Maximum distance separable (MDS) and almost maximum distance separable (AMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes because of their algebraic properties and excellent error-correcting capabilities. In this paper, we construct a class of extended twisted generalized Reed-Solomon (TGRS) codes and determine the necessary and sufficient conditions for these codes to be MDS or AMDS. Additionally, we prove that these codes are not equivalent to generalized Reed-Solomon (GRS) codes. As an application, under certain circumstances, we compute the covering radii and deep holes of these codes.

翻译：最大距离可分（MDS）码与几乎最大距离可分（AMDS）码因其代数性质和优异的纠错能力，已广泛应用于通信系统、数据存储和量子编码等领域。本文构造了一类扩展扭曲广义Reed-Solomon（TGRS）码，并给出了此类码为MDS或AMDS的充要条件。此外，我们证明了这些码与广义Reed-Solomon（GRS）码不等价。作为应用，在特定条件下，我们计算了这些码的覆盖半径与深洞。