Maximum 2-scattered subspaces of $V(r,q^6)$ for every $r\ge 3$

Abstract

We prove that for every prime power $q$ and every integer $r\ge 3$ there exists a maximum $2$-scattered ${\mathbb{F}}_q$-subspace of $V(r,q^6)$, that is, a $2r$-dimensional ${\mathbb{F}}_q$-subspace meeting every $2$-dimensional ${\mathbb{F}}_{q^6}$-subspace in dimension at most $2$. Since an example in dimension three is known and direct sums preserve the property, it suffices to treat $r=4$ and $r=5$. For $r=4$ we study the subspaces $U_b=\{(x,x^q+b{x}^{q^5}):{\mathrm{Tr}}_{q^{12}/q^4}(x)=0\}$ of ${\mathbb{F}}_{q^{12}}^2$, $b\in {\mathbb{F}}_{q^{12}}$. We describe the parameters $b$ for which $U_b$ is not maximum $2$-scattered in terms of trace conditions and Dickson determinants, and deduce that at least $\frac{1}{2}{q}^{12}-q^{11}+O(q^{10})$ parameters work; this number is positive for $q\ge 3$, and a computer search settles $q=2$. For every $r\ge 5$ coprime to $6$ we show that $U_s=\{x+\epsilon s{x}^{q^r}:x\in {\mathbb{F}}_{q^{2r}}\}\subseteq {\mathbb{F}}_{q^{6r}}$ is maximum $2$-scattered for all $\epsilon \in {\mathbb{F}}_{q^6}\setminus {\mathbb{F}}_{q^2}$ and $s\in {\mathbb{F}}_{q^r}\setminus {\mathbb{F}}_q$. Consequently, ${\mathbb{F}}_{q^6}$-linear $[2r,r,4{]}_{q^6/q}$ MRD codes exist for every $r\ge 3$; for $r$ coprime to $6$, the codes obtained from $U_s$ are equivalent to their duals.

arXiv comment24 pages. Python code included in the appendix for the case q=2