Orthogonal trades in complete sets of MOLS

TL;DR

This paper investigates properties of Latin trades in the Latin square $B_p$, focusing on orthogonality preservation, bounds on symbol occurrences, and implications for orthomorphisms and transversals, revealing new structural insights.

Contribution

It introduces bounds on Latin trades preserving orthogonality in $B_p$, and demonstrates the existence of specific Latin squares with orthogonal properties and small subsquares.

Findings

Lower bounds on symbol occurrences in trades are logarithmic in $p$.

Existence of orthomorphisms differing from linear ones by a small amount.

Any transversal hits the main diagonal either $p$ or at most $p - ext{log}_2 p - 1$ times.

Abstract

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 $B_p$.