{
  "abstract": "Guruswami, Li, Mosheiff, Resch, Silas and Wootters conjectured that, for every fixed prime power $q$ and $p\\in(0,1-1/q)$, random linear codes of rate $1-h_q(p)-\\varepsilon$ have list size $\\frac{h_q(p)}{\\varepsilon}\\cdot(1+o(1))$ as $\\varepsilon\\to0$, both for list decoding and average-radius list decoding. The conjecture was known for $q=2$ and remained open for larger alphabets. We prove the conjecture for list decoding for every prime power $q$, with the sharper bound $L=\\frac{h_q(p)}{\\varepsilon}+O_{p,q}(1).$ In contrast, we disprove the conjecture for average-radius list decoding. We show that the required list-size leading constant is at least $p\\left(1+\\log_q\\frac{(q-1)(1-p)}{p}\\right)$, which is strictly larger than $h_q(p)$ whenever $q^p(1-p)>1$. Moreover, for $p=1-λ/q$ with fixed $λ>1$, the required leading constant can be as large as $\\left(\\frac{\\logλ}{eλ}+o_q(1)\\right)\\frac{q}{\\log q}.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01070",
  "author_total": 2,
  "authors": [
    {
      "name": "Chen Yuan"
    },
    {
      "name": "Ruiqi Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6p4pmnnar0cmv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01070",
  "fulltext_url": null,
  "id": "6p4pmnnar0cmv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01070",
  "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|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7206203087",
  "pdf_url": "https://arxiv.org/pdf/2609.01070",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6p4pmnnar0cmv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.01070",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01070"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "List Size of Random Linear Codes: Optimality and the Limits of Average-Radius List Decoding",
  "title_text": "List Size of Random Linear Codes: Optimality and the Limits of Average-Radius List Decoding",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/6p4pmnnar0cmv",
  "venue": "arXiv",
  "versions": []
}