Vectorial dual-bent functions and association schemes

Official URL: https://dx.doi.org/10.1007/s10623-026-01946-3

Pott et al. [24] showed that weakly regular p-ary bent functions, which are l-forms, or equivalently vectorial dual-bent, induce p-class association schemes. Using a criterion based on the Bannai–Muzychuk theorem, we describe all Maiorana–McFarland functions Tr1m(xϕ(y)) inducing p-class association schemes. Our result confirms that for this class of functions, the property of being vectorial dual-bent is also necessary. We then, more generally, show that vectorial dual-bent functions F from Vn(p) to Vm(p) for which all components are regular (weakly regular but not regular), give rise to pm-class, (pm-1)-class and sometimes to (pm+1)-class association schemes on Vn(p). We investigate some fusions of these association schemes. In particular, we explain some classes of (pk+1)-class amorphic association schemes on V2m(p), which were recently obtained from bent partitions, as fusions of (non-amorphic) (pm+1)-class association schemes on V2m(p).