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 $1\le h\le k<hn$ and coprime $h,k+1$ with $n(k-h+1)\ge k+1$, we construct spanning $(h,k)$-evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^n}^r$ 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 $m\ge3$ 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 $1\leτ\le m-2$ 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 $K\ge32$ and $K\ge59$, respectively.