Generalized Rank Weights of Reducible Codes, Optimal Cases, and Related Properties

Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes, and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error correction in network coding. In this paper, we study their security behavior against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst case information leakage to the wire tapper; 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal) and use the previous bounds to estimate their higher GRWs; 3) we show that all linear (over the extension field) codes, whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence; and 4) we show that the information leaked to a wire tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: conditions to be rank equivalent to Cartesian products of linear codes and conditions to be rank degenerate, duality properties, and MRD ranks.