{
  "abstract": "Let $k>t\\ge 2$ be fixed integers, and let $G$ be a permutation group on a set of $v$ points. A $G$-invariant packing is free if every block has $|G|$ distinct images under $G$. We give two conditions on $G$ under which, as $v\\to\\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\\binom{v}{t}/\\binom{k}{t}$ blocks. This is a $G$-invariant form of the Erdős--Hanani theorem on asymptotically optimal packings. For semiregular groups $G$, packings with these properties exist if and only if all but $o(v^t)$ of the $t$-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for $t\\ge 3$, and by every cyclic semiregular group for $t=2$. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight $w$ and constant correlation $λ$ with $w\\geλ+2$, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10973",
  "author_total": 1,
  "authors": [
    {
      "name": "Yeow Meng Chee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8aq4adf3ad170.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10973",
  "fulltext_url": null,
  "id": "8aq4adf3ad170",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10973",
  "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.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221184025",
  "pdf_url": "https://arxiv.org/pdf/2610.10973",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8aq4adf3ad170.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10973",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10973"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A $G$-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes",
  "title_text": "A G-Invariant Erdős--Hanani Theorem with Applications to Optical Orthogonal and Constant-Weight Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/8aq4adf3ad170",
  "venue": "arXiv",
  "versions": []
}