Perfect Mendelsohn designs with equal-sized holes and block size four

Let M = {m₁, m₂, …, m_h} and X be a v-set (of points). A holey perfect Mendelsohn designs (briefly (v, k, λ) - HPMD), is a triple (X, H, B), where H is a collection of subsets of X (called holes) with sizes M and which partition X, and B is a collection of cyclic k-tuples of X (called blocks) such that no block meets a hole in more than one point and every ordered pair of points not contained in a hole appears t-apart in exactly λ blocks, for 1 ≤ t ≤ k − 1. The vector (m₁, m₂, …, m_h) is called the type of the HPMD. If m₁ = m₂ = … = m_h = m, we write briefly m^h for the type. In this article, it is shown that the necessary condition for the existence of a (v, 4, λ) - HPMD of type m^h, namely, is also sufficient with the exception of types 2⁴ and 1⁸ with λ = 1, and type m⁴ for odd m with odd λ. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 203–213, 1997