A geometric approach to the density of rank-metric codes

2026-09-17

AI Summary

本文通过几何方法研究了有限域上几何不可约射影簇的无F_q点线性截面的渐近密度，并将其应用于秩度量码，解决了先前未知情况下的密度极限问题。

Abstract

We study the asymptotic density of $\mathbb{F}_q$-point-free linear sections of geometrically irreducible projective varieties over finite fields. We then apply these results to rank-metric codes via determinantal varieties. Our approach recovers the known cases in which the density tends to $0$ or $1$ and determines the limit in the cases where it was previously unknown. To compute these previously unknown limits, we extend the notion of quasireflexivity to higher-dimensional varieties and show that determinantal varieties satisfy this property. This allows us to invoke the Chebotarev density theorem for varieties over finite fields to obtain the desired estimate.

Problem

• rank-metric codes
• asymptotic density
• geometrically irreducible projective varieties
• determinantal varieties
• finite fields

Innovation

• geometric approach
• rank-metric codes
• determinantal varieties
• quasireflexivity
• Chebotarev density theorem