  Let $\phi(x, y)\colon \mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}$ be a
function. We say $\phi$ is a Mattila--Sj\"{o}lin type function of index
$\gamma$ if $\gamma$ is the smallest number satisfying the property that for
any compact set $E\subset \mathbb{R}^d$, $\phi(E, E)$ has a non-empty interior
whenever $\dim_H(E)>\gamma$. The usual distance function, $\phi(x, y)=|x-y|$,
is conjectured to be a Mattila--Sj\"{o}lin type function of index
$\frac{d}{2}$. In the setting of finite fields $\mathbb{F}_q$, this definition
is equivalent to the statement that $\phi(E, E)=\mathbb{F}_q$ whenever $|E|\gg
q^{\gamma}$. The main purpose of this paper is to prove the existence of such
functions with index $\frac{d}{2}$ in the vector space $\mathbb{F}_q^d$.
