Further results on LCD generalized Gabidulin codes

Xubo Zhao; Xiaoping Li; Tongjiang Yan; Yuhua Sun; Xubo Zhao; Xiaoping Li; Tongjiang Yan; Yuhua Sun

doi:10.3934/math.2021812

AIMS Mathematics

2021, Volume 6, Issue 12: 14044-14053. doi: 10.3934/math.2021812

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Further results on LCD generalized Gabidulin codes

College of Sciences, China University of Petroleum, Qingdao 266580, China

Received: 14 September 2021 Accepted: 18 October 2021 Published: 19 October 2021
MSC : 94B05

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 $.
- generalized Gabidulin codes,
- LCD codes,
- MRD codes,
- rank metric
Citation: Xubo Zhao, Xiaoping Li, Tongjiang Yan, Yuhua Sun. Further results on LCD generalized Gabidulin codes[J]. AIMS Mathematics, 2021, 6(12): 14044-14053. doi: 10.3934/math.2021812

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Abstract

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 $.

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