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Can one find three $10\times 10$ mutually orthogonal Latin squares?

Does anyone know if there is a mathematical "trick" in finding mutually orthogonal Latin squares? Or is it basically trial and error?

hardmath
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mmh0015
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  • Are you asking about orthogonal latin squares of order 10? – hardmath Jan 24 '14 at 03:52
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    Yes I'm sorry, Latin squares of order 10. – mmh0015 Jan 24 '14 at 04:00
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    @user121087 I've edited the question to reflect this. – Jim Belk Jan 24 '14 at 05:59
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    If $n$ is a product of prime powers $>k$, then there is a general construction for $k$ MOLS of size $n\times n$ by something that could be called Kronecker product (or tensor product) of MOLS of those prime power orders (that in turn can be constructed using arithmetic of finite fields). This is the reason why sizes $2\ell$, $\ell$ odd, are the troubling ones - this general construction is not available. – Jyrki Lahtonen Jan 24 '14 at 06:49

3 Answers3

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Although pairs of MOLS of order 10 are known ("Euler spoilers"), the existence of three mutually orthogonal latin squares of order 10 is an open problem.

You may find some details of computer-based searches for these in Erin Delise's M.Sc. thesis (2005). Note the first two bibliographic entries, for Bose, Chakravarti, and Knuth (1960-61).

There are "tricks" to reducing the general search space, but it remains of a formidable size despite its numerous symmetries. The first row of all three such matrices may be assumed to be $(1,2,3,\ldots,10)$, and the number of individual latin squares of order 10 with this fixed first row is 2750892211809148994633229926400. We may further restrict the first column of the first (of the three) latin squares to be $(1,2,3,\ldots,10)^T$, saving a factor of $9!$ in size of search space.

Once an intial pair of MOLS are fixed, it becomes tractable to make an exhaustive computer search for the third MOLS of order 10. Many attempts along this line have failed, although an "almost orthogonal" triple was reported by Franklin (1983); see Mohan, Lee, and Pokhrel (2006) for some references to the literature.

A set of 9 mutually orthogonal latin squares of order 10 would amount to the existence of a finite projective plane of order 10, but Lam (1991) reported the results of an extensive computer search proved this impossible.

hardmath
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Can one find three 10×10 mutually orthogonal Latin squares?

Practically, no. Not at the moment. Theoretically, we haven't excluded the possibility that they exist. I'd guess they do exist, but I wouldn't be surprised if they don't.

To my knowledge, the most up-to-date computer search is by McKay, Meynert, and Myrvold, 2006 (link):

We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups.

This is a problem Latin-square researchers discuss from time to time, but it's generally regarded as a very hard problem. One of those research problems it's best to avoid if you want to remain a functional mathematician.

Rebecca J. Stones
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http://en.wikipedia.org/wiki/Orthogonal_matrix#Elementary_constructions

There's a section on Elementary Constructions in everyone's favorite reference. For example, the identity matrix is orthogonal, and by exchanging two rows you still have an orthogonal matrix.

Jeff Snider
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