the associated lattices, and derive contradictions through Gauss sums and determinant congruences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36725",
  "author_total": 1,
  "authors": [
    {
      "name": "Zili Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/52cbxjmt1a55e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36725",
  "fulltext_url": null,
  "id": "52cbxjmt1a55e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36725",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214939283",
  "pdf_url": "https://arxiv.org/pdf/2609.36725",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/52cbxjmt1a55e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.36725",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36725"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs",
  "title_text": "Arithmetic Nonexistence Conditions for Tight Spherical $5$-Designs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/52cbxjmt1a55e",
  "venue": "arXiv",
  "versions": []
}