Geometry of the Minimum Distance

TL;DR

This paper establishes new lower bounds for the minimum distance of evaluation codes associated with point sets in projective space, using algebraic invariants like initial degree and socle degree, generalizing previous results.

Contribution

It introduces novel bounds for evaluation code minimum distances based on algebraic properties, applicable to arbitrary point sets and those in general linear position.

Findings

Lower bounds for evaluation code distances using initial degree.

Enhanced bounds for points in general linear position.

Generalization of existing results to broader settings.

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 \(a\geq 1\) 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.