and conditions to be rank degenerate. We study their duality properties and MRD ranks. Finally, we propose alternative constructions to the classical $ (\mathbf{u}, \mathbf{u} + \mathbf{v}) $ construction.

Definitions and preliminaries
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Rank-metric codes
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Fix a prime power $ q $ and positive integers $ m $ and $ n $, and let $ \mathbb{F}_q $ and $ \mathbb{F}_{q^m} $ denote the finite fields with $ q $ and $ q^m $ elements, respectively. We may identify vectors in $ \mathbb{F}_{q^m}^n $ with $ m \times n $ matrices over $ \mathbb{F}_q $: Fix a basis $ \alpha_1, \alpha_2, \ldots, \alpha_m $ of $ \mathbb{F}_{q^m} $ over $ \mathbb{F}_q $. If $ \mathbf{c} = (c_1, c_2, \ldots, c_n) \in \mathbb{F}_{q^m}^n $, $ c_j = \sum_{i = 1}^m \alpha_i c_{i,j} $, and $ c_{i,j} \in \mathbb{F}_q $, for $ i = 1,2, \ldots, m $ and $ j = 1,2, \ldots, n $, we may identify $ \mathbf{c} $ with the matrix $$M(\mathbf{c}) = \left( c_{i,j} \right)_{1 \leq j \leq n}^{1 \leq i \leq m}.
\label{eq def matrix representation}$$ The rank weight of a vector $ \mathbf{c} \in \mathbb{F}_{q^m}^n $ is defined as the rank of the matrix $ 