Abstract
We present some new conjugate orthogonal Latin squares which are obtained from a direct method of construction of the starter-adder type. Combining these new constructions with earlier results of K. T. Phelps and the first author, it is shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal Latin square of order v exists for all positive integers v ≠ 2, 6. It is also shown that a (3, 2, 1)- (or (1, 3, 2)-) conjugate orthogonal idempotent Latin square of order v exists for all positive integers v ≠ 2, 3, 6 with one possible exception v = 12, and this result can be used to enlarge the spectrum of a certain class of Mendelsohn designs and provide better results for problems on embedding.
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