On defining generalized rank weights

Relinde Jurrius; Ruud Pellikaan

doi:10.3934/amc.2017014

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2017, Volume 11, Issue 1: 225-235. Doi: 10.3934/amc.2017014

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On defining generalized rank weights

Relinde Jurrius^1, and
Ruud Pellikaan^2, ,

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
^* Corresponding author

Received: August 2015

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 94C99; Secondary: 94B27, 94B60.

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References

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