Latin hypercubes and MDS codes

Maximum distance separable (MDS) codes have special properties that give them excellent error correcting capabilities. Counting the number of q-ary MDS codes of length n and distance d, denoted by Dq(n,d)MDSDq(n,d)MDS, is a very hard problem. This paper shows that for d=2d=2, it amounts to counting the number of (n-1)(n-1)-dimensional Latin hypercubes of order q. Thus, Dq(3,2)MDSDq(3,2)MDS is the number of Latin squares of order q, which is known only for a few values of q. This paper proves constructively that D3(n,2)MDS=6·2n-2D3(n,2)MDS=6·2n-2.