In this paper we present six classes of topological quantum codes (TQC) on compactsurfaces with genus g ≥ 2. These codes are derived from self-dual, quasi self-dual anddenser tessellations associated with embeddings of self-dual complete graphs and com-plete bipartite graphs on the corresponding compact surfaces. The majority of the newclasses has the self-dual tessellations as their algebraic and geometric supporting math-ematical structures. Every code achieves minimum distance 3 and its encoding rate issuch that k/n → 1 as n → ∞, except for the one case where k/n → 1/3 as n → ∞.