# American Institute of Mathematical Sciences

February  2017, 11(1): 225-235. doi: 10.3934/amc.2017014

## On defining generalized rank weights

 1 Institut de Mathématiques, Université de Neuchatel, Rue Emilie-Argand 11,2000 Neuchatel, Switzerland 2 Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513,5600 MB Eindhoven, The Netherlands

* Corresponding author

Received  August 2015 Published  February 2017

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite extension of a field $K$. This is a generalization of the case where $K=\mathbb{F}_q$ and $L={\mathbb{F}_{{q^m}}}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu [12,13], Oggier-Sboui [16] and Ducoat [6]. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.

Citation: Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014
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