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 hypergraph.