Vanishing ideals over finite fields

Let ${\mathbb{F}}_q$ be a finite field, let $\mathbb{X}$ be a subset of a projective space ${\mathbb{P}}^{s-1}$, over the field ${\mathbb{F}}_q$, parameterized by rational functions, and let $I(\mathbb{X})$ be the vanishing ideal of $\mathbb{X}$. The main result of this paper is a formula for $I(\mathbb{X})$ that will allows us to compute: (i) the algebraic invariants of $I(\mathbb{X})$, and (ii) the basic parameters of the corresponding Reed-Muller-type code.