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PreprintarXiv licencearXiv

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 𝔽q-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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Cite

BibTeXRIS
BibTeX
@misc{asgarli2026geometric,
  title = {{A geometric approach to the density of rank-metric codes}},
  author = {Shamil Asgarli and Lian Duan and Nathan Kaplan and Kuan-Wen Lai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21189},
  eprint = {2609.21189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21189},
}
RIS
TY  - UNPB
TI  - A geometric approach to the density of rank-metric codes
AU  - Shamil Asgarli
AU  - Lian Duan
AU  - Nathan Kaplan
AU  - Kuan-Wen Lai
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21189
UR  - https://arxiv.org/abs/2609.21189
AB  - We study the asymptotic density of 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.
ER  -