A family of linear codes that are either non-GRS MDS codes or NMDS codes

Both maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes (non-GRS MDS codes) and near MDS (NMDS) codes have nice applications in communication and storage systems. In this paper, we introduce and study a new family of linear codes involving their parameters, weight distributions, and self-orthogonal properties. We prove that such codes are either non-GRS MDS codes or NMDS codes, and hence, they can produce as many of the desired codes as possible. We also completely determine their weight distributions with the help of the solutions to some subset sum problems. 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 new classes of almost self-dual codes.