distance is not worse than that of cartesian products of codes [@silva-universal Sec. VII.C], being actually MRD in some cases, and they can correct many rank errors beyond half of their minimum rank distance (even in the MRD cases). Therefore they seem to be the best known codes for error correction in linear network coding when $ n > m $.

However, on the security side, only the existence of codes with optimal first GRW (MRD codes) has been studied in the case $ n > m $ [@allertonversion Sec. IV-B], but no bounds nor estimates of higher GRWs of rank-metric codes or other properties related to their worst-case information leakage are known when $ n > m $, except for cyclic codes with minimal GRWs [@oggiercyclic].

In this paper, we study the security provided by reducible codes in linear network coding when used for coset coding as in [@silva-universal] by studying their GRWs and showing their optimality in several cases. In particular, we study for the first time the GRWs of a concrete family of rank-metric codes with $ n > m $, which moreover include MRD codes for several parameters.

Main contributions
------------------

Our main contributions are the 