On maximum distance separable and completely regular codes

We investigate when a maximum distance separable (MDS) code over Fq is also completely regular (CR). For lengths n = q + 1 and n = q + 2 we provide a complete classification of the MDS codes that are CR or at least uniformly packed in the wide sense (UPWS). For the more restricted case n <= q with q <= 5 we obtain a full classification (up to equivalence) of all nontrivial MDS codes: there are none for q = 2; only the ternary Hamming code for q = 3; four nontrivial families for q = 4; and