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
$\mathbb {F}_{p}$
,
$n$
is an even positive integer and
$p$
is a prime. The classical examples of bent partitions are obtained from (partial) spreads. In Anbar and Meidl (2022) and Meidl and Pirsic (2021), two classes of bent partitions which are not obtained from (partial) spreads were presented. In Anbar et al. (2023), more bent partitions
$\Gamma _{1}, \Gamma _{2}, \Gamma _{1}^{\bullet }, \Gamma _{2}^{\bullet }, \Theta _{1}, \Theta _{2}$
were presented from (pre)semifields, including the bent partitions given in Anbar and Meidl (2022) and Meidl and Pirsic (2021). 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
$\mathcal {C}$
) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when
$p$
is an odd prime, we show that bent partitions satisfying Condition
$\mathcal {C}$
one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that
$\Gamma _{1}, \Gamma _{2}, \Gamma _{1}^{\bullet }, \Gamma _{2}^{\bullet }, \Theta _{1}, \Theta _{2}$
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
$f: V_{n}^{(3)}\rightarrow \mathbb {F}_{3}$
(
$n$
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 Anbar and Meidl (2022). 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
$\mathcal {C}$
in terms of partial difference sets.