Geometry of the minimum distance

Let K K be any field, let X⊂ Pᵏ⁻¹ X ⊂ P k - 1 be a set of n n distinct K K -rational points, and let a≥ 1 a ≥ 1 be an integer. In this paper we find lower bounds for the minimum distance d(X)ₐ d ( X ) a of the evaluation code of order a a associated to X X . The first results use α (X) α ( X ) , the initial degree of the defining ideal of X X , and the bounds are true for any set X X . In another result we use $$s(X)$$ s ( X ) , the minimum socle degree, to find a lower bound for the case when X X is in general linear position. In both situations we improve and generalize known results.