Non-GRS type MDS and AMDS codes from extended TGRS codes
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.