Generalized rank weights : a duality statement

We consider linear codes over some fixed finite field extension Fqm/Fq, where Fq is an arbitrary finite field. In [1], Gabidulin introduced rank metric codes, by endowing linear codes over Fqm with a rank weight over Fq and studied their basic properties in analogy with linear codes and the classical Hamming distance. Inspired by the characterization of the security in wiretap II codes in terms of generalized Hamming weights by Wei [8], Kurihara et al. defined in [3] some generalized rank weights and showed their relevance for secure network coding. In this paper, we derive a statement for generalized rank weights of the dual code, completely analogous to Wei’s one for generalized Hamming weights and we characterize the equality case of the rth-generalized Singleton bound for the generalized rank weights, in terms of the rank weight of the dual code.