{
  "abstract": "The $\\mathbb{Z}_{p^s}$-additive codes are subgroups of $\\mathbb{Z}_{p^s}^n$. A $\\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\\mathbb{Z}_p$ which is the Gray map image of a $\\mathbb{Z}_{p^s}$-additive code. A recursive construction gives such codes $H_p^{t_1,\\dots,t_s}$, of type $(n;t_1,\\dots,t_s)$ and length $p^t$, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime $p$. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code $H_p^{t_1,\\dots,t_s}$ with $t_1\\geq2$, the derived code is a $\\mathbb{Z}_{p^{s-1}}$-linear GH code of the same family, although the mechanism for odd $p$ differs from the binary one. As a consequence, for $t\\geq3$, there are $\\mathcal P(t+1)-\\mathcal P(t)$ nonequivalent such codes of length $p^t$ if $p$ is odd, and $\\mathcal P(t+1)-\\mathcal P(t)-1$ if $p=2$, where $\\mathcal P(n)$ is the number of partitions of $n$. In particular, this completes the classification of the $\\mathbb{Z}_{2^s}$-linear Hadamard codes. We also prove a conjecture on the codes with $t_1\\geq2$ and $t_s=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18409",
  "author_total": 1,
  "authors": [
    {
      "name": "Dipak K. Bhunia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyg8zhgfhv3ey.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18409",
  "fulltext_url": null,
  "id": "eyg8zhgfhv3ey",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18409",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221829286",
  "pdf_url": "https://arxiv.org/pdf/2610.18409",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyg8zhgfhv3ey.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18409",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18409"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Full classification of $\\mathbb{Z}_{p^s}$-linear generalized Hadamard codes",
  "title_text": "Full classification of ℤ_p^s-linear generalized Hadamard codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/eyg8zhgfhv3ey",
  "venue": "arXiv",
  "versions": []
}