  Current work presents a new approach to quantum color codes on compact
surfaces with genus $g \geq 2$ using the identification of these surfaces with
hyperbolic polygons and hyperbolic tessellations. We show that this method may
give rise to color codes with a very good parameters and we present tables with
several examples of these codes whose parameters had not been shown before. We
also present a family of codes with minimum distance $d=4$ and the encoding
rate asymptotically going to 1 while $n \rightarrow \infty$.
