{
  "abstract": "Fix integers $e\\ge2$ and $r\\ge1$. In this paper we study the $r$-th generalized covering radius $ρ_r\\left(BCH(e,m)\\right)$ of the binary primitive $e$-error-correcting BCH code $BCH(e,m)$. By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove that \\[ρ_r\\bigl(\\BCH(e,m)\\bigr)\\le(r+1)e-1\\] for all sufficiently large $m$. For $e\\ge7$, this improves a recent result of Belinsky--Zabokritskiy. Our proof gives a substantially simpler geometric approach to this upper bound. In particular it implies that \\[ρ_2\\bigl(BCH(e,m)\\bigr)=3e-1\\] for all sufficiently large $m$. Previously it was only known that \\[ρ_2\\bigl(\\BCH(e,m)\\bigr) \\in \\left\\{3e-1,3e\\right\\}\\] for all sufficiently large $m$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30961",
  "author_total": 3,
  "authors": [
    {
      "name": "Maosheng Xiong"
    },
    {
      "name": "Chi Hoi Yip"
    },
    {
      "name": "Ferdinando Zullo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4m4rbf512nsrx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30961",
  "fulltext_url": null,
  "id": "4m4rbf512nsrx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30961",
  "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.IT"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7204864067",
  "pdf_url": "https://arxiv.org/pdf/2608.30961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4m4rbf512nsrx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes",
  "title_text": "Asymptotic Bounds on Generalized Covering Radii of Binary Primitive BCH Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/4m4rbf512nsrx",
  "venue": "arXiv",
  "versions": []
}