Generalized rank weights of reducible codes, optimal cases and related properties

Reducible codes for the rank metric were introduced for cryptographic purposes [1]. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes when Gabidulin codes [2] may not be applied and can correct many rank errors beyond half of their minimum rank distance, which make them suitable for network coding [3]. In this paper, we give lower and upper bounds on their generalized rank weights (GRWs), which measure information leakage on the network [4]. We give conditions for them to be rank equivalent to cartesian products and conditions to be rank degenerate. We study their duality properties and MRD ranks. Finally, we obtain codes with optimal GRWs for all possible fixed packet and code sizes, and prove that they are the unique optimal codes up to rank equivalence. Moreover, we see that all of them have explicit polynomial-time decoding algorithms using any of their bases.