wt_R}(\mathbf{c}) = d_{R,1}(C) $, for $ \mathbf{c} \in C $. It holds that $ \mathbf{c} = \mathbf{c}_1^\prime + \mathbf{c}_2^\prime + \cdots + \mathbf{c}_l^\prime $, with $ \mathbf{c}_i^\prime \in C_i^\prime $, and $ \mathbf{c}_i^\prime = \mathbf{x}_i G_{i,i}^\prime $ (recall (\[row components\])), for some $ \mathbf{x}_i \in \mathbb{F}_{q^m}^{k_i} $ and all $ i = 1,2, \ldots, l $.

We may assume without loss of generality that $ \mathbf{x}_1 \neq \mathbf{0} $. We just need to define $ \overline{G}_{i,i} = G_{i,i} $ and choose matrices $ A_{1,j} \in \mathbb{F}_{q^m}^{k_1 \times k_j} $ and $ \overline{G}_{i,j} \in \mathbb{F}_{q^m}^{k_i \times n_j} $, for $ 1 \leq i \leq l-1 $ and $ i+1 \leq j \leq l $, such that the $ k \times k $ matrix $$A = \left(
\begin{array}{cccccc}
I & A_{1,2} & A_{1,3} & \ldots & A_{1, l-1} & A_{1,l} \\
0 & I & 0 & \ldots & 0 & 0 \\
0 & 0 & I & \ldots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ldots & I & 0 \\
0 & 0 & 0 & \ldots & 0 & I \\
\end{array} 