Maximum $2$-scattered subspaces of $V(r,q^6)$

Abstract

For every prime power $q$ and every integer $r\geq5$ coprime to $6$, we construct a maximum $2$-scattered $\mathbb{F}_q$-subspace of $V(r,q^6)$. The construction is a two-dimensional $\mathbb{F}_{q^r}$-subspace of $\mathbb{F}_{q^{6r}}$, viewed as an $r$-dimensional vector space over $\mathbb{F}_{q^6}$. A trace argument reduces the proof to the complementarity of two $\mathbb{F}_{q^r}$-subspaces. We establish this complementarity by separating two cases, which lead to a cubic polynomial obstruction and a quadratic norm obstruction. The associated rank-metric code is a $[2r,r,4]_{q^6/q}$ MRD code equivalent to its dual. Taking the cases $r=5,7$ and direct sums with the standard dimension-three construction gives maximum $2$-scattered subspaces of $V(r,q^6)$ for every $q$ and every $r\geq3$ except $r=4$. When $q$ is an odd power of $2$, the known dimension-four construction also covers this remaining case.