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Uniform semi-Latin squares and their Schur-optimality. (English) Zbl 1248.05022

Summary: Let \(n\) and \(k\) be integers, with \(n> 1\) and \(k> 0\). An \((n\times n)/k\) semi-Latin square \(S\) is an \(n \times n\) array, whose entries are \(k\)-subsets of an \(nk\)-set, the set of symbols of \(S\), such that each symbol of \(S\) is in exactly one entry in each row and exactly one entry in each column of \(S\). Semi-Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an \((n\times n)/k\) semi-Latin square \(S\) is uniform if there is a constant \(\mu\) such that any two entries of \(S\), not in the same row or column, intersect in exactly \(\mu\) symbols (in which case \(k=\mu(n-1)\)). We prove that a uniform \((n\times n)/k\) semi-Latin square is Schur-optimal in the class of \((n\times n)/k\) semi-Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria.
We give a simple construction to make an \((n\times n)/k\) semi-Latin square \(S\) from a transitive permutation group \(G\) of degree \(n\) and order \(nk\), and show how certain properties of \(S\) can be determined from permutation group properties of \(G\). If \(G\) is 2-transitive then \(S\) is uniform, and this provides us with Schur-optimal semi-Latin squares for many values of \(n\) and \(k\) for which optimal \((n\times n)/k\) semi-Latin squares were previously unknown for any optimality criterion. The existence of a uniform \((n\times n)/((n- 1)\mu)\) semi-Latin square for all integers \(\mu>0\) is shown to be equivalent to the existence of \(n-1\) mutually orthogonal Latin squares (MOLS) of order \(n\). Although there are not even two MOLS of order 6, we construct uniform, and hence Schur-optimal, \((6\times 6)/(5\mu)\) semi-Latin squares for all integers \(\mu> 1\).

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Software:

DESIGN; GAP
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References:

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