Direct Constructions for General Families of Cyclic Mutually Nearly Orthogonal Latin Squares. Issue 5 (29th April 2014) - Demirkale, Fatih; Donovan, Diane; Khodkar, Abdollah
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 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 48 k + 14, 48 k + 22, 48 k + 38, and 48 k + 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.