Further results on LCD generalized Gabidulin codes

<abstract><p>Linear complementary dual (abbreviated LCD) generalized Gabidulin codes (including Gabidulin codes) have been recently investigated by Shi and Liu et al. (Shi et al. IEICE Trans. Fundamentals E101-A(9):1599-1602, 2018, Liu et al. Journal of Applied Mathematics and Computing 61(1): 281-295, 2019). They have constructed LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ by using self-dual bases of $ \mathbb{F}_{q^{n}} $ over $ \mathbb{F}_{q} $ when $ q $ is even or both $ q $ and $ n $ are odd. Whereas for the case of odd $ q $ and even $ n $, whether LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ exist or not is still open. In this paper, it is shown that one can always construct LCD generalized Gabidulin codes of length $ n $ over $ \mathbb{F}_{q^{n}} $ for the case of odd $ q $ and even $ n $.</p></abstract>