Galois Hull Dimensions of Gabidulin Codes

For a prime power q, an integer m and 0 ≤ e ≤ m − 1 we study the e-Galois hull dimension of Gabidulin codes Gk(α) of length m and dimension k over ${\mathbb{F}_{{q^m}}}$. Using a self-dual basis α of ${\mathbb{F}_{{q^m}}}$ over ${\mathbb{F}_q}$, we first explicitly compute the hull dimension of Gk(α). Then a necessary and sufficient condition of Gk(α) to be linear complementary dual (LCD), self-orthogonal and self-dual will be provided. We prove the existence of e-Galois (where $e = \frac{m}{2}$) self-dual Gabidulin codes of length m for even q, which is in contrast to the known fact that Euclidean self-dual Gabidulin codes do not exist for even q. As an application, we construct two classes of MDS entangled-assisted quantum error-correcting codes (MDS EAQECCs) whose parameters have more flexibility compared to known codes in this context.