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arXiv · 2609.21189

A geometric approach to the density of rank-metric codes

Shamil Asgarli·Lian Duan·Nathan Kaplan·Kuan-Wen Lai
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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.

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Shamil Asgarli, Lian Duan, Nathan Kaplan, Kuan-Wen Lai. 2026-09-18. A geometric approach to the density of rank-metric codes. https://arxiv.org/abs/2609.21189

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