  In this paper, we address the following problem due to Frankl and F\"uredi
(1984). What is the maximum number of hyperedges in an $r$-uniform hypergraph
with $n$ vertices, such that every set of $r+1$ vertices contains $0$ or
exactly $2$ hyperedges? They solved this problem for $r=3$. For $r=4$, a
partial solution is given by Gunderson and Semeraro (2017) when $n=q+1$ for
some prime power number $q\equiv3\pmod{4} $. Assuming the existence of
skew-symmetric conference matrices for every order divisible by $4$, we give a
solution for $n\equiv0\pmod{4} $ and for $n\equiv3\pmod{4}$.
