  The spectrum of a graph is closely related to many graph parameters. In
particular, the spectral gap of a regular graph which is the difference between
its valency and second eigenvalue, is widely seen an algebraic measure of
connectivity and plays a key role in the theory of expander graphs. In this
paper, we extend previous work done for graphs and bipartite graphs and present
a linear programming method for obtaining an upper bound on the order of a
regular uniform hypergraph with prescribed distinct eigenvalues. Furthermore,
we obtain a general upper bound on the order of a regular uniform hypergraph
whose second eigenvalue is bounded by a given value. Our results improve and
extend previous work done by Feng-Li (1996) on Alon-Boppana theorems for
regular hypergraphs and by Dinitz-Schapira-Shahaf (2020) on the Moore or
degree-diameter problem. We also determine the largest order of an $r$-regular
$u$-uniform hypergraph with second eigenvalue at most $\theta$ for several
parameters $(r,u,\theta)$. In particular, orthogonal arrays give the structure
of the largest hypergraphs with second eigenvalue at most $1$ for every
sufficiently large $r$. Moreover, we show that a generalized Moore geometry has
the largest spectral gap among all hypergraphs of that order and degree.
