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,. ., n - 1}, are said to be nearly orthogonal if the superimposition of L on M yields an n × n array A = [(l(i, j ),m(i, j ))] in which each ordered pair (x, y), 0 ≤ x, y ≤ n - 1 and x ≠ y, 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 48κ + 14, 48κ + 22, 48κ + 38, and 48κ + 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