Galois Hull Dimensions of Gabidulin Codes

Abstract

For a prime power q, an integer m and 0 ≤ e ≤ m − 1 we study the e-Galois hull dimension of Gabidulin codes G k (α) of length m and dimension k over F q m . Using a self-dual basis α of F q m over F q , we first explicitly compute the hull dimension of G k (α). Then a necessary and sufficient condition of G k (α) to be linear complementary dual (LCD), self-orthogonal and selfdual will be provided. We prove the existence of e-Galois (where e = m2 ) 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 entangled-assisted quantum error-correcting codes (EAQECCs) whose parameters have more flexibility compared to known codes in this context.