Properties of Codes with the Rank Metric

Abstract

In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic forms of these two bounds and the Singleton bound. Based on the asymptotic bounds, we observe that asymptotically Gilbert-Varsharmov bound is exceeded by MRD codes and sphere-packing bound cannot be attained. We also establish bounds on the rank covering radius of maximal codes, and show that all MRD codes are maximal codes and all the MRD codes known so far achieve the maximum rank covering radius.

5 pages, 4 figures, accepted to Globecom 2006

Cited by in corpus (7)

• A Rank-Metric Approach to Error Control in Random Network Coding
• Generalized Twisted Gabidulin Codes
• Security for Wiretap Networks via Rank-Metric Codes
• A new family of MRD codes in ${\mathbb{F}}_q^{2n\times 2n}$ with right and middle nuclei ${\mathbb{F}}_{q^n}$
• MacWilliams Identity for Codes with the Rank Metric
• Constant-Rank Codes and Their Connection to Constant-Dimension Codes
• Packing and Covering Properties of Rank Metric Codes