New results on vectorial dual-bent functions and partial difference sets
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Bent functions f: V_n→𝔽_p with certain additional properties play an important role in constructing partial difference sets, where V_n denotes an n-dimensional vector space over 𝔽_p, p is an odd prime. In <cit.>, the so-called vectorial dual-bent functions are considered to construct partial difference sets. In <cit.>, Çeşmelioǧlu et al. showed that for vectorial dual-bent functions F: V_n→ V_s with certain additional properties, the preimage set of 0 for F forms a partial difference set. In <cit.>, Çeşmelioǧlu et al. showed that for a class of Maiorana-McFarland vectorial dual-bent functions F: V_n→𝔽_p^s, the preimage set of the squares (non-squares) in 𝔽_p^s^* for F forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for vectorial dual-bent functions F: V_n→𝔽_p^s with certain additional properties, the preimage set of the squares (non-squares) in 𝔽_p^s^* for F and the preimage set of any coset of some subgroup of 𝔽_p^s^* for F form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non)-quadratic vectorial dual-bent functions. In this paper, we illustrate that almost all the results of using weakly regular p-ary bent functions to construct partial difference sets are special cases of our results.
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