McKay, Meynert, Myrvold (2006) (Small latin squares, quasigroups, and loops, DOI:10.1002/jcd.20105, author copy) computationally eliminate the possibility of set of 3 mutually orthogonal Latin squares of order 10, i.e., a 3-MOLS(10), where one of the Latin squares has a non-trivial autoparatopism group (i.e., it has some symmetry).

I'm unsure of what the current status for 3-MOLS(10) in which one is self-orthogonal (i.e., orthogonal to its transpose), and another is its transpose. It seems like this could be within the realm of computational feasibility.

Question: Has the existence of a 3-MOLS(10) containing a self-orthogonal Latin square and its transpose been eliminated?

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    $\begingroup$ I added a link to a free copy of the paper, and the title :-) $\endgroup$ Aug 24 '18 at 6:02

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