Some Constructions of Sets of Mutually kn-Orthogonal Latin Squares of Order n

Abstract

Generalizing a Bose-Chakravarti-Knuth’s method of constructing sets of mutually orthogonal latin squares [3], we give a method of constructing sets of mutually kn-orthogonal latin squares of order n with 1 <k≤n. With the same method, suitably adjusted, we can construct sets of mutually kn-orthogonal n×m latin rectangles (with n≥ m and 1 <k ≤ m), and we generalize a method of constructing sets of mutually orthogonal latin rectangles given in [9]. This method yields some constructions of sets of mutually kn-orthogonal latin squares and we present some partial solutions to the research problem proposed by J. Denes in [6]. We also give a simple construction of sets of n– 1 mutually orthogonal n’ × n latin rectangles with n prime number and n’ an integer such that n’≥n and n’∈{m∈N | m ≥2n–3}⋃{2n–5, 2n–7}.