Nearly Orthogonal Latin Squares

Abstract

A Latin square of order $n$ is an $n$ by $n$ array in which every row and column is a permutation of a set $N$ of $n$ elements. Let $L=[l_{i,j}]$ and $M=[m_{i,j}]$ be two Latin squares of even order $n$, based on the same $N$-set. Define the superposition of $L$ onto $M$ to be the $n$ by $n$ array $A=(l_{i,j},m_{i,j})$. When $n$ is even, $L$ and $M$ are said to be $nearlyorthogonal$ if the superposition of $L$ onto $M$ has every ordered pair $(i,j)$ appearing exactly once except for $i=j$, when the ordered pair appears $0$ times and except for $i–j=\frac{n}{2}(\mathrm{mod}n)$, when the ordered pair appears $2$ times. A set of $t$ Latin squares of order $2m$ is called a set of $mutuallynearlyorthogonalLatinsquares$ (MNOLS($2m$)) if the $t$ Latin squares are pairwise nearly orthogonal. We provide two elementary proofs for results that were stated and proved earlier. We also provide some computer results and prove two recursive constructions for MNOLS. Using these results we show that there always exist $3$ mutually nearly orthogonal Latin squares of order $2m$, for $2m\ge 358$.