Direct constructions for general families of cyclic mutually nearly orthogonal Latin squares

Two Latin squares $L=[l(i,j)]$ and $M=[m(i,j)]$, of even order $n$ with entries $\{0,1,2,\dots ,n-1\}$, are said to be nearly orthogonal if the superimposition of $L$ on $M$ yields an $n\times n$ array $A=[(l(i,j),m(i,j))]$ in which each ordered pair $(x,y)$, $0\le x,y\le n-1$ and $x\mathrm{\ne }y$, occurs at least once and the ordered pair $(x,x+n\mathrm{/}2)$ occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders $48k+14$, $48k+22$, $48k+38$ and $48k+46$. The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi-difference” sets for these orders.