Freedman-Meyer-Luo code[1] 

Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom [2]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [3]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology.

The Freedman-Meter-Luo code has been generalized to a family with rate of order \(O(1/\sqrt{\log n})\) and minimum distance of order \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates [4].