\(\lambda\)-fold \(K_{2,5}\)-Designs

In this note, we present necessary conditions for decomposing \(\lambda K_n\) into copies of \(K_{2,5}\), and show that these conditions are sufficient except for \(\lambda = 5\) and \(n = 8\), and possibly for the following cases: \(\lambda = 1\) and \(n = 40\); and \(\lambda = 3\) and \(n = 16\) or \(20\).