Geometry of the Minimum Distance
John Pawlina, Stefan Tohaneanu
TL;DR
Geometry of the Minimum Distance investigates how algebraic invariants of point configurations in projective space control the minimum distance of evaluation codes. The authors derive a universal lower bound $d(X)_a \geq \binom{\alpha(X)-1-a+k-1}{k-1}$ for $1 \le a \le \alpha(X)-1$, and a complementary bound in general linear position involving the minimum socle degree $s(X)$, namely either $d(X)_a \le k-1$ or $d(X)_a \ge (k-1)(s(X)-1-a)+2$; these results generalize previous Ballico-Fontanari-type bounds and improvements via homological invariants. The work uses evaluation maps, Hilbert functions, free resolutions, and Artinian reductions to connect code parameters with geometric and algebraic properties of $X$, extending to finite fields without requiring algebraic closure. The findings illuminate how the distribution of points across hyperplanes and the tails of free resolutions govern distance properties, with constructed examples showing sharpness and guiding code-design implications.
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 \(α(X)\), 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.