Geometry of the Minimum Distance

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