Generalized Gabidulin codes over fields of any characteristic

We generalise Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the automorphism to be able to have a proper notion of rank metric which is in coherence with linearized polynomials. Under these conditions, we generalize Gabidulin codes and provide a decoding algorithm which decode both errors and erasures. Then, we focus on codes over integer rings and how to decode them. We are then faced with the problem of the exponential growth of intermediate values, and to circumvent the problem, it is natural to propose to do computations modulo a prime ideal. For this, we study the reduction of generalized Gabidulin codes over number ideals codes modulo a prime ideal, and show they are classical Gabidulin codes. As a consequence, knowing side information on the size of the errors or the message, we can reduce the decoding problem over the integer ring to a decoding problem over a finite field. We also give examples and timings.