{
  "abstract": "The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that $k$ divides $v$ is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5$-$(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5$-$(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5$-$(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36613",
  "author_total": 3,
  "authors": [
    {
      "name": "Huijiao Hu"
    },
    {
      "name": "Zheng Huang"
    },
    {
      "name": "Shouqiang Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3v4z3xqagjt9h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36613",
  "fulltext_url": null,
  "id": "3v4z3xqagjt9h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36613",
  "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",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7214949353",
  "pdf_url": "https://arxiv.org/pdf/2609.36613",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3v4z3xqagjt9h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36613",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36613"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Block-Transitive $5$-$(v,k,2)$ Designs with $k$ divides $v$",
  "title_text": "Block-Transitive $5−(v,k,2)$ Designs with k divides v",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/3v4z3xqagjt9h",
  "venue": "arXiv",
  "versions": []
}