to measure the worst-case information leakage for a given number of wire-tapped links. Later, GRWs were extended in [@ravagnaniweights] and [@allertonversion] to codes that are linear over the base field, where they are called Delsarte generalized weights and generalized matrix weights, respectively. We will use the term GRWs for the latter parameters, which were also found to measure the worst-case information leakage for codes that are linear over the base field [@allertonversion Th. 3].

Gabidulin codes [@gabidulin] constitute a family of maximum rank distance (MRD) codes that cover all cases when the number of outgoing links $ n $ is not larger than the packet length $ m $, and all of their GRWs are optimal (meaning largest possible).

Cartesian products of these codes are proposed in [@silva-universal Sec. VII.C] for the case $ n > m $ both for error correction and security against information leakage. A generalization of these codes, called reducible codes, were introduced earlier in [@reducible] as an alternative to Gabidulin codes [@gabidulin] to improve the security of rank-based public key cryptosystems [@ideals]. On the error correction side, it was shown in [@reducible] that reducible codes have fast encoding and rank error-correcting algorithms, their minimum rank 