Families of codes of topological quantum codes from tessellations tessellations {4i+2,2i+1}, {4i,4i}, {8i-4,4} and ${12i-6,3}

In this paper we present some classes of topological quantum codes on surfaces with genus $g \geq 2$ derived from hyperbolic tessellations with a specific property. We find classes of codes with distance $d = 3$ and encoding rates asymptotically going to 1, $\frac{1}{2}$ and $\frac{1}{3}$, depending on the considered tessellation. Furthermore, these codes are associated with embedding of complete bipartite graphs. We also analyze the parameters of these codes, mainly its distance, in addition to show a class of codes with distance 4. We also present a class of codes achieving the quantum Singleton bound, possibly the only one existing under this construction.