On the existence of latin squares with special distribution properties

An n × n latin square ( a r , c | 0 ≤ r , c ≤ n − 1 ) is an ( s , t ) latin square if every subrectangle R i , j , 0 ≤ i , j ≤ n − 1 , consisting of cells { a i + k , j + ℓ | 0 ≤ k < t , 0 ≤ ℓ < s } , where the addition of indices is performed modulo n , contains s t different elements. We show that an ( s , t ) latin square exists if and only if n ≥ s t + t or n > s t and the product of the greatest common divisors g c d ( s , n ) g c d ( t , n ) is a divisor of n .