Recent zbMATH articles in MSC 05B10 https://zbmath.org/atom/cc/05B10 2021-06-15T18:09:00+00:00 Werkzeug Graphs of vectorial plateaued functions as difference sets. https://zbmath.org/1460.05028 2021-06-15T18:09:00+00:00 "Çeşmelioğlu, Ayça" https://zbmath.org/authors/?q=ai:cesmelioglu.ayca "Olmez, Oktay" https://zbmath.org/authors/?q=ai:olmez.oktay Summary: A function $$F:\mathbb{F}_{p^n}\to\mathbb{F}_{p^m}$$, is a vectorial $$s$$-plateaued function if for each component function $$F_b(\mu)=Tr_n(bF(x))$$, $$b\in\mathbb{F}_{p^m}^\ast$$ and $$\mu\in\mathbb{F}_{p^n}$$, the Walsh transform value $$|\hat{F_b}(\mu)|$$ is either 0 or $$p^{\frac{n+s}{2}}$$. In this paper, we explore the relation between (vectorial) $$s$$-plateaued functions and partial geometric difference sets. Moreover, we establish the link between three-valued cross-correlation of $$p$$-ary sequences and vectorial $$s$$-plateaued functions. Using this link, we provide a partition of $$\mathbb{F}_{3^n}$$ into partial geometric difference sets. Conversely, using a partition of $$\mathbb{F}_{3^n}$$ into partial geometric difference sets, we construct ternary plateaued functions $$f:\mathbb{F}_{3^n}\to\mathbb{F}_3$$. We also give a characterization of $$p$$-ary plateaued functions in terms of special matrices which enables us to give the link between such functions and second-order derivatives using a different approach.