{
  "abstract": "We investigate the optimal asymptotic rate $R_q$ of perfect $q$-hash codes, a fundamental extremal problem in combinatorics, information theory, and computer science. Let $B_q=-\\log_2(1-q!/q^q)/(q-1)$ denote the classical probabilistic lower bound. We prove that $R_q=B_q$ if and only if $q=2$, establishing an explicit strict improvement for every $q\\ge3$ and settling the first assertion of a conjecture of Xing and Yuan (Combinatorica 43 (2023), 347--366). More significantly, we show that, for every $η>0$, $R_q\\ge B_q+\\exp(-(2+η)q)$ for all sufficiently large integers $q$. This improves the previously known increments of scale $\\exp(-3q+O(\\ln q))$ by an exponential factor. We also establish a structural limitation of the concatenation method. For pairwise uniform inner codes of subexponential length, the resulting rate bound exceeds $B_q$ by at most $\\exp(-3q+o(q))$. Thus, exponentially long inner codes are necessary to achieve the new improvement scale within this framework. Our proofs combine Fourier analysis, high-dimensional simplex codes, additive permutation counts, and entropy bounds for hypergraph matchings. We further strengthen lower bounds from short inner codes and improve several known numerical estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08230",
  "author_total": 2,
  "authors": [
    {
      "name": "Shitao Li"
    },
    {
      "name": "Minjia Shi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/emsvw2xncfgkj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08230",
  "fulltext_url": null,
  "id": "emsvw2xncfgkj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08230",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08230",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/emsvw2xncfgkj.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08230",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08230"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The probabilistic bound for perfect hashing is sharp only for binary alphabets",
  "title_text": "The probabilistic bound for perfect hashing is sharp only for binary alphabets",
  "type": "preprint",
  "updated_at": "2026-11-08T21:20:42Z",
  "url": "https://research.draconitic.com/paper/emsvw2xncfgkj",
  "venue": "arXiv",
  "versions": []
}