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.
Categories: math.AC cs.IT math.AG math.CO math.IT