rank $n-i$. Also, for any $f \in U_i$ and $\alpha\in{\mathbb{F}_{q^n}}$, we have $f \circ \omega_\alpha \in U_i$ if and only if $\alpha \in \ker f$. Thus, $$A_{n-i}= \frac{|U_i| (q^n-1)}{q^i-1}.$$ By Lemma \[lemma:complete weight\] $A_{n-\iota}\ne0$, and follows. Furthermore, $$|U|=1+ A_{n-1}\frac{q-1}{q^n-1}+\ldots+A_{n-\iota}\frac{q^\iota -1}{q-1},$$ i.e. $$(q^n-1)(|U|-1)= A_{n-1}(q-1)+\ldots+A_{n-\iota}(q^\iota -1).$$ By Theorem \[th:dualrelations\] applied to $\mathcal{C}'$ with $\nu=1$, we get $$A_{n-1}(q-1)+\ldots+A_{n-\iota}(q^\iota -1)=(q^n-1)(q^{\frac{\iota rn}{\iota+1}}-1),$$ whence $\dim_{\mathcal{F}_{n,q}}(U)=\frac{\iota rn}{\iota +1}$.

Finally, choosing $G:\mathcal{C}^\prime\to W$, $f\mapsto f(1)$ and recalling that $\tau_{f}\colon {{\mathbb F}}_{q^n}\rightarrow \operatorname{\mathcal{C}}'$, $\alpha \mapsto f\circ \omega_{\alpha}$ for any $f \in \operatorname{\mathcal{C}}^\prime$, we obtain $\operatorname{\mathcal{C}}'=\operatorname{\mathcal{C}}_{U,G}$.

Theorems \[th:MRDiff\] and \[th:MRDconverse\] provide a correspondence between:

-   ${{\mathbb F}_{q}}$-subspaces $U=V(\frac{\iota rn}{\iota+1},q)$ of $V=V(r,q^n)$ such that $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in V^*\}$; and

-   ${{\mathbb F}_{q}}$-linear MRD codes $\mathcal{C}$ with parameters $\left(\frac{rn}{\iota+1},n,q;n-\iota\right)$ and right idealiser isomorphic to ${\mathbb{F}_{q^n}}$.

When $W={{\mathbb F}}_{q^{{nr}/{(\iota+1)}}}$ and $R(\mathcal{C})=\mathcal{F}_n$, Theorem \[th:MRDconverse\] reads as follows.

Let $\iota,r,n$ be positive integers such that $\iota<n$, $\,\iota<r$ and $(\iota+1)\mid rn$. Let $f_1,\ldots,f_r\colon {\mathbb{F}_{q^n}}\rightarrow {{\mathbb F}}_{q^{{nr}/{(\iota+1)}}}$ be $\mathcal{F}_n$-linearly independent (on the right) ${{\mathbb F}_{q}}$-linear maps. Then the RM code $$\operatorname{\mathcal{C}}_{f_1,\ldots,f_r}=\{f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r} \colon \alpha_1,\ldots,\alpha_r \in {\mathbb{F}_{q^n}}\}$$ is an MRD code if and only if $$\dim_{{{\mathbb F}_{q}}} (\ker(f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r})) \leq \iota$$ for every $\alpha_1,\ldots,\alpha_r \in {\mathbb{F}_{q^n}}$. In this case, $\mathcal{C}_{f_1,\ldots,f_r}$ has parameters $\left(\frac{rn}{\iota+1},n,q;n-\iota\right)$ and $R(\operatorname{\mathcal{C}}_{f_1,\ldots,f_r})=\mathcal{F}_n$. Also, 