Orthogonal trades in complete sets of MOLS

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Cavenagh, Nicholas J, Donovan, Diane M, Demirkale, Fatih
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.07.2016
Subjects
Fields (mathematics)
Integers
Lower bounds
Orthogonality
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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 \(p\equiv 1\mod{6}\) we show the existence of Latin square containing a \(2\times 2\) subsquare which is orthogonal to \(B_p\).
ISSN:2331-8422