Almost self-dual MDS codes and NMDS codes from twisted generalized Reed–Solomon codes

Twisted generalized Reed–Solomon (TGRS) codes are generalized from generalized Reed–Solomon (GRS) codes by adding some monomials. It is well known that TGRS codes contain maximum distance separable (MDS) codes which are not equivalent to GRS codes in general. In this paper, we study a class of TGRS codes and their extended codes (ETGRS), which are generated by the polynomial set [Formula: see text] where [Formula: see text] and [Formula: see text]. Through the parity-check matrix of these codes, we first show that these codes cannot be self-dual for large [Formula: see text]. Then, we give an explicit construction of [Formula: see text] self-orthogonal MDS (NMDS) codes with [Formula: see text] and [Formula: see text] from TGRS and ETGRS, which are non-GRS codes.