# Freedman-Meyer-Luo code[1]

## Description

Hyperbolic surface code constructed using cellulation of a Riemann Manifold \(M\) exhibitng 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.

## Protection

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

## Rate

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.

## Notes

See thesis by Fetaya for pedagogical exposition [4].

## Parent

## Cousin

- Ramanujan-complex product code — Ramanujan codes broke 20-year record on minimum code distance set by Freedman-Meyer-Luo codes.

## Zoo code information

## References

- [1]
- “Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002). DOI
- [2]
- M. H. Freedman, “Z2–Systolic-Freedom”, Proceedings of the Kirbyfest (1999). DOI
- [3]
- E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, 062202 (2012). DOI
- [4]
- Ethan Fetaya, “Homological Error Correcting Codes and Systolic Geometry”. 1108.2886

## Cite as:

“Freedman-Meyer-Luo code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/freedman_meyer_luo