In this article, we continue to study the existence of large sets of partitioned incomplete Latin squares (LSPILS). We complete the determination of the spectrum of an LSPILS ( g n ) and prove that there exists an LSPILS ( g n ) if and only if g ≥ 1 , n ≥ 3, and ( g , n ) ≠ ( 1 , 6 ). We also start the investigation of LSPILS with two group sizes and prove that there exists an LSPILS + ( g n MathClass-open( 2 g MathClass-close) 1 ) for all g ≥ 1 and n ≥ 3 except possibly for n ≡ 2 , 100.3em ( mod0.3em 12 ) and n ≥ 14. Furthermore, we obtain a pair of orthogonal LSPILS + ( 1 p 2 1 )s, where p is a prime and p ≡ 10.3em ( mod0.3em 6 ).

In this article, we focus on the existence of orthogonal large sets of partitioned incomplete Latin squares (OLSPILS) of type u 1 p 1 . We prove that there exists an