Indecomposable Triple Systems with \(\lambda = 6\)

A triple system \(B[3, \lambda; v]\) is indecomposable if it is not the union of two triple systems \(B[3, \lambda_1; v]\) and \(B[3, \lambda_2; v]\) with \(\lambda = \lambda_1 + \lambda_2\). We prove that indecomposable triple systems with \(\lambda = 6\) exist for \(v = 8, 14\) and for all \(v \geq 17\).