A note on $GDD(1, n, n , 4;\lambda_{1},\lambda_{2})$

Abstract

The present note is motivated by two papers on group divisible designs (GDDs) with the same block size three but different number of groups: three and four where one group is of size $1$ and the others are of the same size $n$. Here we present some interesting constructions of GDDs with block size 4 and three groups: one of size $1$ and other two of the same size $n$. We also obtain necessary conditions for the existence of such GDDs and prove that they are sufficient in several cases. For example, we show that the necessary conditions are sufficient for the existence of a GDD$(1,n,n,4;\lambda_1,\lambda_2)$ for $n\equiv0,1,4,5,8,9\pmod{12}$ when $\lambda_1\ge \lambda_2$.