Plateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the ${\mathbb{Z}}_2$‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved.

中文翻译：

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平稳函数，部分几何差异集和部分几何设计

近年来，已经在许多论文中研究了有限域上的平稳函数。作为有限域上平稳函数的推广，我们介绍了有限阿贝尔群上的平稳函数的概念。我们将根据与该函数关联的矩阵方程式给出平稳函数的特征。然后我们建立了一对一的对应关系${\mathbb{ž}}_2$有限阿贝尔群中的值稳定函数和部分几何差异集（具有特定参数）。我们还将讨论从（abelian或nonabelian）有限组中的旧部分构造新的部分几何差异集的两种通用方法（扩展和提升），并构造许多部分几何差异集和平稳函数。将证明部分几何差异集（在任意有限组中）与部分几何设计之间的一对一对应关系。