Completing the spectrum of \(r\)-orthogonal Latin squares (Q1398282)

Two Latin squares of order \(n\) are \(r\)-orthogonal if placing one upon the other yields precisely \(r\) distinct ordered pairs. \textit{G. B. Belyavskaya} [Ann. Discrete Math. 46, 169-202 (1991; Zbl 0754.05016)], \textit{C. J. Colbourn} and \textit{L. Zhu} [Math. Appl., Dordr. 329, 49-75 (1995; Zbl 0836.05012)] and the authors [Discrete Math. 238, 183-191 (2001; Zbl 0987.05025)] have previously shown the existence of pairs of \(r\)-orthogonal Latin squares for all \(n\geq 7\) if and only if \(n\leq r\leq n^2\) and \(r\neq n+1\) or \(n^2-1\), with the possible exception of \(n= 14\) and \(r= 14^2- 3\). In this paper, the authors provide a pair of \((14^2- 3)\)-orthogonal Latin squares of order \(14\). Also, since the values of \(r\) are known for which there is no pair of \(r\)-orthogonal Latin squares of orders \(2\) through \(6\), the spectrum is completely determined. Based on experimental work on small orders, the authors also conjecture the existence of a pair of \(r\)-self-orthogonal Latin squares, with \(r\) satisfying the same conditions as above, for all \(n\geq n_0\), with \(n_0\) yet to be determined.