Some Indecomposable t-Designs

The existence of large sets of 5-(14,6,3) designs is in doubt. There are five simple 5-(14,6,6) designs known in the literature. In this note, by the use of a computer program, we show that all of these designs are indecomposable and therefore they do not lead to large sets of 5-(14,6,3) designs. Moreover, they provide the first counterexamples for a conjecture on disjoint t-designs which states that if there exists a t-(v, k, λ) design (X, D) with minimum possible value of λ, then there must be a t-(v, k, λ) design (X, D′) such that D∩ D′ = Ø.