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Orthogonal trades in complete sets of MOLS. (English) Zbl 1369.05026

Summary: Let \(B_p\) be the Latin square given by the addition table for the integers modulo an odd prime \(p\). Here we consider the properties of Latin trades in \(B_p\) which preserve orthogonality with one of the \(p-1\) MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in \(p\) for the number of times each symbol occurs in such a trade, with an overall lower bound of \((\log{p})^2/\log\log{p}\) for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orthomorphism by a small amount. We also show that any transversal in \(B_p\) hits the main diagonal either \(p\) or at most \(p-\log_2{p}-1\) times. Finally, if \(p\equiv 1\pmod{6}\) we show the existence of a Latin square which is orthogonal to \(B_p\) and which contains a \(2\times 2\) subsquare.

MSC:

05B15 Orthogonal arrays, Latin squares, Room squares
05D15 Transversal (matching) theory
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References:

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