On the vanishing ideal of an algebraic toric set and its parametrized linear codes

摘要

Let K be a finite field and let X be a subset of a projective space, over the field K, which is parametrized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parametrized linear codes along with certain estimates for the algebraic invariants of I(X).