Modified planar functions and their components

Abstract

Zhou (J. Combin. Des. 21(12), 563–584, 2013) introduced modified planar functions in order to describe (2n, 2n, 2n, 1) relative difference sets R as a graph of a function on the finite field \(\mathbb {F}_{2^{n}}\), and pointed out that projections of R are difference sets that can be described by negabent or bent4 functions, which are Boolean functions given in multivariate form. One of the objectives of this paper is to contribute to the understanding of these component functions of modified planar functions. We identify the versions of the Walsh transforms that apply to modified planar functions on \(\mathbb {F}_{2^{n}}\) and their components, obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function, and point out some further properties of these components. Finally we introduce vectorial bent4 functions (over finite fields), which correspond to relative difference sets in certain groups. We give conditions under which Maiorana-McFarland functions are such functions, hence give rise to relative difference sets in \(\mathbb {Z}_{2}^{n/2}\times \mathbb {Z}_{4}^{n/2}\).