  In this paper we are interested in the {\it{Cage Problem}} that consists in
constructing regular graphs of given girth $g$ and minimum order. We focus on
girth $g=5$, where cages are known only for degrees $k \le 7$. We construct
regular graphs of girth $5$ using techniques exposed by Funk [Note di
Matematica. 29 suppl.1, (2009) 91 - 114] and Abreu et al. [Discrete Math. 312
(2012), 2832 - 2842] to obtain the best upper bounds known hitherto. The tables
given in the introduction show the improvements obtained with our results.
