Computing Vanishing Ideals for Toric Codes

Motivated by applications to the theory of error-correcting codes, we give methods for computing a generating set for the ideal generated by $\beta$-graded polynomials vanishing on certain subsets of a simplicial complete toric variety $X$ over a finite field ${\mathbb{F}}_q$, 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 $\mathbb{N}\beta =⟨w_1,\dots ,w_r⟩$, $X$ is a weighted projective space and generators of the relevant vanishing ideal is given using generators of defining (toric) ideals of numerical semigroup rings corresponding to semigroups generated by subsets of $\{w_1,\dots ,w_r\}$.