论文标题
部分可观测时空混沌系统的无模型预测
New results on vectorial dual-bent functions and partial difference sets
论文作者
论文摘要
弯曲函数$ f：v_ {n} \ rightarrow \ mathbb {f} _ {p} $具有某些其他属性在构造部分差异集中起着重要作用，其中$ v_ {n} $表示$ n $ n $ - dimensional vector vector act of $ n $ dimensional vector Space over $ \ \ m m mathbb {f} $ {p $} $，p $} $ {p $}在\ cite {cesmelioglu1，cesmelioglu2}中，所谓的矢量双键函数被认为是构建部分差异集。在\ cite {cesmelioglu1}中，çeşmelioǧlu\ emph {et al。}表明，对于矢量双键函数$ f：v_ {n} \ rightArrow v_ {s} $ f：v_ {n} \ rightArrow v_ {s} $，带有某些其他属性，具有$ f $ f $差异的$ f $差异a部分。 In \cite{Cesmelioglu2}, Çeşmelioǧlu \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 $ for $ f $ fors forme for n of partial差异集。在本文中，我们进一步研究了矢量双弯曲功能和部分差异集。我们证明，对于矢量双凸函函数$ f：v_ {n} \ rightarrow \ mathbb {f} _ {p^s} $具有某些附加属性，$ \ \ \ \ m i \ {f} _ {p^s} $ sere in $ \ mathbb {pre cos of squares（non-squares）的预先映像集（non-squares） $ \ mathbb {f} _ {p^s}^{*} $的子组，用于$ f $形式的部分差异集。此外，某些（非）次数矢量双弯曲函数得出了部分差异集的显式结构。在本文中，我们说明，几乎所有使用弱规则的$ p $ ary弯曲函数来构建部分差异集的结果都是我们结果的特殊情况。
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}, Çeşmelioǧlu \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}, Çeşmelioǧlu \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.
