{
  "abstract": "The $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive codes are subgroups of $\\mathbb{Z}_p^{α_1}\\times\\mathbb{Z}_{p^2}^{α_2}$, and they can be seen as linear codes over $\\mathbb{Z}_p$ when $α_2=0$, as $\\mathbb{Z}_{p^2}$-additive codes when $α_1=0$, and as $\\mathbb{Z}_2\\mathbb{Z}_4$-additive codes when $p=2$. A $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\\mathbb{Z}_p$ which is the Gray map image of a $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-additive code. For every prime $p$, there are recursive constructions of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear GH codes $H_p^{t_1,t_2}$ with $α_1\\neq0$ and $α_2\\neq0$, and of $\\mathbb{Z}_{p^s}$-linear GH codes $\\bar H_p^{t_1,\\dots,t_s}$. We show that the kernel blocks of a nonlinear code $H_p^{t_1,t_2}$ give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of $H_p^{t_1,t_2}$. We also determine the ranks of the codes $H_p^{t_1,t_2}$ and $\\bar H_p^{t_1,t_2}$. For odd $p$, the rank of $H_p^{t_1,t_2}$ depends on the Bernoulli numbers modulo $p$. By comparing the derived codes and the ranks, we show that, for odd $p$, no code $H_p^{t_1,t_2}$ is equivalent to any of these $\\mathbb{Z}_{p^s}$-linear GH codes. For $p=2$, a nonlinear code $H_2^{t_1,t_2}$ is equivalent to a $\\mathbb{Z}_{2^s}$-linear Hadamard code if and only if $t_2\\geq2$. Finally, we count the nonequivalent codes in the union of both families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28306",
  "author_total": 1,
  "authors": [
    {
      "name": "Dipak K. Bhunia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1z3bqczas2d74.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28306",
  "fulltext_url": null,
  "id": "1z3bqczas2d74",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28306",
  "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": "W7223115675",
  "pdf_url": "https://arxiv.org/pdf/2610.28306",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1z3bqczas2d74.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28306",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28306"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Nonequivalence of $\\mathbb{Z}_p\\mathbb{Z}_{p^2}$-linear and $\\mathbb{Z}_{p^s}$-linear generalized Hadamard codes for odd primes",
  "title_text": "Nonequivalence of ℤ_pℤ_p^2-linear and ℤ_p^s-linear generalized Hadamard codes for odd primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/1z3bqczas2d74",
  "venue": "arXiv",
  "versions": []
}