the $\mathcal{F}_{n,q}$-subspace $U_{f_1,\ldots,f_r}$ of $C_{f_1,\ldots,f_r}$ given by $$U_{f_1\ldots,f_r}=\{ f_1\circ \omega_{\alpha_1}+\ldots+f_r\circ \omega_{\alpha_r} \in \operatorname{\mathcal{C}}_{f_1,\ldots,f_r} \colon f_1(\alpha_1)+\ldots+f_r(\alpha_r)=0 \}$$ has dimension $\frac{\iota rn}{\iota +1}$, and $\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U_{f_1,\ldots,f_r}\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})\colon \mathbf{v}\in \operatorname{\mathcal{C}}_{f_1,\ldots,f_r}^*\}$.

Let $\iota,n,r$ be positive integers such that $\iota<n$ and $(\iota+1)\mid r$. Define $t=r/(\iota+1)$. The code $$\mathcal{C}=\left\{x\in{{\mathbb F}}_{q^{nt}}\mapsto a_0 x+a_1 x^q+\ldots+a_{\iota}x^{q^\iota}\in{{\mathbb F}}_{q^{nt}}\;\colon\; a_0,\ldots,a_{\iota}\in{{\mathbb F}}_{q^{nt}}\right\}$$ is an MRD code with parameters $(nt,nt,q;nt-\iota)$, known as Gabidulin code; see Section \[sec:noGab\] below. Consider the code $$\mathcal{C}|_{{\mathbb{F}_{q^n}}}=\left\{f|_{{\mathbb{F}_{q^n}}}:{\mathbb{F}_{q^n}}\to{{\mathbb F}}_{q^{nt}}\colon f\in\mathcal{C}\right\}.$$ By Theorem \[th:punct\], $\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ is an MRD code with parameters $(nt,n,q;n-\iota)$. Also, $R(\mathcal{C}|_{{\mathbb{F}_{q^n}}})=\mathcal{F}_n$ and an $\mathcal{F}_n$-basis of $\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ (seen as a right vector space) is $$\left\{ f_{j,i} \colon x\in{\mathbb{F}_{q^n}}\mapsto \xi^{i}x^{q^j}\in{{\mathbb F}}_{q^{nt}}\;\mid\; 0\leq i\leq t-1,\,0\leq j\leq \iota \right\},$$ where $\{1,\xi,\ldots,\xi^{t-1}\}$ is an ${\mathbb{F}_{q^n}}$-basis of ${{\mathbb F}}_{q^{nt}}$. Moreover, the set of the elements $f\in\mathcal{C}|_{{\mathbb{F}_{q^n}}}$ vanishing at $1$ is equal to $$U=\left\{x\in{\mathbb{F}_{q^n}}\mapsto -(a_1+\ldots+a_{\iota}) x+a_1 x^q+\ldots+a_{\iota}x^{q^\iota}\in{{\mathbb F}}_{q^{nt}}\;\colon\; a_1,\ldots,a_{\iota}\in{{\mathbb F}}_{q^{nt}} \right\},$$ and $\iota=\max\left\{\dim_{\mathcal{F}_{n,q}}\left(U\cap\langle f\rangle_{\mathcal{F}_n}\right)\colon f\in\mathcal{C}|_{{\mathbb{F}_{q^n}}}^{\,*}\right\}$. Let $$B=\left(f_{j,i}\,\colon\, j=0,\ldots,\iota,\;i=0,\ldots,t-1\right).$$ The coordinates of a vector in $U$ with respect to $B$ are $$\left( -\sum_{k=1}^{\iota}a_{k,0}\;,\ldots,-\sum_{k=1}^{\iota}a_{k,0}\;,\;a_{1,0}^{q^{n-1}},\ldots,a_{1,t-1}^{q^{n-1}}\;,\;\ldots\ldots,\;a_{\iota,0}^{q^{n-\iota}},\ldots,a_{\iota,t-1}^{q^{n-\iota}} \right),$$ where $a_{k,i}\in {\mathbb{F}_{q^n}}$ are such that $a_k=\sum_{i=0}^{t-1}a_{k,i}\xi^{i}$. Denote by $\overline{U}$ the set of the coordinates of the vectors in $U$. Let $\sigma^{\prime}:\mathbb{F}_{q^n}^{t(\iota+1)}\times \mathbb{F}_{q^n}^{t(\iota+1)}\to\mathbb{F}_q$, $(\mathbf{u},\mathbf{v})\mapsto\mathrm{Tr}_{q^n/q}(\langle\mathbf{u},\mathbf{v}\rangle)$, where $\langle\cdot,\cdot\rangle$ is the standard inner product. Then the vectors of $\overline{U}^{\perp^\prime}$ are $$(y_0,\ldots,y_{t-1},y_0^{q^{n-1}},\ldots,y_{t-1}^{q^{n-1}},\ldots \ldots, y_0^{q^{n-\iota}},\ldots,y_{t-1}^{q^{n-\iota}}),$$ where $y_0,\ldots,y_{t-1}\in\mathbb{F}_{q^n}$. 