Let \(v, k,\lambda\) and \(n\) be positive integers. \((x_1, x_2, \ldots, x_k)\) is defined to be \(\{(x_i, x_j) : i \neq j, i,j =1,2,\ldots,k\},\) in which the ordered pair \((x_i, x_j)\) is called \((j-i)\)-apart for \(i > j\) and \((k+j-i)\)-apart for \(i > j\), and is called a cyclically ordered \(k\)-subset of \(\{x_1, x_2, \ldots, x_k\}\).
A perfect Mendelsohn design, denoted by \((v, k, \lambda)\)-PMD, is a pair \((X, B)\), where \(X\) is a \(v\)-set (of points), and \(B\) is a collection of cyclically ordered \(k\)-subsets of \(X\) (called blocks), such that every ordered pair of points of \(X\) appears \(t\)-apart in exactly \(\lambda\) blocks of \(B\) for any \(t\), where \(1 \leq t \leq k-1\).
If the blocks of a \((v, k, \lambda)\)-PMD for which \(v \equiv 0 \pmod{k}\) can be partitioned into \(\lambda(v-1)\) sets each containing \(v/k\) blocks which are pairwise disjoint, the \((v, k, \lambda)\)-PMD is called resolvable, denoted by \((v, k, \lambda)\)-RPMD.
In the paper [14], we have showed that a \((v, 4, 1)\)-RPMD exists for all \(v \equiv 0 \pmod{4}\) except for \(4, 8\) and with at most \(49\) possible exceptions of which the largest is \(336\).
In this article, we shall show that a \((v, 4, 1)\)-RPMD for all \(v \equiv 0 \pmod{4}\) except for \(4, 8, 12\) and with at most \(27\) possible exceptions of which the largest is \(188\).
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