Guth-Lubotzky code[1] 

Hyperbolic surface code based on cellulations of certain four-dimensional manifolds. The manifolds are shown to have good homology and systolic properties for the purposes of code construction, with corresponding codes exhibiting linear rate.

Guth and Lubotzky [1] show that there exists \(\epsilon\), a four-dimensional hyperbolic manifold \(M\), and a sequence of manifolds \(M_i\) such that each \(M_i\) is a finite sheeted covering of \(M\), and the four-dimensional volumes of the manifolds \(\text{Vol}_4(M_i)\) of the sequence tend to infinity. Also, the dimension of the second homology and size of systoles are bounded by \(H_2(M_i, Z_2) \geq \frac{\text{Vol}_4(M_i)}{100}\) and \(\text{Sys}_2(M_i) \geq \text{Vol}_4(M_i)^\epsilon\), respectively.

Then given any cellulation of \(M\), it can naturally be extended to cellulations for each of the manifolds \(M_i\) and used to define CSS codes via the homological construction by choosing the size three chain complex consisting of the \(3,2\) and \(1\)-cells of the cellulations.

For dense cellulations (i.e. large \(n\)) the number of physical qubits for these codes will scale with the volume of the manifolds. Therefore, bounds on the dimension of the second homology and size of systoles are achieved in terms of \(n\) for large \(n\).