On Constructions of GMGDs

Abstract

Modified group divisible designs MGD ( k , λ , m , n ) are extensively studied because of an intriguing combinatorial structure that they possess and their applications. In this paper, we present a generalization of MGDs called GMGD ( k , λ 1 , λ 2 , m , n ) , and we provide some elementary results and constructions of some special cases of GMGDs. In addition, we show that the necessary conditions are sufficient for the existence of a GMGD ( 3 , λ , 2 λ , m , n ) for any positive integer λ , and a GMGD ( 3 , 2 , 3 , m , n ) . Though not a general result, the construction of a GMGD ( 3 , 3 , 2 , 2 , 6 ) given in the paper is worth mentioning in the abstract. Along with another example of a GMGD ( 3 , 3 , 2 , 2 , 4 ) , and n to t n construction, we have families of GMGD ( 3 , 3 λ , 2 λ , 2 , n ) s for n = 4 t or 6 t when t ≡ 0 , 1 ( mod 3 ) , for any positive integer λ .