> 0 $. Hence, these bounds coincide and give the value $ d_{R,r}(C) = n_2 - k_2 + r $.

3.  Assume $ k_2 < r \leq k $: As in the previous case, now it is necessary that $ r_1 > 0 $ and $ r_2 > 0 $, and thus Theorem \[grw bounds\] gives the value $ d_{R,r}(C) = n - k +r $, which is optimal by the Singleton bound [@rgrw Proposition 1].

Finally, it is natural to ask whether different reductions (see Definition \[def reducible\]) may give different bounds in Theorem \[grw bounds\]. In Appendix \[app 1\], we show that all reductions have the same main components, thus (\[lower bound\]) remains unchanged. We now show that (\[upper bound min distance\]) can always be attained by some particular reduction. Other cases where (\[upper bound\]) may be attained by some reduction are open.

\[proposition exact for one red\] With notation as in Subsection \[subsec reducible\], there exists a reduction $ \overline{\mathcal{R}} = (\overline{G}_{i,j})_{1 \leq i \leq l}^{i \leq j \leq l} $ of $ C $ such that the bound (\[upper bound min distance\]) is an equality.

Assume that the minimum rank distance is attained by $ {\rm 