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.


Four-dimensional manifolds with weak systolic freedom yield \([[n,2,\Omega(\sqrt{n \sqrt{\log n}})]]\) surface codes.


Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, \(d=\Omega(\sqrt{n \sqrt{\log n}})\), broken by Ramanujan tensor-product codes.


See thesis by Fetaya for pedagogical exposition [4].




“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
M. H. Freedman, “Z\({}_{\text{2}}\)–Systolic-Freedom”, Proceedings of the Kirbyfest (1999) DOI
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
E. Fetaya, “Homological Error Correcting Codes and Systolic Geometry”, (2011) arXiv:1108.2886
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Zoo Code ID: freedman_meyer_luo

Cite as:
“Freedman-Meyer-Luo code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_freedman_meyer_luo, title={Freedman-Meyer-Luo code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Freedman-Meyer-Luo code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.