Bent partitions and Maiorana-McFarland association schemes

Abstract The recently introduced generalized semifield spreads are partitions of $${\mathbb {F}}_{p^m}\times {\mathbb {F}}_{p^m}$$ F p m × F p m , which are constructed from presemifields with a certain property, called right $${\mathbb {F}}_{p^k}$$ F p k -linearity. These partitions have similar properties as spreads. In particular, they are bent partitions, hence they yield a large number of bent functions, vectorial bent functions and amorphic association schemes. We show that with a slight change of parameters, we obtain inequivalent bent partitions, non-isomorphic divisible designs, and bent functions with various algebraic degrees. This is in contrast to classical spreads of $${\mathbb {F}}_{p^m} \times {\mathbb {F}}_{p^m}$$ F p m × F p m , which yield bent functions all of which have algebraic degree $$(p-1)m$$ ( p - 1 ) m . We show that with right $${\mathbb {F}}_{p^k}$$ F p k -linear presemifields we can obtain a large variety of vectorial dual-bent functions, which yield, not necessarily amorphic, association schemes. We investigate fusions of these association schemes, which reveal information on their inner structure, and may provide a tool to distinguish non-isomorphic association schemes.