Embedding cyclic Latin squares of order \(2^ n\) in a complete set of orthogonal F-squares (Q2266556)

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