Menu
The subject matter for this paper is GDDs with three groups of sizes \(n_1,n(n\geq n_1)\) and \(n+1\), for \(n_1=1\, or\, 2\) and block size four. A block having Configuration \((1,1,2)\) means that the block contains 1 point from two different groups and 2 points from the remaining group. a block having Configuration \((2,2)\) means that the block has exactly two points from two of the three groups. First, we prove that a GDD\((n_1,n,n+1,4;\lambda_1,\lambda_2)\) for \(n_1 = 1\, o\,r 2\) does not exist if we require that exactly halh of the blocks have the Configuration \((1,1,2)\) and the other half of the blocks have the configuration \((2,2)\) except possibly for n=7 when \(n_1=2\). Then we provide necessary conditions for the existence of a GDD\((n_1,n,n+1,4;\lambda_1,\lambda_2)\) for \(n_1=1\, and \,2\), and prove that these conditions are sufficient for several families of GDDs. For \(n_1=2\), these usual necessary conditions are not sufficient in general as we provide a glimpse of challenges which exist even for the case of \(n_1=2\). A general results that a GDD\((n_1,n_2,n_3,4;\lambda_1,\lambda_2)\) exists if \(n_1 + n_2 + n_3=0,4\) \((mod\, 12)\) is also given.