  A $3$-uniform hypergraph is a generalization of simple graphs where each
hyperedge is a subset of vertices of size $3$. The degree of a vertex in a
hypergraph is the number of hyperedges incident with it. The degree sequence of
a hypergraph is the sequence of the degrees of its vertices. The degree
sequence problem for $3$-uniform hypergraphs is to decide if a $3$-uniform
hypergraph exists with a prescribed degree sequence. Such a hypergraph is
called a realization. Recently, Deza \emph{et al.} proved that the degree
sequence problem for $3$-uniform hypergraphs is NP-complete. Some special cases
are easy; however, polynomial algorithms have been known so far only for some
very restricted degree sequences. The main result of our research is the
following. If all degrees are between $\frac{2n^2}{63}+O(n)$ and
$\frac{5n^2}{63}-O(n)$ in a degree sequence $D$, further, the number of
vertices is at least $45$, and the degree sum can be divided by $3$, then $D$
has a $3$-uniform hypergraph realization. Our proof is constructive and in
fact, it constructs a hypergraph realization in polynomial time for any degree
sequence satisfying the properties mentioned above. To our knowledge, this is
the first polynomial running time algorithm to construct a $3$-uniform
hypergraph realization of a highly irregular and dense degree sequence.
