{
  "abstract": "Let $\\mathcal{C}$ be an almost maximum distance separable (AMDS) code over the finite fields with parameters $[n,k,n-k]$ whose minimum-weight codewords support an $(n-k-1)$-$(n,n-k,λ)$ design. We first determine the parameters of the dual code $\\mathcal{C}^{\\bot}$. We also characterize the design and its complementary design supported by the minimum weight codewords of $\\mathcal{C}^{\\bot}$. In particular, we present explicit formulas for the weight enumerators of all subset codes of $\\mathcal{C}^{\\bot}$ with length at least $k+1$. These formulas are completely determined by the cardinality of the subset, the dimension, together with the parameter $λ$, and are independent of the specific choice of subsets. As an application, we answer an open problem posed by Wang, Tang and Ding in 2023 by proving that the $3$-design supported by the minimum-weight codewords in their proposed cyclic dual code is isomorphic to the complementary design of the Witt spherical geometry design.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29744",
  "author_total": 3,
  "authors": [
    {
      "name": "Huakai Wei"
    },
    {
      "name": "Shiyuan Qiang"
    },
    {
      "name": "Shaofang Hong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/594pw3f469adb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29744",
  "fulltext_url": null,
  "id": "594pw3f469adb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29744",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223252521",
  "pdf_url": "https://arxiv.org/pdf/2610.29744",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/594pw3f469adb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29744",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29744"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Almost MDS codes holding $(n-k-1)$-designs and an open problem on Witt design",
  "title_text": "On Almost MDS codes holding (n−k−1)-designs and an open problem on Witt design",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/594pw3f469adb",
  "venue": "arXiv",
  "versions": []
}