New results on vectorial dual-bent functions and partial difference sets

Abstract

Bent functions \(f: V_{n}\rightarrow \mathbb {F}_{p}\) play an important role in constructing partial difference sets, where \(V_{n}\) denotes an n-dimensional vector space over \(\mathbb {F}_{p}\), p is an odd prime. In [2, 3], the so-called vectorial dual-bent functions are considered to construct partial difference sets. In [2], Çeşmelioğlu et al. showed that for certain vectorial dual-bent functions \(F: V_{n}\rightarrow V_{s}\), the preimage set of 0 for F forms a partial difference set. In [3], Çeşmelioğlu et al. showed that for a class of Maiorana-McFarland vectorial dual-bent functions \(F: V_{n}\rightarrow \mathbb {F}_{p^s}\), the preimage set of the squares (non-squares) in \(\mathbb {F}_{p^s}^{*}\) 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 certain vectorial dual-bent functions \(F: V_{n}\rightarrow \mathbb {F}_{p^s}\), the preimage set of the squares (non-squares) in \(\mathbb {F}_{p^s}^{*}\) for F and the preimage set of any coset of some subgroup of \(\mathbb {F}_{p^s}^{*}\) 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 many results of using weakly regular p-ary bent functions to construct partial difference sets are special cases of our results. In [2], the authors considered weakly regular p-ary bent functions f with \(f(0)=0\). They showed that if such a function f is an l-form with \(gcd(l-1, p-1)=1\) for some integer \(1\le l\le p-1\), then f is vectorial dual-bent. We prove that the converse also holds, which answers one open problem proposed in [3].