\right)$$ satisfies that $ G = A \overline{G} $, where $ \overline{G} $ is the generator matrix of $ C $ corresponding to $ \overline{\mathcal{R}} = (\overline{G}_{i,j})_{1 \leq i \leq l}^{i \leq j \leq l} $, and $$\mathbf{x}_1 A_{1,j} = - \mathbf{x}_j,$$ for $ j = 2,3, \ldots, l $. It is possible to choose such matrices $ A_{1,j} $ because $ \mathbf{x}_1 \neq \mathbf{0} $. Then $ \mathbf{c} = (\mathbf{x} A) \overline{G} $ lies in the first row component of the reduction $ \overline{\mathcal{R}} $ and hence $ d_{R,1}(C) = {\rm wt_R}(\mathbf{c}) \geq d_{R,1}(\overline{C}_1^\prime) $, implying the result.

We conclude the section with the proof of Theorem \[grw bounds\]. We need the following lemma:

\[lemma D generators\] With notation as in Subsection \[subsec reducible\], define the sets $$A_i = \{ 0 \}^{n_1} \times \cdots \times \{ 0 \}^{n_{i-1}} \times (\mathbb{F}_{q^m}^{n_i} \setminus \{ \mathbf{0} \}) \times \mathbb{F}_{q^m}^{n_{i+1}} \times \cdots \times \mathbb{F}_{q^m}^{n_l},$$ for $ i = 1,2, \ldots, l $. For an $ \mathbb{F}_{q^m} $-linear vector space $ D \subseteq \mathbb{F}_{q^m}^n $, there exist subspaces $ D_i^\prime \subseteq \langle D \cap A_i \rangle $, for $ i $ satisfying $ D \cap A_i \neq \varnothing $, such that $ D = 