🤖 AI Summary
This work proposes a new class of extended twisted generalized Reed–Solomon (TGRS) codes to broaden the design space of optimal and near-optimal error-correcting codes. It systematically investigates the necessary and sufficient conditions under which these codes attain maximum distance separable (MDS) or almost MDS (AMDS) properties. Leveraging algebraic coding theory and equivalence analysis, the study rigorously establishes—for the first time—that the constructed codes are non-equivalent to classical Reed–Solomon (GRS) codes for specific parameter choices. Furthermore, it precisely determines their covering radii and deep holes. By providing explicit constructions of novel non-GRS MDS and AMDS codes along with clear parameter criteria, this research significantly enriches the theoretical foundations and practical resources available for high-performance error correction.
📝 Abstract
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.