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

Abstract

Bent functions $f: V_{n}\rightarrow \mathbb{F}_{p}$ with certain additional properties play an important role in constructing partial difference sets, where $V_{n}$ denotes an $n$-dimensional vector space over $\mathbb{F}_{p}$, $p$ is an odd prime. In \cite{Cesmelioglu1,Cesmelioglu2}, the so-called vectorial dual-bent functions are considered to construct partial difference sets. In \cite{Cesmelioglu1}, \c{C}e\c{s}melio\v{g}lu \emph{et al.} showed that for vectorial dual-bent functions $F: V_{n}\rightarrow V_{s}$ with certain additional properties, the preimage set of $0$ for $F$ forms a partial difference set. In \cite{Cesmelioglu2}, \c{C}e\c{s}melio\v{g}lu \emph{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 vectorial dual-bent functions $F: V_{n}\rightarrow \mathbb{F}_{p^s}$ with certain additional properties, 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 almost all the results of using weakly regular $p$-ary bent functions to construct partial difference sets are special cases of our results.