On geometrically uniform codes and topological quantum MDS codes

In this paper, we present generalized edge-pairings for the family of hyperbolic tessellations {4 lambda, 4} with the purpose of obtaining 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. We also consider the topological quantum error-correcting codes on surfaces with genus g >= 2, as proposed in other papers. From that, we focus on the class of topological quantum maximum distance separable codes, showing that the only such codes are precisely the ones with minimum distance d <= 2. Published under an exclusive license by AIP Publishing.