Summary: For an integer \(n\) and a prime \(p\), let \(n_{(p)} = \max \{i:p^i\) divides \(n \}\). In this paper, we present a construction for vertex-transitive self-complementary \(k\)-uniform hypergraphs of order \(n\) for each integer \(n\) such that \(p^{n_{(p)}}\equiv 1 (\text{mod }2^{\ell +1})\) for every prime \(p\), where \(\ell =\max\{k_{(2)},(k - 1)_{(2)}\}\), and consequently we prove that the necessary conditions on the order of vertex-transitive self-complementary uniform hypergraphs of rank \(k=2^\ell \) or \(k=^2\ell +1\) due to Potoňick and Šajna are sufficient. In addition, we use Burnside’s characterization of transitive groups of prime degree to characterize the structure of vertex-transitive self-complementary \(k\)-hypergraphs which have prime order \(p\) in the case where \(k=2^\ell \) or \(k=2^\ell +1\) and \(p\equiv 1 (\text{mod }2^{\ell +1})\), and we present an algorithm to generate all of these structures. We obtain a bound on the number of distinct vertex-transitive self-complementary graphs of prime order \(p\equiv 1\)(mod 4), up to isomorphism.