equivalent by means of a diagonal matrix; we denote one of them by $\mathcal{C}_{\mathcal{P}}$.

\[prop:projsyst\] Let $\mathcal{P}$ be a projective $[N,k,d]_q$-system of $\Omega$ and $\mathcal{C}_{\mathcal{P}}$ be a corresponding linear $[N,k,d]_q$-code. Then the weights of $\mathcal{C}_{\mathcal{P}}$ are the values $N-i$, where $i=|\mathcal{P}\cap\mathcal{H}|$ and $\mathcal{H}$ runs over the hyperplanes of $\Omega$. The number $A_i^H$ of codewords of $\mathcal{C}_{\mathcal{P}}$ with weight $i$ is equal to the number of hyperplanes $\mathcal{H}$ of $\Omega$ such that $|\mathcal{P}\cap\mathcal{H}|=i$.

### Rank metric codes {#sec:rank}

Rank metric codes were introduced by Delsarte [@Delsarte] in 1978 and they have been intensively investigated in recent years because of their applications; we refer to [@sheekey_newest_preprint] for a survey on this topic. The set ${{\mathbb F}_{q}}^{m\times n}$ of $m \times n$ matrices over ${{\mathbb F}_{q}}$ may be endowed with a metric, called *rank metric*, defined by $$d(A,B) = \mathrm{rk}\,(A-B).$$ A subset $\operatorname{\mathcal{C}}\subseteq {{\mathbb F}_{q}}^{m\times n}$ equipped with the rank metric is called a *rank metric code* (shortly, an *RM code*). The minimum distance of $\operatorname{\mathcal{C}}$ is defined as $$d = \min\{ d(A,B) \colon A,B \in \operatorname{\mathcal{C}},\,\, A\neq B \}.$$ Denote the parameters of an RM code $\operatorname{\mathcal{C}}\subseteq{{\mathbb F}_{q}}^{m\times n}$ with minimum distance $d$ by $(m,n,q;d)$. We are interested in ${{\mathbb F}_{q}}$-*linear* RM 