Evasive subspaces, minimal rank-metric codes and subspace designs via auxiliary fields
Abstract
We extend the auxiliary-field method for scattered subspaces to evasive subspaces and subspace designs. For 1≤ h≤ k<hn and coprime h,k+1 with n(k−h+1)≥ k+1, we construct spanning (h,k)-evasive 𝔽q-subspaces of 𝔽qnr of dimension rn(k−h+1)/(k+1) in every sufficiently large admissible ambient dimension, with a threshold independent of q. When k<n, this dimension is maximum. The proof combines an intersection lemma, a rank-support density inequality, scalar orbit counting, and a profile preserved by direct sums. The coding consequences include, for every fixed odd extension degree m≥3 and every sufficiently large K, minimal rank-metric codes of optimal length 2K with first two generalized weights (m+1)/2 and m+1. For these codes the stable profile is equivalent to minimality with the largest possible minimum distance; for even m the analogous class consists of minimal MRD codes. More generally, for 1≤τ≤ m−2 and gcd(m,τ+1)=1, we obtain nondegenerate τ-minimal codes of length (τ+1)K for every sufficiently large K, uniformly in q. We also construct maximum h-designs whose number of members grows exponentially with the ambient dimension at fixed q,n,h, and the associated maximum sum-rank distance (MSRD) codes with a growing number of equal long blocks. In extension degree five, the ordinary and $2$-minimal constructions hold for every q and every K≥32 and K≥59, respectively.
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BibTeX
@misc{giannoni2026evasive,
title = {{Evasive subspaces, minimal rank-metric codes and subspace designs via auxiliary fields}},
author = {Alessandro Giannoni},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08627},
eprint = {2611.08627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08627},
}
RIS
TY - UNPB
TI - Evasive subspaces, minimal rank-metric codes and subspace designs via auxiliary fields
AU - Alessandro Giannoni
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08627
UR - https://arxiv.org/abs/2611.08627
AB - We extend the auxiliary-field method for scattered subspaces to evasive subspaces and subspace designs. For 1≤ h≤ k<hn and coprime h,k+1 with n(k−h+1)≥ k+1, we construct spanning (h,k)-evasive 𝔽_q-subspaces of 𝔽_q^n^r of dimension rn(k−h+1)/(k+1) in every sufficiently large admissible ambient dimension, with a threshold independent of q. When k<n, this dimension is maximum. The proof combines an intersection lemma, a rank-support density inequality, scalar orbit counting, and a profile preserved by direct sums. The coding consequences include, for every fixed odd extension degree m≥3 and every sufficiently large K, minimal rank-metric codes of optimal length 2K with first two generalized weights (m+1)/2 and m+1. For these codes the stable profile is equivalent to minimality with the largest possible minimum distance; for even m the analogous class consists of minimal MRD codes. More generally, for 1≤τ≤ m−2 and gcd(m,τ+1)=1, we obtain nondegenerate τ-minimal codes of length (τ+1)K for every sufficiently large K, uniformly in q. We also construct maximum h-designs whose number of members grows exponentially with the ambient dimension at fixed q,n,h, and the associated maximum sum-rank distance (MSRD) codes with a growing number of equal long blocks. In extension degree five, the ordinary and $2$-minimal constructions hold for every q and every K≥32 and K≥59, respectively.
ER -