(6,3)-MDS Codes over an Alphabet of Size 4

An (n,k)_q-MDS code C over an alphabet (of size q) is a collection of q^k n–tuples over 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 must also be equivalent to linear.