of $1\times p^m$ all-ones matrix $j$ and an incidence matrix $N$ of a partial geometric design.

Let $j$ be the $1\times p^m$ all-ones matrix. Let $D$ be the block design associated with $G_f$ where the point set is $\mathbb{F}_{p^{n}} \times \mathbb{F}_{p} $ and the blocks are the translates of the graph of $f$. Suppose $\textbf{B}$ is a block in $D$. Note that $\textbf{B}=(u,v)+G_f$ for some $(u,v) \in \mathbb{F}_{p^{n}} \times \mathbb{F}_{p}.$ Then for each $a\in \Lambda$ we have $$\textbf{B}+(a,f(a))=\textbf{B}$$ since $$(u,v)+G_f=\{(x+u+a,f(x+a)+v): x \in \mathbb{F}_{p^{n}}\}.$$ Thus each block is repeated $p^m$ many times. Therefore, $A=j\otimes N$ for some incidence matrix $N$. Now we are going to show that $N$ is an incidence matrix of a partial geometric design. Since $A$ is an incidence matrix of a partial geometric design, $$\begin{split}
    AA^tA&=(\beta-\alpha) j \otimes N+\alpha J_{p^n \times p^n}\\
    &=j \otimes (\beta-\alpha) N+\alpha j \otimes J_{p^n \times p^{n-m}}\\
    &= j\otimes (\beta-\alpha) N+\alpha J_{p^n \times p^{n-m}}.\\
    \end{split}$$ We also have

$$\begin{split}
    AA^tA&=jj^tj\otimes NN^tN\\
    &=p^m j\otimes NN^tN\\
    &=j\otimes p^m NN^tN.\\
    \end{split}$$ By comparing the left hand sides we can conclude that the equation $$NN^tN=\frac{(\beta-\alpha)}{p^m} N+\frac{\alpha}{p^m} J_{p^n \times p^{n-m}}$$ holds.

Let $f$ be an $s$-plateaued function from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$. Then the set $D_f$ is a PGDS 