i = 1,2, \ldots, l $, and its column components as the $ \mathbb{F}_{q^m} $-linear codes $ \widehat{C}_j \subseteq \mathbb{F}_{q^m}^{n_j} $ generated by the matrices $$\label{column components}
\widehat{G}_j = (G_{1,j}, G_{2,j}, \ldots, G_{j,j})^T,$$ for $ j = 1,2, \ldots, l $, which need not have full rank.

It holds that $ k_i = \dim(C_i^\prime) $, $ \widehat{k}_j = \dim(\widehat{C}_j) \geq k_j $, $ C = C_1^\prime \oplus C_2^\prime \oplus \cdots \oplus C_l^\prime $ and $ C \subseteq \widehat{C} = \widehat{C}_1 \times \widehat{C}_2 \times \cdots \times \widehat{C}_l $.

Different reductions always have the same main components if their block sizes are the same. See Appendix \[app 1\] for a discussion on the uniqueness of reductions of a reducible code.

Bounds on GRWs of reducible codes and exact values {#sec bounds on GRWs}
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With notation as in Subsection \[subsec reducible\], it is proven in [@reducible Lemma 2] that $$\label{bound min distance}
d_{R,1}(C) \geq \min \{ d_{R,1}(C_1), d_{R,1}(C_2), \ldots, d_{R,1}(C_l) \}.$$ We now present the main result of this section, which generalizes (\[bound min distance\]) to higher GRWs and also gives upper bounds. As observed below, it gives the exact values for cartesian products.

\[grw bounds\] With notation as in Subsection \[subsec 