The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between p-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions \(F:{\mathbb {F}}_{p^n}\rightarrow {\mathbb {F}}_{p^s}\) are linear equivalent to l-forms, i.e., to functions satisfying \(F(\beta x) = \beta ^lF(x)\) for all \(\beta \in {\mathbb {F}}_{p^s}\), we investigate properties of partial difference sets obtained from l-forms. We show that they are unions of cosets of \({\mathbb {F}}_{p^s}^*\), which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from \({\mathbb {F}}_{p^n}\) to \({\mathbb {F}}_{p^s}\), we show that the preimage set of the squares of \({\mathbb {F}}_{p^s}\) forms a partial difference set. This extends earlier results on p-ary bent functions.

本文的目的是拓宽对弯曲函数和偏差分集之间联系的理解。最近，前两位作者表明，具有某些附加性质的向量对偶弯曲函数映射到 0 的元素形成偏差分集，概括了布尔弯曲函数与 Hadamard 差分集之间的联系，以及一些后来建立的联系p 元弯曲函数和偏差分集到矢量弯曲函数。我们讨论坐标变换的影响。由于所有目前已知的向量双弯曲函数\(F:{\mathbb {F}}_{p^n}\rightarrow {\mathbb {F}}_{p^s}\)线性等价于l -forms , 即函数满足\(F(\beta x) = \beta ^lF(x)\)对于所有\(\beta \in {\mathbb {F}}_{p^s}\)，我们研究了获得的部分差异集的性质从l -形式。我们证明它们是\({\mathbb {F}}_{p^s}^*\)的陪集的并集，也可以看作是某些分圆类。我们在分圆法的部分差异集上绘制与已知结果的联系。受实验结果启发，对于一类从\({\mathbb {F}}_{p^n}\)到\({\mathbb {F}}_{p^s}\)，我们证明\({\mathbb {F}}_{p^s}\)的平方的原像集形成了一个部分差分集。这扩展了p 元弯曲函数的早期结果。