Properties of codes in rank metric

Abstract

We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric. We study the asymptotic behavior of the minimum rank distance of codes satisfying GV. We derive the probability distribution of minimum rank distance for random and random $\F{q}$-linear codes. We give an asymptotic equivalent of their average minimum rank distance and show that random $\F{q}$-linear codes are on GV bound for rank metric. We show that the covering density of optimum codes whose codewords can be seen as square matrices is lower bounded by a function depending only on the error-correcting capability of the codes. We show that there are quasi-perfect codes in rank metric over fields of characteristic 2.

18 pages, 1 figure

Cited by in corpus (11)

• Generalized Twisted Gabidulin Codes
• Relative Generalized Rank Weight of Linear Codes and Its Applications to Network Coding
• A new family of MRD codes in ${\mathbb{F}}_q^{2n\times 2n}$ with right and middle nuclei ${\mathbb{F}}_{q^n}$
• MRD codes with maximum idealizers
• Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
• On the complexity of the Rank Syndrome Decoding problem
• On the hardness of the decoding and the minimum distance problems for rank codes
• Properties of Rank Metric Codes
• On the similarities between generalized rank and Hamming weights and their applications to network coding
• On deep-holes of Gabidulin codes
• Rank Distance Bicodes and their Generalization