{
  "abstract": "We show that for every prime power $q$ and $ρ\\in(0,1-\\frac1q)$ there exists $\\varepsilon_0(q, ρ) > 0$ such that for all positive $\\varepsilon < \\varepsilon_0(q, ρ)$, a random linear code over $\\mathbb{F}_q$ of rate $1-h_q(ρ)-\\varepsilon$ has list size $\\lfloor h_q(ρ)/\\varepsilon\\rfloor$ or $\\lfloor h_q(ρ)/\\varepsilon\\rfloor+1$ with probability $1-o(1)$. Before this work, the list size was known only up to a constant factor for $q\\ge3$, and for $q=2$ it was known to be either $\\lfloor h_2(ρ)/\\varepsilon\\rfloor-1$, $\\lfloor h_2(ρ)/\\varepsilon\\rfloor$ or $\\lfloor h_2(ρ)/\\varepsilon\\rfloor+1$. We also show that the same two-point concentration holds for the list size of random LDPC codes. This follows from a general theorem establishing that random LDPC codes and random linear codes have the same threshold rate for distance, list decoding, list recovery from errors and any other \\emph{convex} property, which answers a question of Mosheiff, Resch, Ron-Zewi, Silas and Wootters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06570",
  "author_total": 1,
  "authors": [
    {
      "name": "Shashwat Silas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fsbt22cxe1wza.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06570",
  "fulltext_url": null,
  "id": "fsbt22cxe1wza",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06570",
  "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.DM"
    ],
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      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
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      "arxiv",
      "math.IT"
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      "openalex",
      "1702|Coding theory and cryptography"
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  "openalex": "W7212033351",
  "pdf_url": "https://arxiv.org/pdf/2609.06570",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fsbt22cxe1wza.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2609.06570",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06570"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The list size of random linear and LDPC codes at capacity",
  "title_text": "The list size of random linear and LDPC codes at capacity",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/fsbt22cxe1wza",
  "venue": "arXiv",
  "versions": []
}