  Let $\phi(x, y)\colon \mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}$ be a
function. We say $\phi$ is a \textit{Mattila--Sj\"{o}lin type function} if for
any compact set $E\subset \mathbb{R}^d$, we have $\phi(E, E)$ has non-empty
interior whenever $\dim_H(E)>\frac{d}{2}$. The usual distance function,
$\phi(x, y)=|x-y|$, is conjectured to be a Mattila--Sj\"{o}lin type function.
In the setting of finite fields $\mathbb{F}_q$, this definition is equivalent
to the statement that $\phi(E, E)=\mathbb{F}_q$ whenever $|E|\gg
q^{\frac{d}{2}}$. The main purpose of this paper is to prove the existence of
such functions in the vector space $\mathbb{F}_q^d$. In light of our results,
it seems that one can not expect a function to be Mattila--Sj\"{o}lin type in
all dimensions.
