Constructing transpose-orthogonal Latin squares (Q1118599)

In [Proc. Louisiana Conf. Combinat. Graph Theory Computing, Louisiana State Univ., Baton Rouge, Congr. Numerantium 1, 213-226 (1970; Zbl 0235.05006)] \textit{R. C. Mullin} and \textit{E. Nemeth} gave a construction involving starters and skew adders for Latin squares that are self- orthogonal, i.e. orthogonal to their transpose. In the present paper it is shown that their construction does not always produce self-orthogonal Latin squares. In this paper the authors give a condition on the cycles for a given starter and skew adder that is both necessary and sufficient for the construction to give self-orthogonal Latin squares. The details are too complicated to state here.