Properties of codes in rank metric

TL;DR

This paper explores the properties of rank metric codes over finite fields, establishing bounds, asymptotic behaviors, and the existence of quasi-perfect codes, while showing that perfect codes do not exist in this metric.

Contribution

It provides new bounds, probabilistic analyses, and existence results for rank metric codes, including the first demonstration of quasi-perfect codes in characteristic 2.

Findings

Perfect codes do not exist in rank metric.

Random linear codes are on the Gilbert--Varshamov bound for rank metric.

Existence of quasi-perfect codes in rank metric over characteristic 2 fields.

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.