A note on self-complementary 4-uniform hypergraphs

We prove that a permutation \(\theta\) is complementing permutation for a \(4\)-uniform hypergraph if and only if one of the following cases is satisfied: (i) the length of every cycle of \(\theta\) is a multiple of \(8\), (ii) \(\theta\) has \(1\), \(2\) or \(3\) fixed points, and all other cycles have length a multiple of \(8\), (iii) \(\theta\) has \(1\) cycle of length \(2\), and all other cycles have length a multiple of \(8\), (iv) \(\theta\) has \(1\) fixed point, \(1\) cycle of length \(2\), and all other cycles have length a multiple of \(8\), (v) \(\theta\) has \(1\) cycle of length \(3\), and all other cycles have length a multiple of \(8\). Moreover, we present algorithms for generating every possible \(3\) and \(4\)-uniform self-complementary hypergraphs