New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions

Codes in the rank metric have been studied for the last four decades. For linear codes a Singleton-type bound can be derived for these codes. In analogy to MDS codes in the Hamming metric, we call rank-metric codes that achieve the Singleton-type bound MRD (maximum rank distance) codes. Since the works of Delsarte [3] and Gabidulin [4] we know that linear MRD codes exist for any set of parameters. The codes they describe are called Gabidulin codes. Moreover, Berger in [1] and Morrison in [8] showed what the linear and semi-linear isometries of rank-metric codes are. It is an open question if there are other general constructions of MRD codes that are not equivalent (under the isometries) to Gabidulin codes. Recently several results have been established in this direction, e.g. in [2, 9], where many of the derived codes are not linear over the underlying field but only linear over some subfield of it. Hence it is still an open question to find other constructions of non-Gabidulin MRD codes. In this paper we want to derive criteria for MRD and Gabidulin codes and use these to come up with new non-Gabidulin MRD codes that are linear over the original field, not only a subfield. Moreover, we want to give a classification of these codes and investigate how many different equivalence classes of MRD codes we get for small parameters. This paper is structured as follows. In Section 2 we give some preliminaries on finite fields, rank-metric codes and Gabidulin codes. In Section 3 we present a new criterion for MRD codes, in Section 4 we derive a criterion for Gabidulin codes. In Section 5 we use the results of Sections 3 and 4 to find new non-Gabidulin MRD codes for small parameters. We conclude in Section 6.