Constructions of optimal rank-metric codes from automorphisms of rational function fields

Abstract

<p style='text-indent:20px;'>We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.</p>

Antipodal two-weight rank metric codes

“…The first construction of MRD codes was independently presented in [6,8]. Later, many constructions of new classes of MRD codes were obtained, e.g., in [9,15,18] to name a few.…”

“…To see this, let C be an [n, 2, n−1] MRD codes over F q m /F q , where n ≤ m. Then, for any c ∈ C such that c = 0, it is clear that n − 1 ≤ rank(c) ≤ n. Hence, the two dimensional MRD codes are [n, 2, n − 1] ATW rank metric codes. As, we explained earlier, there are many constructions of MRD codes [8,9,15,18] and those already provide inequivalent classes of ATW rank metric codes of dimension 2.…”