{
  "abstract": "We give new bounds for list sizes of random linear codes at capacity, max loads of linear hash functions, and Furstenberg sets, over every finite field $\\mathbb{F}_q$. 1. Random linear codes over $\\mathbb{F}_q$ with rate $1 - H_q(p) - ε$ are $(p, O(q H_q(p)/ε))$-list decodable with high probability for all values of $p, q, ε$, including the high error regime. This nearly matches the list size lower bound of $H_q(p)/ε$ due to Guruswami, Li, Mosheiff, Resch, Silas, and Wootters [IEEE Trans. Inf. Theory 2022]. Our bound is the first uniform improvement for $q > 2$ since Guruswami, Håstad, and Kopparty [STOC 2010]. 2. Linear hash functions over $\\mathbb{F}_q$ hashing $n$ balls to $n$ bins achieve maximum load $O(q \\ln \\ln q / {\\ln q}) \\cdot \\ln n / {\\ln \\ln n}$, both in expectation and with probability $1-o(1)$. This nearly matches the lower bound of $\\ln n / {\\ln \\ln n}$. Previously, only a polylogarithmic upper bound was known for $q > 2$, due to Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos [J. ACM 1999]. We reduce list decodability and linear hashing to strong Furstenberg set lower bounds, which we prove using a new polynomial method of multiplicity gaps. While previous polynomial methods analyze a set $S$ by studying polynomials that vanish on it, we consider polynomials that vanish everywhere, but with higher multiplicity inside $S$ than outside.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17020",
  "author_total": 2,
  "authors": [
    {
      "name": "Vinayak M. Kumar"
    },
    {
      "name": "Geoffrey Mon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4tarfp1jq0xpk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17020",
  "fulltext_url": null,
  "id": "4tarfp1jq0xpk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17020",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
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      "arxiv",
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    ],
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      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
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      "arxiv",
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      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7213407277",
  "pdf_url": "https://arxiv.org/pdf/2609.17020",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4tarfp1jq0xpk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17020",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17020"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "List Decoding, Linear Hashing, and Furstenberg over $\\mathbb{F}_q$",
  "title_text": "List Decoding, Linear Hashing, and Furstenberg over 𝔽_q",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/4tarfp1jq0xpk",
  "venue": "arXiv",
  "versions": []
}