Recent zbMATH articles in MSC 05B10https://zbmath.org/atom/cc/05B102021-06-15T18:09:00+00:00WerkzeugGraphs of vectorial plateaued functions as difference sets.https://zbmath.org/1460.050282021-06-15T18:09:00+00:00"Çeşmelioğlu, Ayça"https://zbmath.org/authors/?q=ai:cesmelioglu.ayca"Olmez, Oktay"https://zbmath.org/authors/?q=ai:olmez.oktaySummary: 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.