---
abstract: |
    A function $F:\mathbb{F}_{p^n}\rightarrow \mathbb{F}_{p^m},$ is a vectorial $s$-plateaued function if for each component function $F_{b}(\mu)=Tr_n(\alpha F(x)), b\in \mathbb{F}_{p^m}^*$ and $\mu \in \mathbb{F}_{p^n}$, the Walsh transform value $|\widehat{F_{b}}(\mu)|$ 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 constructed ternary plateaued functions $f:\mathbb{F}_{3^n}\rightarrow \mathbb{F}_3$. 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.

    partial geometric designs, partial geometric difference sets, plateaued functions, three-valued cross-corelation function.\
address: 'Department of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara, 06100, Turkey.'
author:
- Ayça Çeşmelioğlu
- Oktay Olmez
title: Graphs of Vectorial Plateaued Functions as Difference Sets
---

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Introduction
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A (block) design is a pair $(\mathcal{P}, \mathcal{B})$ consisting of a finite set $\mathcal{P}$ of points and a finite collection $\mathcal{B}$ of nonempty subsets of 