{
  "abstract": "A subspace design is a collection of subspaces $H_1,\\ldots,H_n$ of $\\mathbb{F}_q^k$ with the property that no low-dimensional subspace $W$ intersects the collection \"too much\". Subspace designs and related objects in linear-algebraic pseudorandomness have found a broad range of applications, ranging from list decoding, to derandomizing algorithms. We construct explicit strong subspace designs over every finite field. In the extremal case where the co-dimension $t$ of each $H_i$ is equal to the dimension of $W$, for every constant field size our construction attains $n=Ω(k)$ and matches the probabilistic intersection bound up to a constant factor. All previous constructions required the field size to grow with $t$ (or $k$). Our subspace designs also imply new construction of rank condensers over arbitrary finite fields. This result is the first to achieve an optimal dependence on $k$ while maintaining both a constant output entropy rate and a constant field size. As an application, we construct lossless rank extractors for linear sources of rank $r$, for all $r < q$, with parameters matching those of Guo, Raj, Shangguan and Zhang (FOCS '26), thereby generalizing their result to prime fields and smaller field sizes. Our construction is based on an algebraic-geometric version of the Parvaresh-Vardy codes (Parvaresh-Vardy FOCS '05, Guruswami ECCC '05), extending the framework underlying the condensers of Guruswami, Umans and Vadhan (JACM '09). We view our construction as a linear-algebraic analysis - tailored to affine sources - of the GUV construction, generalized to functions over algebraic curves. More specifically, inspired by Ta-Shma and Umans (CCC 12') we develop a two-level evaluation scheme, where we first evaluate a function on a curve at extension-field points, and then evaluate a corresponding affine-linear polynomial to obtain outputs over the base field.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32834",
  "author_total": 3,
  "authors": [
    {
      "name": "Gil Cohen"
    },
    {
      "name": "Dean Doron"
    },
    {
      "name": "Noam Goldgraber"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cxtyt8j35c85e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32834",
  "fulltext_url": null,
  "id": "cxtyt8j35c85e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32834",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214709369",
  "pdf_url": "https://arxiv.org/pdf/2609.32834",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cxtyt8j35c85e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32834",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32834"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Algebraic-Geometric Parvaresh--Vardy Subspace Designs and Rank Condensers",
  "title_text": "Algebraic-Geometric Parvaresh--Vardy Subspace Designs and Rank Condensers",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:04Z",
  "url": "https://research.draconitic.com/paper/cxtyt8j35c85e",
  "venue": "arXiv",
  "versions": []
}