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In this paper we present generalized edge-pairings for the family of hyperbolic tessellations $$\{10\lambda ,2\lambda \}$$ , with the purpose to obtain the corresponding discrete group of isometries. These tessellations have greater density packing than the self-dual tessellations $$\{4\lambda ,4\lambda \}$$ implying that the associated codes achieve the least error probability, or equivalently, that these codes are optimum codes.

In this paper we generalize the process of identifying arithmetic Fuchsian groups in a quaternion order over an extension of the rational as shown in [3]. This procedure enable us to construct new lattices in the hyperbolic plane which are associated with signal constellations. In addition to this, the procedure also enable us to realize the labelling of the signals belonging to such signal constellations.