Small Graphs and Hypergraphs of Given Degree and Girth

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