Isometries for rank distance and permutation group of Gabidulin codes

The rank distance was introduced by E.M. Gabidulin (see Probl. Pered. Inform., vol.21, p.1-12, 1985). He determined an upper bound for the minimum rank distance of a code. Moreover, he constructed a class of codes which meet this bound: the so-called Gabidulin codes. We first characterize the linear and semilinear isometries for the rank distance. Then we determine the isometry group and the permutation group of Gabidulin codes of any length. We give a characterization of equivalent Gabidulin codes. Finally, we prove that the number of equivalence classes of Gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in GF(p^m) under some particular relations.