On deep holes of Gabidulin codes

In this paper, we study deep holes of Gabidulin codes in both rank and Hamming metrics. Specifically, first, we give a tight lower bound for the distance of any word to a Gabidulin code and a sufficient and necessary condition for achieving this lower bound as well. Then, a class of deep holes of a Gabidulin code are discovered. Furthermore, we obtain some other deep holes for certain Gabidulin codes.