Existence of IPBDs of Block Size Four

In this paper we prove that a \((v, u; \{4\}, 3)\)-IPBD exists when \(v, u \equiv 2\) or \(3 \pmod{4}\) and \(v \geq 3u + 1\), and then solve the problem of the existence of \((v, u; \{4\}, \lambda)\)-IPBD completely, which generalizes the result of \([7]\).