Non-GRS type MDS and AMDS codes from extended TGRS codes
This work addresses the need for non-GRS type MDS and AMDS codes in fields like communication and data storage, offering incremental improvements in code construction.
The paper constructs extended twisted generalized Reed-Solomon (TGRS) codes and establishes conditions for them to be maximum distance separable (MDS) or almost maximum distance separable (AMDS), proving they are not equivalent to generalized Reed-Solomon (GRS) codes, with applications including computing covering radii and deep holes.
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.