On some automorphisms of rational functions and their applications in rank metric codes

Recently, there is a growing interest in the study of rank metric codes. These codes have applications in network coding and cryptography. In this paper, we investigate some automorpshisms on polynomial rings over finite fields. We show how the linear operators from these automorphisms can be used to construct some maximum rank distance (MRD) codes. First we work on rank metric codes over arbitrary extension and then we reduce these to finite fields extension. Some particular constructions give MRD codes which are not equivalent to the twisted Gabidulin codes. Another application is to use these linear operators to construct some optimal rank metric codes for some Ferrers diagrams. In fact we give some examples of Ferrers diagrams for which there was no known construction of optimal rank metric codes.