Embedding cyclic latin squares of order 2n in a complete set of orthogonal F-squares

Abstract A cyclic Latin square of order 2n, which has no orthogonal Latin square mate, is shown to have (2n−1)(2n−2) mutually orthogonal F(2n;2n−1,2n−1)-squares. This is a complete set of F-squares for the cyclic Latin square. Row and column operations areused to construct this complete set of F-squares from a Hadamard matrix and 2n−1 OF(2n;2n−1,2n−1)-squares into which the Latin square is decomposed. Tables of complete sets of mutually orthogonal F(2n;2n−1,2n−1)-squares are given for n=2 and 3, i.e., for cyclic Latin squares of orders 4 and 8.