Orthogonal trades in complete sets of MOLS

• Main Authors: Cavenagh, Nicholas J, Donovan, Diane M, Demirkale, Fatih
• Format: Journal Article
• Language: English
• Published: 15-07-2016
• Subjects: Mathematics - Combinatorics
• DOI: 10.48550/arxiv.1607.04429

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 $B_p$.