On some automorphisms of rational functions and their applications in rank metric codes
Tovohery Hajatiana Randrianarisoa, Rakhi Pratihar · arXiv (Cornell University) · 2019
In this paper, we define some automorphisms of rational functions over finite fields. We employ a similar technique to Gabidulin's to define linear operators on rational functions using these automorphisms and construct some maximum rank distance (MRD) codes. The construction works for arbitrary field extensions. Reducing these to finite fields extensions we construct MRD codes over finite fields and in some particular cases, those MRD codes are not equivalent to the twisted Gabidulin codes. Using these linear operators we construct some optimal Ferrers diagram codes for some pattern conjectured by Etzion and Silberstein but our construction also provides optimal Ferrers diagrams rank metric codes for more general parameters and patterns. The codes over the rational functions are also used to construct maximum sum-rank distance codes which generalize both the Gabidulin and Reed-Solomon codes.