Generalized rank weights : a duality statement

We consider linear codes over some fixed finite field extension Fq m/Fq, where Fq is an arbitrary finite field. In [1], Gabidulin introduced rank metric codes, by endowing linear codes over Fq m with a rank weight over Fq and studied their basic properties in analogy with linear codes and the classical...

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Bibliographic Details
Main Author: Ducoat, Jérôme
Other Authors: School of Physical and Mathematical Sciences
Format: Conference or Workshop Item
Language:English
Published: 2015
Subjects:
Online Access:https://hdl.handle.net/10356/100886
http://hdl.handle.net/10220/25511
http://www.ams.org/books/conm/632/
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Institution: Nanyang Technological University
Language: English
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Summary:We consider linear codes over some fixed finite field extension Fq m/Fq, where Fq is an arbitrary finite field. In [1], Gabidulin introduced rank metric codes, by endowing linear codes over Fq m 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.