In particular, it is shown in [@rgrw Cor. 1] that $ d_{R,1}(C) $ is the minimum rank distance of the code $ C $ (also denoted by $ d_R(C) $). Thus the minimum rank distance is of particular importance, since it gives the maximum number of wire-tapped links that guarantee zero information leakage, and we may evaluate the code’s optimality among all rank-metric codes (linear and non-linear) in this sense using the Singleton bound [@delsartebilinear Th. 6.3]: $$\label{singleton bound}
\# C \leq q^{\max \{ m,n \} (\min \{ m,n \} - d_R(C) + 1)},$$ where $ C \subseteq \mathbb{F}_{q^m}^n $ is an arbitrary rank-metric code. Codes attaining this bound are called maximum rank distance (MRD) codes.

Existing MRD code constructions {#existing code constructions}
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We briefly revisit two existing code constructions that have already been considered in the literature:

1.  Assume $ n \leq m $ and $ 1 \leq k \leq n $: Take elements $ \beta_1, \beta_2, \ldots, \beta_n \in \mathbb{F}_{q^m} $ that are linearly independent over $ \mathbb{F}_q $. The $ \mathbb{F}_{q^m} $-linear code $ C_{Gab} \subseteq \mathbb{F}_{q^m}^n $ generated by the matrix $$\left( \begin{array}{cccc}
    \beta_1 & \beta_2 & \ldots & \beta_n \\
    \beta_1^{[1]} & \beta_2^{[1]} & 