Existence of self-conjugate self-orthogonal diagonal latin squares

In this article we give some new constructions of self-conjugate self-orthogonal diagonal Latin squares (SCSODLS). As an application of such constructions, we give a conclusive result regarding the existence of SCSODLS and show that there exists an SCSODLS of order n if and only if n ≡ 0, 1 (mod 4), except for n = 5. This result completely disproves a conjecture of Danhof, Phillips, and Wallis about SCSODLS in Danhof, Philips, and Wallis, JCMCC, 8 (1990), 3–8. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:51–62, 1998