d_{R,r_l}(C_l) \\
 & \mid r = r_1 + r_2 + \cdots + r_l \}.
\end{split}$$ \[grw product\]

Now we illustrate Theorem \[grw bounds\] with the following example that includes the MRD $ \mathbb{F}_{q^m} $-linear codes in Subsection \[existing code constructions\], item 2, for $ l = 2 $:

\[example computation\] With notation as in Theorem \[grw bounds\], assume that $ l = 2 $, $ n_1, n_2 \leq m $, $ k_1 \leq k_2 $ and take $ C_1 $ and $ C_2 $ as MRD codes (the matrix $ G_{1,2} $ can be arbitrary). In particular, $ d_{R,r_i}(C_i) = n_i - k_i + r_i $ [@rgrw] as in Subsection \[existing code constructions\], $ 1 \leq r_i \leq k_i $, $ i = 1,2 $. We estimate $ d_{R,r}(C) $ considering three cases:

1.  Assume $ 1 \leq r \leq k_1 $: The bounds (\[lower bound\]) and (\[upper bound\]) give $$\min \{ n_1 - k_1, n_2 - k_2 \} + r \leq d_{R,r}(C) \leq n_2 - k_2 + r.$$

2.  Assume $ k_1 < r \leq k_2 $ (if $ k_1 < k_2 $): In this case, in both bounds in Theorem \[grw bounds\], it is necessary that $ r_2 