Generalized rank weights: A duality statement
Jérôme Ducoat · Contemporary mathematics - American Mathematical Society · 2015
We consider linear codes over some fixed finite field extension F q m / F q \mathbb F_{q^m}/\mathbb F_q , where F q \mathbb F_q is an arbitrary finite field. Gabidulin (1985) introduced rank metric codes, by endowing linear codes over F q m \mathbb F_{q^m} with a rank weight over F q \mathbb F_q 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, Kurihara et al. defined 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 r t h r^{th} -generalized Singleton bound for the generalized rank weights, in terms of the rank weight of the dual code.