Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular…

Abstract

Let p be an odd prime, q=p^k, and r=p^s with s| k. We study 𝔽_r-valued generalized Maiorana-McFarland functions H(x,y,z)=P^(z)(x)+Tr_s^k(yz) on V_n×𝔽_q×𝔽_q. The ingredients P^(z) are vectorial dual-bent functions with weakly regular components, constant on the cosets of 𝔽_r^* in 𝔽_q^*, and, for z≠ 0, of inversion type with respect to t↦ t⁻¹. Under a mild condition on P⁽⁰⁾, we prove that the partition into the 𝔽_r-valued level sets of H, refined according to the 𝔽_r-lines of the relevant coordinates, is Fourier-reflexive. It therefore induces a translation association scheme with (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 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.

General constructions of normal bent partitions related to vectorial dual-bent functions
Alexander Kholosha, Mohit Pal
arXiv2 Oct 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
Neighbourhood-Based Generalized Dynamic Principal Components for Spatial Functional Data
Dharini Pathmanathan, Teck Xiang Seow, Philipp Otto +1
arXiv14 Sep 2026PreprintCC BY-NC-ND 4.0Full textStatisticsarXiv
Rhaly and generalized Hilbert operators between spaces of analytic functions
Petros Galanopoulos, Daniel Girela
arXiv3 Sep 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
Generalized p-ary cPS Bent Functions
Alexander Kholosha, Mohit Pal
arXiv16 Sep 2026PreprintCC BY 4.0Full textGeneral mathematicsarXiv
The non-spherical sup-norm problem for GL(n) and generalized spherical functions
Valentin Blomer, Gergely Harcos, Péter Maga +1
arXiv16 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Analysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{pelen2026association,
  title = {{Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components}},
  author = {Rumi Melih Pelen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33107},
  eprint = {2609.33107},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33107},
}

RIS

TY  - UNPB
TI  - Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components
AU  - Rumi Melih Pelen
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33107
UR  - https://arxiv.org/abs/2609.33107
AB  - Let p be an odd prime, q=p^k, and r=p^s with s| k. We study 𝔽_r-valued generalized Maiorana-McFarland functions H(x,y,z)=P^(z)(x)+Tr_s^k(yz) on V_n×𝔽_q×𝔽_q. The ingredients P^(z) are vectorial dual-bent functions with weakly regular components, constant on the cosets of 𝔽_r^* in 𝔽_q^*, and, for z≠ 0, of inversion type with respect to t↦ t^−1. Under a mild condition on P^(0), we prove that the partition into the 𝔽_r-valued level sets of H, refined according to the 𝔽_r-lines of the relevant coordinates, is Fourier-reflexive. It therefore induces a translation association scheme with (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 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.
ER  -

Abstract

Related

Cite