{
  "abstract": "The quaternary Preparata codes of Hammons, Kumar, Calderbank, Sloane and Solé are $\\mathbb{Z}_4$-linear codes of length $2^m$ with $4^{2^m-m-1}$ codewords. They are usually studied through their binary Gray images, that is, in the Lee metric. We consider them instead as codes over a 4-letter alphabet with the Hamming metric. We show that for odd $m$ they have covering radius 2 and minimum distance 4, so they are quasi-perfect. For even $m$ the same construction has covering radius 3. The proof is a short computation with Teichmüller representatives in the Galois ring $\\mathrm{GR}(4,m)$, and it also gives the coset weight distribution. The covering density of these codes tends to $9/8$. The best previously known infinite families of quaternary codes with covering radius 2 have covering density about 1.5. Consequently $K_4(n,2)\\le 4^{n-m-1}$ for all $n\\ge 2^m$ and odd $m$, which improves the best available bounds for a range of lengths above each $2^m$. For $m=3$ the code is the octacode, and it gives $K_4(8,2)\\le 256$; the previous bound was 352, from 2005, and the best lower bound is 251. We also give a quaternary code showing $K_4(6,2)\\le 48$, where the previous bound was 52, together with consequences for larger alphabets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03760",
  "author_total": 1,
  "authors": [
    {
      "name": "Tamal Maharaj"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eze29v1xyx24h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03760",
  "fulltext_url": null,
  "id": "eze29v1xyx24h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03760",
  "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"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7220660782",
  "pdf_url": "https://arxiv.org/pdf/2610.03760",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eze29v1xyx24h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.03760",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03760"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric",
  "title_text": "Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/eze29v1xyx24h",
  "venue": "arXiv",
  "versions": []
}