all of them in Section \[estimates on their\].

The two previous constructions are particular cases of reducible codes, introduced in [@reducible], which we will study in the rest of the paper.

In [@allertonversion], MRD $ \mathbb{F}_q $-linear codes obtained by transposing the matrix representations of codewords in a Gabidulin code are proposed for the case $ n > m $. However no exact values or lower bounds are known for any code in this case ($ n > m $).

Reducible codes and reductions {#subsec reducible}
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Consider positive integers $ l, n_1, n_2, \ldots, n_l $ and $ \mathbb{F}_{q^m} $-linear codes $ C_1 \subseteq \mathbb{F}_{q^m}^{n_1}, C_2 \subseteq \mathbb{F}_{q^m}^{n_2}, \ldots, C_l \subseteq \mathbb{F}_{q^m}^{n_l} $ of dimensions $ k_1 $, $ k_2, \ldots, k_l $, respectively. Consider matrices $ G_{i,j} \in \mathbb{F}_{q^m}^{k_i \times n_j} $, for $ i =1,2, \ldots, l $ and $ j = i,i+1, \ldots, l $, where $ G_{i,i} $ generates $ C_i $.

\[**[@reducible]**\] \[def reducible\] We say that an $ \mathbb{F}_{q^m} $-linear code $ C \subseteq \mathbb{F}_{q^m}^n $ is reducible with reduction $ \mathcal{R} = (G_{i,j})_{1 \leq i \leq l}^{i \leq j \leq l} $ if it has a generator matrix of the form $$G 