New constructions of MDS and NMDS self-dual codes

Recently, a 2 × 2 matrix variant of the Roth-Lempel code was introduced in (Des. Codes Cryptogr. 2025). In this paper, we present some explicit constructions of MDS (NMDS) codes by the generalized code. Then we characterize a sufficient and necessary condition for the generalized code to be self-dual and give some new explicit constructions of MDS (NMDS) self-dual codes. Furthermore, we prove that our constructions are non-GRS type.