Note that $\overline{U}^{\perp^\prime}$ is the direct sum of $t$ copies of $$\{ (z,z^{q},\ldots,z^{q^\iota})\colon z\in\mathbb{F}_{q^n} \}$$ which is a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^{\iota+1}$.

Therefore, when $\iota+1$ divides $r$, the restriction to ${\mathbb{F}_{q^n}}$ of a Gabidulin code is associated with the direct sum of $t$ copies of a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^{\iota+1}$ (which is a $\iota$-scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}^r$); see Section \[sec:twocharact\].

Previously known connections {#sec:uni}
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In this section, we show that the connection between ${{\mathbb F}_{q}}$-vector spaces and ${{\mathbb F}_{q}}$-linear MRD codes established in Section \[sec:subMRD\] generalizes those presented in [@Sheekey; @ShVdV; @Lunardon2017; @CSMPZ2016].

Sheekey’s connection {#sec:sheekey}
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The first connection was pointed out by Sheekey in its seminal paper [@Sheekey]. Let $U$ be an ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}\times{\mathbb{F}_{q^n}}$, so that $$U=U_{f_1,f_2}=\{(f_1(x),f_2(x))\colon x\in{\mathbb{F}_{q^n}}\}$$ for some $f_1(x),f_2(x)$ in $\mathcal{L}_{n,q}$. Consider the ${{\mathbb F}_{q}}$-linear RM code $$\mathcal{S}_{f_1,f_2}=\{a_1 f_1(x)+a_2 f_2(x)\colon a_1,a_2\in{\mathbb{F}_{q^n}}\}\subset\mathcal{L}_{n,q},$$ whose left idealiser is isomorphic to ${\mathbb{F}_{q^n}}$. Then $U_{f_1,f_2}$ is a maximum scattered ${{\mathbb F}_{q}}$-subspace of ${\mathbb{F}_{q^n}}\times{\mathbb{F}_{q^n}}$ if and only if $\mathcal{S}_{f_1,f_2}$ is an MRD code with parameters $(n,n,q;n-1)$; see [@Sheekey Section 5].

A generalization to maximum $(r-1)$-scattered ${{\mathbb F}_{q}}$-subspaces of ${\mathbb{F}_{q^n}}^r$ {#sec:r-1}
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Sheekey’s connection was extended by Sheekey and Van de Voorde in [@ShVdV] as follows; see 