(6,3)-MDS codes over an alphabet of size 4

AAn (n, k)(q)-MDS code C over an alphabet A (of size q) is a collection of q(k) n-tuples over A such that no two words of C agree in as many as k coordinate positions. It follows that n <= q + k - 1. By elementary combinatorial means we show that every (6, 3)(4)-MDS code, linear or not, turns out to be a linear (6, 3)(4)-MDS code or else a code equivalent to a linear code with these parameters. It follows that every (5, 3)(4)-MDS code over A must also be equivalent to linear.