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In 2000, Rees and Shalaby constructed simple indecomposable two-fold cyclic triple systems for all \(v \equiv 0, 1, 3, 4, 7, \text{ and } 9 \pmod{12}\) where \(v = 4\) or \(v \geq 12\), using Skolem-type sequences.
We construct, using Skolem-type sequences, three-fold triple systems having the properties of being cyclic, simple, and indecomposable for all admissible orders \(v\), with some possible exceptions for \(v = 9\) and \(v = 24c + 57\), where \(c \geq 2\) is a constant.