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

Abstract

Two Latin squares L=[l(i,j)] and M=[m(i,j)], of even order n with entries {0,1,2,...,n-1}, are said to be nearly orthogonal if the superimposition of L on M yields an nxn array A=[(l(i,j),m(i,j))] in which each ordered pair (x,y), 0x,yn-1 and xy, occurs at least once and the ordered pair (x,x+n/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.