Plateaued functions,partial geometric difference sets,and partial geometric designs

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.