$\mathcal{F}_n$ isomorphic to ${{\mathbb F}_{q}}$.

-   $\tau_{\mathbf{v}}:{\mathbb{F}_{q^n}}\to W_1$, $\lambda\mapsto \lambda \mathbf{v}$, for any $\mathbf{v}\in W_1$.

We define a family of ${{\mathbb F}_{q}}$-linear RM codes associated with an ${{\mathbb F}_{q}}$-vector space $U$.

Let $n,r,k$ be positive integers with $k<rn$, $U$ be a $k$-dimensional ${{\mathbb F}_{q}}$-subspace of an $r$-dimensional ${\mathbb{F}_{q^n}}$-vector space $V$, $W$ be an $(rn-k)$-dimensional ${{\mathbb F}_{q}}$-vector space, and $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with kernel $U$. For any $\mathbf{v} \in V$ define the ${{\mathbb F}_{q}}$-linear map $\Gamma_{\mathbf{v}}=G\circ\tau_{\mathbf{v}}$.

\[th:construction\] Let $V=V(r,q^n)$ and $W=V(rn-k,q)$. Let $U=V(k,q)$ be an ${{\mathbb F}_{q}}$-subspace of $V$, and $G:V\to W$ be an ${{\mathbb F}_{q}}$-linear map with $\ker(G)=U$. Define $$\iota=\max\{\dim_{{{\mathbb F}_{q}}}(U\cap\langle\mathbf{v}\rangle_{{\mathbb{F}_{q^n}}}) \colon \mathbf{v}\in V^*  \}.$$ If $\iota<n$, then the pair $(U,G)$ defines an ${{\mathbb F}_{q}}$-linear RM code $$\label{eq:rd}
    \mathcal{C}_{U,G}=\left\{ \Gamma_{\mathbf{v}}=G\circ \tau_{\mathbf{v}} \colon \mathbf{v} \in V \right\}$$ of dimension $rn$ with parameters $(rn-k,n,q;n-\iota)$, whose right idealiser contains $\mathcal{F}_n$.

For any $\mathbf{v},\mathbf{w}\in V$ and $\alpha\in{{\mathbb F}_{q}}$ we have $\Gamma_{\mathbf{v}}+\Gamma_{\mathbf{w}}=\Gamma_{\mathbf{v}+\mathbf{w}}$ and $\alpha\,\Gamma_{\mathbf{v}}=\Gamma_{\alpha\mathbf{v}}$, and hence $\mathcal{C}_{U,G}$ is an ${{\mathbb F}_{q}}$-vector space.

For any $\mathbf{v}\in V$, let $R_{\mathbf{v}}=\{ \lambda \in {\mathbb{F}_{q^n}}\colon \lambda \mathbf{v} \in U \}$. Clearly $\ker (\Gamma_{\mathbf{v}})=R_{\mathbf{v}}$ and, when $\mathbf{v}\ne\mathbf{0}$, $\dim_{{{\mathbb F}_{q}}}(R_{\mathbf{v}})=\dim_{{{\mathbb F}_{q}}}(U\cap\langle \mathbf{v}\rangle_{{\mathbb{F}_{q^n}}})$. Then $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{v}}))\leq\iota$ and there exists $\mathbf{u}\in V^*$ such that $\dim_{{{\mathbb F}_{q}}}(\ker(\Gamma_{\mathbf{u}}))=\iota$, so that the 