  Let $X$ be a complete $n$-dimensional simplicial toric variety with
homogeneous coordinate ring $S$. We study the multigraded Hilbert function
$H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit
formulas and prove non-decreasing and stabilization properties of $H_Y$ when
$Y$ is a $0$-dimensional complete intersection in $X$. We apply our results to
computing the dimension of some evaluation codes on $0$-dimensional complete
intersection in simplicial toric varieties.
