CZ\in \operatorname{\mathcal{C}}\hspace{0.1cm} \text{for all}\hspace{0.1cm} C \in \operatorname{\mathcal{C}}\}.$$ The notion of idealisers have been introduced by Liebhold and Nebe in [@LN2016 Definition 3.1]; they are invariant under equivalences of rank metric codes. Further invariants have been introduced in [@GiuZ; @NPH2]. In [@LTZ2], idealisers have been studied in details and the following result has been proved.

\[th:propertiesideal\] Let $\mathcal{C}$ and $\mathcal{C}^\prime$ be ${{\mathbb F}_{q}}$-linear RM codes of ${{\mathbb F}_{q}}^{m\times n}$.

-   If $\mathcal{C}$ and $\mathcal{C}^\prime$ are equivalent, then their left and right idealisers are isomorphic as ${{\mathbb F}_{q}}$-algebras ([@LTZ2 Proposition 4.1]).

-   $L(\operatorname{\mathcal{C}}^\top)=R(\operatorname{\mathcal{C}})^\top$ and $R(\operatorname{\mathcal{C}}^\top)=L(\operatorname{\mathcal{C}})^\top$ ([@LTZ2 Proposition 4.2]).

-   Let $\mathcal{C}$ have minimum distance $d>1$. If $m \leq n$, then $L(\operatorname{\mathcal{C}})$ is a finite field with $|L(\operatorname{\mathcal{C}})|\leq q^m$. If $m \geq n$, then $R(\operatorname{\mathcal{C}})$ is a finite field with $|R(\operatorname{\mathcal{C}})|\leq q^n$. In particular, when $m=n$, $L(\operatorname{\mathcal{C}})$ and $R(\operatorname{\mathcal{C}})$ are both finite fields ([@LTZ2 Theorem 5.4 and Corollary 5.6]).

Let $\mathcal{C}$ be an RM code in ${{\mathbb F}_{q}}^{n\times n}$, and $A\in{{\mathbb F}_{q}}^{m\times n}$ be a matrix of rank $m\leq n$. The RM code $A\mathcal{C}=\{AM\colon M\in\mathcal{C}\}\subseteq{{\mathbb F}_{q}}^{m\times n}$ is a *punctured code* obtained by *puncturing $\mathcal{C}$ with $A$*.

\[th:punct\]([@BR Corollary 35], [@CsS Theorem 3.2]) Let $\operatorname{\mathcal{C}}$ be an MRD code with parameters $(n,n,q;d)$, $A 