New non-GRS type MDS codes and NMDS codes

In this paper, we study a class of special linear codes involving their parameters, weight distributions, and self-orthogonal properties. On one hand, we prove that such codes must be maximum distance separable (MDS) or near MDS (NMDS) codes and completely determine their weight distributions with the help of the solutions to some subset sum problems. Based on the well-known Schur method, we also show that such codes are non-equivalent to generalized Reed-Solomon codes. On the other hand, a sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two classes of almost self-dual codes.