Binomial Coefficient Finite Field

Question

Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{E}$ an extension field of degree $n$ of $\mathbb{F}$. Let $S(x) \in \mathbb{E}[x]$. How I will be able to reduce a expression: $\sum_{i,j=0}^{r-1}p_{ij}(\sum_{k=0}^{n-1}s_kx^{q^k})^{q^i+q^j}$?