{
  "abstract": "In the Hamming space $\\mathbb{F}_q^n$, a $d$-anticode is a subset in which every pair of points has Hamming distance at most $d$, or equivalently, a set of diameter at most $d$. Let $m_q(n,d)$ denote the maximum size of a $d$-anticode and let $\\mathcal{A}_q(n,d)$ be the family of all $d$-anticodes. We show that for fixed $q$, as $n\\to\\infty$, the total number of $d$-anticodes satisfies \\[ |\\mathcal{A}_q(n,d)|=2^{(1+o(1))m_q(n,d)} \\] for $4\\le d\\le n-1$, except when $q=2$ and $d=n-1$. We also count linear $d$-anticodes, that is, subspaces of $\\mathbb F_q^n$ with diameter at most $d$. For fixed $q$ and $d$, their number is polynomial in $n$. For growing $d$, we determine the exponential order as $q^{Θ_q(d^2+d\\log_q(n/d))}$ in the sublinear regime and $q^{Θ(n^2)}$ when $d/n\\toα\\in(0,1)$. In the near-full regime $n-d=o(n)$, we obtain the sharper count $q^{(1/4+o(1))n^2}$. A random-generator construction further improves the binary lower bound for every fixed linear diameter.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31618",
  "author_total": 2,
  "authors": [
    {
      "name": "Anna-Lena Horlemann"
    },
    {
      "name": "Ananthakrishnan Ravi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7w0j9mkt7888.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31618",
  "fulltext_url": null,
  "id": "a7w0j9mkt7888",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31618",
  "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",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223416882",
  "pdf_url": "https://arxiv.org/pdf/2610.31618",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7w0j9mkt7888.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31618",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31618"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Asymptotic Number of Anticodes in Hamming Spaces",
  "title_text": "The Asymptotic Number of Anticodes in Hamming Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/a7w0j9mkt7888",
  "venue": "arXiv",
  "versions": []
}