Bent Partitions, Vectorial Dual-Bent Functions and Partial Difference Sets

Bent partitions of V n (p) are quite powerful in constructing bent functions, vectorial bent functions and generalized bent functions, where V n (p) is an n -dimensional vector space over F p , n is an even positive integer and p is a prime. The classical examples of bent partitions are obtained from (partial) spreads. In [5], [19], two classes of bent partitions which are not obtained from (partial) spreads were presented. In [3], more bent partitions Γ 1 , Γ 2 , Γ * 1 , Γ * 2 , Θ 1 , Θ 2 were presented from (pre)semifields, including the bent partitions given in [5], [19]. In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime p, we show that one can generate certain bent partitions (called bent partitions satisfying Condition C ) from certain vectorial dualbent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when p is an odd prime, we show that bent partitions satisfying Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that Γ 1 , Γ 2 , Γ * 1 , Γ * 2 , Θ 1 are bent partitions in terms of vectorial dual-bent functions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any weakly regular ternary bent function f : V n (3) → F 3 ( n is even) of 2-form can generate a bent partition. When such f is weakly regular but not regular, the generated bent partition from f is not coming from a normal bent partition, which answers an open problem proposed in [5]. We give a sufficient condition on constructing partial difference sets from bent partitions, and when p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of partial difference sets.