  We consider $4$-uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of $5$ vertices spans either $0$ or
exactly $2$ hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.
