  Motivated by applications to the theory of error-correcting codes, we give an
algorithmic method for computing a generating set for the ideal generated by
$\beta$-graded polynomials vanishing on a subset of a simplicial complete toric
variety $X$ over a finite field $\mathbb{F}_q$, parameterized by rational
functions, where $\beta$ is a $d\times r$ matrix whose columns generate a
subsemigroup $\mathbb{N}\beta$ of $\mathbb{N}^d$. We also give a method for
computing the vanishing ideal of the set of $\mathbb{F}_q$-rational points of
$X$. When $\beta=[w_1 \cdots w_r]$ is a row matrix corresponding to a numerical
semigroup $\mathbb{N}\beta=\langle w_1,\dots,w_r \rangle$, $X$ is a weighted
projective space and generators of its vanishing ideal is given using
generators of defining (toric) ideals of numerical semigroup rings
corresponding to semigroups generated by subsets of $\{w_1,\dots,w_r\}$.
