{
  "abstract": "The connection between coding theory and combinatorial $t$-designs is an important research topic at the intersection of coding theory and combinatorics. Let $q=p^m$, where $p$ is an odd prime and $m\\geq 2$. In this paper, we investigate a class of linear codes $\\mathcal{C}$ over $\\mathbb{F}_{q^2}$ and their connection with combinatorial $3$-designs. By analyzing the relevant structural properties of $\\mathcal{C}$ and $\\mathcal{C}^{\\perp}$, we show that the supports of the codewords of every nonzero weight in $\\mathcal{C}$ and {the supports of the codewords of weight $4$ in $\\mathcal{C}^{\\perp}$} form $3$-designs. We further investigate a class of nonlinear codes $\\mathcal{C}_2$ associated with $\\mathcal{C}$, and prove that the supports of the codewords of every nonzero Hamming weight in $\\mathcal{C}_2$ also form $3$-designs. These results identify further classes of linear and nonlinear codes whose codewords support combinatorial $3$-designs. In particular, the nonlinear case provides additional examples of codes supporting $3$-designs, a topic for which relatively few results are currently available. As applications, we construct from $\\mathcal{C}^{\\perp}$ an entanglement-assisted quantum error-correcting code with parameters $[[q+1,q-3,4;4]]_q$. We also prove that $\\mathcal{C}$ is an all-symbol locally repairable code with locality $3$. Furthermore, we show that the code $\\mathcal{C}$ {meets the Singleton-type bound for locally repairable codes} and hence is optimal in some cases.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15525",
  "author_total": 4,
  "authors": [
    {
      "name": "Shiyan Xiong"
    },
    {
      "name": "Xiaoqiang Wang"
    },
    {
      "name": "Dabin Zheng"
    },
    {
      "name": "Qinqin Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/312rb0302cvyz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15525",
  "fulltext_url": null,
  "id": "312rb0302cvyz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15525",
  "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.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7213288477",
  "pdf_url": "https://arxiv.org/pdf/2609.15525",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/312rb0302cvyz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15525",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15525"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Infinite families of 3-designs from linear and nonlinear codes",
  "title_text": "Infinite families of 3-designs from linear and nonlinear codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/312rb0302cvyz",
  "venue": "arXiv",
  "versions": []
}