{
  "abstract": "We show that random linear codes (RLCs) possess nearly optimal discrepancy-type properties in a broad range of settings. Our main results are two general discrepancy theorems: one controls all translates of a fixed test, and the other controls large families of Fourier-pseudorandom tests. Two motivating examples follow: First, RLCs behave essentially like unstructured random codes for list-decoding from errors above capacity. More precisely, an RLC $C\\subseteq \\mathbb{F}_q^n$ of rate $1 - \\frac{1}{n}\\log_q|B_ρ| + \\varepsilon$, where $|B_ρ|$ is the volume of a radius-$ρ$ Hamming ball in $\\mathbb{F}_q^n$, satisfies $|C \\cap B| = (1\\pm o(1)) \\frac{|C|\\cdot |B|}{q^n}$ simultaneously for all radius-$ρ$ Hamming balls $B$ with high probability. This vastly generalizes the previously best known fact that RLCs of this rate have covering radius at most $ρn$ with high probability (Blinovsky, 1987). Second, over prime fields, RLCs behave essentially like unstructured random codes for zero-error list-recovery, and list-recovery from erasures, above capacity. More precisely, for a prime $q>2$ and input list size $2\\leq \\ell\\leq q-1$, an RLC $C\\subseteq \\mathbb{F}_q^n$ of rate $1-\\log_q \\ell+\\varepsilon$ will satisfy $|C \\cap S| = (1\\pm o(1)) \\frac{|C|\\cdot \\ell^n}{q^n}$ simultaneously for all combinatorial rectangles $S=S_1\\times S_2\\times\\cdots\\times S_n$, where $|S_i|=\\ell$ for all $i$, with high probability. An analogous result also holds when we can bound $|S_i|$ only for some of the $i$'s. We use this to show the abundance of locally leakage-resilient $n$-party linear ramp secret sharing schemes with any linear reconstruction threshold and sublinear threshold gap $O(n/\\log n)$ over fields of polynomial size $q=Θ(n^γ)$ for a constant $γ\\in(0,1/5)$. Prior work was stuck at reconstruction thresholds above $n/2$ for both threshold and ramp schemes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.24471",
  "author_total": 6,
  "authors": [
    {
      "name": "Dean Doron"
    },
    {
      "name": "Tal Leonov"
    },
    {
      "name": "Jonathan Mosheiff"
    },
    {
      "name": "Henrique Navas"
    },
    {
      "name": "Nicolas Resch"
    },
    {
      "name": "João Ribeiro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1k594mqdfmhtj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.24471",
  "fulltext_url": null,
  "id": "1k594mqdfmhtj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.24471",
  "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.CC"
    ],
    [
      "arxiv",
      "cs.CR"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7165789812",
  "pdf_url": "https://arxiv.org/pdf/2606.24471",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1k594mqdfmhtj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2606.24471",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.24471"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Discrepancy for Random Linear Codes",
  "title_text": "Discrepancy for Random Linear Codes",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/1k594mqdfmhtj",
  "venue": "arXiv",
  "versions": []
}