Galois Hull Dimensions of Gabidulin Codes
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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 self-dual will be provided. We prove the existence of e-Galois (where e=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 entangled-assisted quantum error-correcting codes (EAQECCs) whose parameters have more flexibility compared to known codes in this context.
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