A quadratic Welch-Berlekamp algorithm to decode generalized Gabidulin codes, and some variants

Gabidulin codes are Maximum Rank Distance (MRD) codes. They have been recently generalized to cyclic Galois extension fields. The unique decoding problem is equivalent to the linear reconstruction problem. The aim of this article is the study of an algorithm to solve this reconstruction problem. We prove that the output of our algorithm is a solution of the reconstruction problem. Then we give some variants. We also establish that (one of the variant of) the algorithm is quadratic.