GDD\(n_{1}+ n_{2}, 3; \lambda_{1}, \lambda_{2}\) with Equal Number of Blocks of two Configurations

A \(\mathrm{GDD}(n_1+n_2,3; \lambda_1,\lambda_2)\) is a group divisible design with two groups of sizes \(n_1\) and \(n_2\), where \(n_1 < n_2\), with block size \(3\) such that each pair of distinct elements from the same group occurs in \(\lambda_1\) blocks and each pair of elements from different groups occurs in \(\lambda_2\) blocks. We prove that the necessary conditions are sufficient for the existence of group divisible designs \(\mathrm{GDD}(n_1+n_2, 3; \lambda_1, \lambda_2)\) with equal number of blocks of configuration \((1, 2)\) and \((0, 3)\) for \(n_1 + n_2 \leq 20\), \(n_1 \neq 2\) and in general for \(n_1 = 1, 3, 4, n_2 – 1\), and \(n_2 – 2\).