Generalized Gabidulin codes over fields of any characteristic

We generalize Gabidulin codes to a large family of fields, non necessarily finite, possibly with characteristic zero. We consider a general field extension and any automorphism in the Galois group of the extension. This setting enables one to give several definitions of metrics related to the rank-metric, yet potentially different. We provide sufficient conditions on the given automorphism to ensure that the associated rank metrics are indeed all equal and proper, in coherence with the usual definition from linearized polynomials over finite fields. Under these conditions, we generalize the notion of Gabidulin codes. We also present an algorithm for decoding errors and erasures, whose complexity is given in terms of arithmetic operations. Over infinite fields the notion of code alphabet is essential, and more issues appear that in the finite field case. We first focus on codes over integer rings and study their associated decoding problem. But even if the code alphabet is small, we have to deal with the growth of intermediate values. A classical solution to this problem is to perform the computations modulo a prime ideal. For this, we need study the reduction of generalized Gabidulin codes modulo an ideal. We show that the codes obtained by reduction are the classical Gabidulin codes over finite fields. As a consequence, under some conditions, decoding generalized Gabidulin codes over integer rings can be reduced to decoding Gabidulin codes over a finite field.