{
  "abstract": "Let $p$ be an odd prime, $q=p^k$, and $r=p^s$ with $s\\mid k$. We study $\\mathbb{F}_r$-valued generalized Maiorana-McFarland functions $H(x,y,z)=P^{(z)}(x)+\\operatorname{Tr}_s^k(yz)$ on $V_n\\times\\mathbb{F}_q\\times\\mathbb{F}_q$. The ingredients $P^{(z)}$ are vectorial dual-bent functions with weakly regular components, constant on the cosets of $\\mathbb{F}_r^*$ in $\\mathbb{F}_q^*$, and, for $z\\ne 0$, of inversion type with respect to $t\\mapsto t^{-1}$. Under a mild condition on $P^{(0)}$, we prove that the partition into the $\\mathbb{F}_r$-valued level sets of $H$, refined according to the $\\mathbb{F}_r$-lines of the relevant coordinates, is Fourier-reflexive. It therefore induces a translation association scheme with $\\frac{q-1}{r-1}(r+1)+r$ classes (one fewer in a degenerate case), symmetric when the ingredients are even. For $s=1$ this recovers recent scalar constructions, while for $s=k$ it yields $(2q+1)$-class schemes from vectorial bent functions whose components may all be non-weakly regular. We also construct fusions along $\\mathbb{F}_r$-subspaces, obtaining $(2r+1)$- and $(3r+2)$-class schemes and connections between different levels. Counterexamples show that the inversion, coset-constancy, subspace, and weak-regularity hypotheses cannot in general be omitted. To our knowledge, these are the first association schemes obtained from vectorial bent functions with non-weakly regular components.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33107",
  "author_total": 1,
  "authors": [
    {
      "name": "Rumi Melih Pelen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eg1x4qnx8qep7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33107",
  "fulltext_url": null,
  "id": "eg1x4qnx8qep7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33107",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214732530",
  "pdf_url": "https://arxiv.org/pdf/2609.33107",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eg1x4qnx8qep7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33107",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components",
  "title_text": "Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/eg1x4qnx8qep7",
  "venue": "arXiv",
  "versions": []
}