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, self-orthogonal properties, deep holes, and the existence of error-correcting pairs. We prove that such codes must be maximum distance separable (MDS) codes or near MDS codes and completely determine their weight distributions with the help of the solutions to some subset sum problems. Based on the Schur method, we show that such codes are not equivalent to generalized Reed-Solomon (GRS) codes. A sufficient and necessary condition for such codes to be self-orthogonal is also 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. Additionally, we find a class of deep holes of such codes and determine the existence of their error-correcting pairs in most cases, which also reveal more connections between such codes and GRS codes.