\in {{\mathbb F}}_q^{m\times n}$ be a matrix of rank $m$, and $n-d\leq m\leq n$. Then the punctured code $A\operatorname{\mathcal{C}}$ is an MRD code with parameters $(m,n,q;d+m-n)$ and $(A\operatorname{\mathcal{C}})^\top$ is an MRD code with parameters $(n,m,q;d+m-n)$.

In the literature equivalent representations of RM codes are used, other than the matrix representation that has been described above, and some of them will be used in this paper. In particular, we see the elements of an ${{\mathbb F}_{q}}$-linear RM code $\mathcal{C}$ with parameters $(m,n,q;d)$ as:

-   matrices of ${{\mathbb F}}_q^{m\times n}$ having rank at least $d$;

-   ${{\mathbb F}_{q}}$-linear maps $V\to W$ where $V=V(n,q)$ and $W=V(m,q)$, having usual map rank at least $d$;

-   when $m=n$, elements of the ${{\mathbb F}_{q}}$-algebra $\mathcal{L}_{n,q}$ of $q$-polynomials over ${\mathbb{F}_{q^n}}$ modulo $x^{q^n}-x$, having rank at least $d$ as an ${{\mathbb F}_{q}}$-linear map ${\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$.

Connection between ${{\mathbb F}_{q}}$-vector spaces and rank metric codes {#sec:subMRD}
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In this section, an ${{\mathbb F}_{q}}$-linear RM code with parameters $(m,n,q;d)$ is regarded as a set of ${{\mathbb F}_{q}}$-linear maps $W_1=V(n,q)\to W_2=V(m,q)$. The following notation will be used.

-   $\omega_{\alpha}:{\mathbb{F}_{q^n}}\to{\mathbb{F}_{q^n}}$, $x\mapsto\alpha x$, for any $\alpha\in{\mathbb{F}_{q^n}}$.

-   $\mathcal{F}_n=\{\omega_{\alpha}\colon\alpha\in{\mathbb{F}_{q^n}}\}$, which is a field isomorphic to ${\mathbb{F}_{q^n}}$.

-   $\mathcal{F}_{n,q}=\{\omega_{\alpha}\colon \alpha\in{{\mathbb F}_{q}}\}$, which is a subfield of 