{
  "abstract": "We study the binomial channel and the structure of its capacity-achieving input and output distributions. It is known that the capacity-achieving input distribution is discrete and supported on finitely many points. The best previously known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $\\sqrt n$ and upper-bounded by a term of order $n/2$, where $n$ is the number of trials. In this work, we derive a new lower bound on the support size of order $\\sqrt{n\\log\\log n}$, up to explicit constants. The proof consists of three main steps. First, we derive new upper and lower bounds on the capacity with a gap that vanishes as $n\\to\\infty$, which yields $C(n)=\\frac12\\log\\frac{nπ}{2e}+o(1)$. Second, we show that the Beta-binomial output distribution induced by the reference input $X_r\\sim\\mathrm{Beta}(1/2,1/2)$ is asymptotically optimal: it approaches the capacity-achieving output distribution in relative entropy and, after a comparison step, in $χ^2$ divergence. Third, we prove a quantitative $χ^2$ approximation lower bound showing that this Beta-binomial output cannot be approximated too well by the output induced by a $K$-point input. Combining these ingredients forces the capacity-achieving input distribution to have at least order $\\sqrt{n\\log\\log n}$ mass points.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.12472",
  "author_total": 4,
  "authors": [
    {
      "name": "Mohammadamin Baniasadi"
    },
    {
      "name": "Antonino Favano"
    },
    {
      "name": "Luca Barletta"
    },
    {
      "name": "Alex Dytso"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4td015y7n10vg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.12472",
  "fulltext_url": null,
  "id": "4td015y7n10vg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.12472",
  "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.IT"
    ],
    [
      "openalex",
      "1705|Error Correcting Code Techniques"
    ]
  ],
  "openalex": "W7160987611",
  "pdf_url": "https://arxiv.org/pdf/2605.12472",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4td015y7n10vg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2605.12472",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.12472"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "An Improved Lower Bound on Support Size of Capacity-Achieving Inputs for the Binomial Channel: Extended version",
  "title_text": "An Improved Lower Bound on Support Size of Capacity-Achieving Inputs for the Binomial Channel: Extended version",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/4td015y7n10vg",
  "venue": "arXiv",
  "versions": []
}