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

Source 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.