Indecomposable Triple Systems with \(\lambda= 5\)

An \(\text{NB}[k, \lambda; v]\) is a \(\text{B}[b, \lambda; v]\) which has no repeated blocks. In this paper we prove that there exists an indecomposable \(\text{NB}[3,5; v]\) for \(v \geq 7\) and \(v \equiv 1 \text{ or } 3 \pmod{6}\), with the exception of \(v = 7\) and \(9\), and the possible exception of \(v = 13, 15\).