Generalized Latin squares II Original Research Article

This paper is a continuation of a study on a new class of combinatorial structures called generalized Latin squares or 〈k,l〉-Latin squares. Here, we study the existence question of perfect 〈k,l〉-Latin squares. The main result is an existence theorem for the case M = 2, where M is the multiplicity of the generalized Latin squares. Specifically, we show that for given positive integers N, k = ip, l = iq, where i = gcd(k,l), there exists a perfect 〈k,l〉-Latin square of order N with M = 2 if and only if (i) N = 4k, (ii) pq(p + q)| N, and (iii) q|2.