Geometry of the minimum distance

Abstract

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 \(a\ge 1\) 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.