In Section 4, we focus on $p-$ary plateaued function. We provide several characteristics which are obtained from Butson-Hadamard-like matrices. This section also provides results concerning partially bent functions and partial geometric designs.

Preliminaries
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Let $G$ be a group of order $v$ and let $S \subset G$ be a $k$-subset. For each $g \in G$, we define $$\delta(g):=|\{(s,t)\in S \times S \colon g=st^{-1}\}|.$$ Next we define the difference sets of our interest.

Let $v, k$ be positive integers with $v>k>2$. Let $G$ be a group of order $v$. A $k$-subset $S$ of $G$ is called a partial geometric difference set (PGDS) in $G$ with parameters $(v, k; \alpha,\beta)$ if there exist constants $\alpha$ and $\beta$ such that, for each $x\in G$, $$\sum\limits_{y\in S}\delta(xy^{-1})=\left \{\begin{array}{ll} \alpha & \mbox{if } x\notin S,\\
    \beta & \mbox{if } x\in S.\\ \end{array} \right .$$

There are two subclasses of PGDS namely difference sets and semiregular relative difference sets which have deep connections with coding theory, and cryptography [@assmus1992; @dingbook; @dingbook2]:

-   A $(v,k,\lambda)$-difference set (DS) in a finite group $G$ of order $v$ is a $k$-subset $D$ with the property that $\delta(g)=\lambda$ for all nonzero elements of $G$.

-   A $(m,u,k,\lambda)$-relative difference set (RDS) in 