generalization of $1$-scattered subspaces, which are simply known as scattered subspaces and were originally presented in [@BL2000]. Since then, the theory of scattered subspaces has constantly increased its importance, mainly because of their applications to several algebraic and geometric objects, such as finite semifields, blocking sets, two-intersection sets; see [@Lavrauw; @LVdV2015; @Polverino]. After the seminal paper [@Sheekey] by Sheekey, the interest towards scattered subspaces was also boosted by their connections with the theory of rank metric codes, whose relevance in communication theory relies on its applications to random linear network coding and cryptography.

A $h$-scattered ${{\mathbb F}_{q}}$-subspace of highest dimension in $V(r,q^n)$ is called *maximum $h$-scattered*; its dimension is upper bounded by $\frac{rn}{h+1}$. This bound is known to be achieved in the following cases: $h=1$, $h=r-1$, $(h+1)\mid r$ or $h=n-3$; see Section \[sec:h-scatt\]. When $h=1$ or $h=r-1$, maximum $h$-scattered subspaces are strongly related to rank metric codes having the greatest correcting and detecting capabilities for fixed dimension and ambient space, that is, to maximum rank distance (MRD) codes. This has been shown in [@Sheekey; @CSMPZ2016; @PZ] for $h=1$ and in [@Lunardon2017; @ShVdV] for $h=r-1$, while no relation was known for $1<h<r-1$. In this paper we establish a connection between ${{\mathbb 