On Orthogonality of Latin Squares

Abstract: A Latin square arrangement is an arrangement of s symbols in s rows and s columns, such that every symbol occurs once in each row and each column. When two Latin squares of same order superimposed on one another, then in the resultant array every ordered pair of symbols occurs exactly once, then the two Latin squares are said to be orthogonal. A frequency square M of type F (n; λ) is an n x n, matrix over an m-set S, where n=mλ, such that every element of S occurs exactly λ times in each row and each column of M. Two frequency squares of the same type over S are orthogonal if one is superimposed on the other each element of S x S appears λ times. These two concepts lead to a third concept that is if t-orthogonal Latin squares of order n, from a set S of Latin squares, are superimposed, then in the resultant array, each t-tuple occurs exactly once. If at all it is possible then how to construct them and that is the genesis of the paper. In this paper while generating Latin square (Latin rectangles), a new concept called t-orthogonality over the set of Latin structure has been discussed and their constructions have been given. Mathematics Subject Classification: 05B05