{
  "abstract": "Universal coding of integers (UCI) provides prefix codes for positive integers such that, for every nonincreasing source distribution $P$, the average codeword length under $P$ stays within $K$ times $\\max\\{1,H(P)\\}$. For a given UCI $\\mathcal{C}$, the smallest such constant $K$ is called the minimum expansion factor of $\\mathcal{C}$, denoted $C_{\\mathcal{C}}^{*}$. Let $C^{*}\\triangleq \\inf \\{C_{\\mathcal{C}}^{*}\\}$ over all UCIs. A UCI attaining $C^{*}$, or a family of UCIs attaining $C^{*}$ in the limit, is called optimal. It is currently known that $2\\le C^{*}\\le 2.0386$. In this paper, we construct a family of one-point plus uniform-tail distributions and prove that, for every UCI, a distribution in this family attains the worst-case ratio; thus, the family is least favorable for the UCI problem. We further establish an inequality, called the \\emph{UCI inequality}, which plays the same role for UCI as the Kraft inequality does for prefix codes: for any real number $B\\ge 1$, it decides whether $B$ lies below or above $C^*$. The UCI inequality gives an equivalent definition of $C^*$. By numerical computation, we determine $C^* = 2.00012475703610197861701176297579886254\\cdots$, the first $38$ decimal digits being certified. Once $C^*$ is known, we can theoretically construct the optimal UCI.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01563",
  "author_total": 3,
  "authors": [
    {
      "name": "Wei Yan"
    },
    {
      "name": "Yunghsiang S. Han"
    },
    {
      "name": "Leqian Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ag1trnd4ty62k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01563",
  "fulltext_url": null,
  "id": "ag1trnd4ty62k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01563",
  "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.IT"
    ],
    [
      "openalex",
      "1702|Algorithms and Data Compression"
    ]
  ],
  "openalex": "W7216055567",
  "pdf_url": "https://arxiv.org/pdf/2610.01563",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ag1trnd4ty62k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.01563",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01563"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Optimal Universal Coding of Integers",
  "title_text": "Optimal Universal Coding of Integers",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/ag1trnd4ty62k",
  "venue": "arXiv",
  "versions": []
}