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.

翻译：设 \({\mathbb K}\) 为任意域，\(X\subset {\mathbb P}^{k-1}\) 为 \(n\) 个互异 \({\mathbb K}\)-有理点构成的集合，且 \(a\geq 1\) 为整数。本文给出了与 \(X\) 关联的 \(a\) 阶评估码最小距离 \(d(X)_a\) 的下界。第一组结果利用 \(X\) 的定义理想的初始次数 \(\alpha(X)\)，所得下界对任意点集 \(X\) 均成立。另一结果中，我们利用最小余基次数 \(s(X)\) 给出了 \(X\) 处于一般线性位置情形下的下界。两种情形均改进并推广了已有结果。