Topological quantum codes on compact surfaces with genus g≥2

In this paper we propose a construction procedure of a class of topological quantum error-correcting codes on surfaces with genus g >= 2. This generalizes the toric codes construction. We also tabulate all possible surface codes with genus 2-5. In particular, this construction reproduces the class of codes obtained when considering the embedding of complete graphs K(s), for s equivalent to 1 mod 4, on surfaces with appropriate genus. We also show a table comparing the rate of different codes when fixing the distance to 3-5.