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.
@article{10_37236_6338,
author = {Nicholas Cavenagh and Diane Donovan and Fatih Demirkale},
title = {Orthogonal trades in complete sets of {MOLS}},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {3},
doi = {10.37236/6338},
zbl = {1369.05026},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6338/}
}
TY - JOUR
AU - Nicholas Cavenagh
AU - Diane Donovan
AU - Fatih Demirkale
TI - Orthogonal trades in complete sets of MOLS
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/6338/
DO - 10.37236/6338
ID - 10_37236_6338
ER -
%0 Journal Article
%A Nicholas Cavenagh
%A Diane Donovan
%A Fatih Demirkale
%T Orthogonal trades in complete sets of MOLS
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/6338/
%R 10.37236/6338
%F 10_37236_6338
Nicholas Cavenagh; Diane Donovan; Fatih Demirkale. Orthogonal trades in complete sets of MOLS. The electronic journal of combinatorics, Tome 24 (2017) no. 3. doi: 10.37236/6338
