An [n, k, d] linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) ifd = n−k+1 ord=n−k, respectively. If a code and its dual code are both AMDS, then the code is said to be near maximum distance separable (NMDS). Fork= 3 andk= 4, there are many constructions of NMDS codes by adding some suitable projective points to arcs in PG(k− 1,q). In this paper, we consider the monomial equivalence problem for some NMDS codes with the same weight distributions and present new constructions of NMDS codes.