Generalized Hamming weights of codes arising from complete intersection

2026-08-20

AI Summary

本文解决了Tohčeanu和Van Tuyl关于完全交集点集编码的最小距离的猜想，通过一个不太为人知的经典Bézout界改进版本，并进一步得到了广义汉明重量的界限。

Abstract

We provide a positive answer to a conjecture proposed by Tohǎneanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical Bézout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.

Problem

• Generalized Hamming Weights
• Complete Intersection
• Minimum Distance

Innovation

• Generalized Hamming Weights
• Complete Intersection
• Bézout Bound