In recent years, the construction of non-GRS type linear codes has attracted considerable attention due to that they can effectively resist both the Sidelnikov-Shestakov attack and the Wieschebrink attack. Constructing linear complementary dual (LCD) codes and determining the hull of linear codes have long been important topics in coding theory, as they play the crucial role in constructing entanglement-assisted quantum error-correcting codes (EAQECCs), certain communication systems and cryptography. In this paper, by utilizing a class of non-GRS type linear codes, namely, generalized Roth-Lempel (in short, GRL) codes, we firstly construct several classes of Euclidean LCD codes, Hermitian LCD codes, and linear codes with small-dimensional hulls, generalized the main results given by Wu et al. in 2021. We also present an upper bound for the number of a class of Euclidean GRL codes with 1-dimensional hull, and then for several classes of Hermitian GRL codes, we firstly derive an upper bound for the dimension of the hull, and prove that the bound is attainable. Secondly, as an application, we obtain several families of EAQECCs. Thirdly, we prove that the GRL code is non-GRS for $k >\ell$. Finally, some corresponding examples for LCD MDS codes and LCD NMDS codes are presented.

翻译：近年来，非GRS型线性码的构造因其能有效抵抗Sidelnikov-Shestakov攻击与Wieschebrink攻击而受到广泛关注。构造线性互补对偶码以及确定线性码的核一直是编码理论中的重要课题，它们在构建纠缠辅助量子纠错码、特定通信系统及密码学中起着关键作用。本文利用一类非GRS型线性码——广义Roth-Lempel码，首先构造了若干类欧几里得LCD码、厄米特LCD码以及具有小维数核的线性码，推广了Wu等人于2021年给出的主要结果。我们还给出了一类具有1维核的欧几里得GRL码数量的上界，并针对若干类厄米特GRL码，首次推导了其核维数的上界，并证明了该界可达。其次，作为应用，我们得到了多个EAQECC族。再次，我们证明了当$k >\ell$时GRL码为非GRS型。最后，给出了若干LCD MDS码与LCD NMDS码的对应实例。