A (k,g)-graph is a k-regular graph with girth g and a (k,g)-cage is a (k,g)-graph with the fewest possible number of vertices. The cage problem consists of constructing (k,g)-graphs of minimum order n(k,g). We focus on girth g=5, where cages are known only for degrees k=7. We construct (k,5)-graphs using techniques exposed by Funk (2009) and Abreu et al. (2012) to obtain the best upper bounds on n(k,5) known hitherto. The tables given in the introduction show the improvements obtained with our...