Mutually Orthogonal Latin Squares and Self-complementary Designs

Suppose that n is even and a set of n/2 -1 mutually orthogonal Latin squares of order n exists. Then we can construct a strongly regular graph with parameters (n², n/2 (n-1), n/2 ( n/2-1), n/2 ( n/2 -1)), which is called a Latin square graph. In this paper, we give a sufficient condition of the Latin square graph for the existence of a projective plane of order n. For the existence of a Latin square graph under the condition, we will introduce and consider a self-complementary 2-design (allowing repeated blocks) with parameters (n, n/2 , n/2 ( n/2 -1)). For n ≡ 2 (mod 4), we give a proof of the non-existence of the design.