  The search for the smallest possible $d$-regular graph of girth $g$ has a
long history, and is usually known as the cage problem. This problem has a
natural extension to hypergraphs, where we may ask for the smallest number of
vertices in a $d$-regular, $r$-uniform hypergraph of given (Berge) girth $g$.
We show that these two problems are in fact very closely linked. By extending
the ideas of Cayley graphs to the hypergraph context, we find smallest known
hypergraphs for various parameter sets. Because of the close link to the cage
problem from graph theory, we are able to use these techniques to find new
record smallest cubic graphs of girths 23, 24, 28, 29, 30, 31 and 32.
