Description
Encodes states (codewords) into coordinates in the hyperbolic plane.Protection
Designed to communicate information over channels for which a Lorentzian metric is appropriate [3]. Linear programming (LP) bounds exist for hyperbolic surfaces [4–8].Cousins
- Pulse-amplitude modulation (PAM) code— Hyperbolic PAM constellations may yield improved performance over Euclidean ones [1].
- Phase-shift keying (PSK) code— Hyperbolic PSK constellations may yield improved performance over Euclidean ones [9].
- Quadrature-amplitude modulation (QAM) code— Hyperbolic QAM constellations may yield improved performance over Euclidean ones [10].
- Hyperbolic tesselation code— Hyperbolic tesselation codes are quantum analogues of hyperbolic sphere packings because they store information in quantum superpositions of points on the hyperbolic plane.
Member of code lists
Primary Hierarchy
Parents
Hyperbolic space in \(D\) dimensions is a symmetric space \(G/H\) for \(G = SO(D,1)\) the proper Lorentz group and \(H = O(D)\). The hyperbolic plane is the case \(D=2\).
Hyperbolic sphere packing
References
- [1]
- E. B. Da Silva, R. Palazzo, and S. R. Costa, “Improving the performance of asymmetric M-PAM signal constellations in Euclidean space by embedding them in hyperbolic space”, 1998 Information Theory Workshop (Cat. No.98EX131) 98 DOI
- [2]
- E. B. da Silva, M. Firer, S. R. Costa, and R. Palazzo Jr., “Signal constellations in the hyperbolic plane: A proposal for new communication systems”, Journal of the Franklin Institute 343, 69 (2006) DOI
- [3]
- M. E. Gertsenshtein and V. B. Vasil’ev, “Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane”, Theory of Probability & Its Applications 4, 391 (1959) DOI
- [4]
- L. Bowen, “Periodicity and Circle Packings of the Hyperbolic Plane”, Geometriae Dedicata 102, 213 (2003) DOI
- [5]
- H. Cohn and Y. Zhao, “Sphere packing bounds via spherical codes”, Duke Mathematical Journal 163, (2014) arXiv:1212.5966 DOI
- [6]
- M. F. Bourque and B. Petri, “Linear programming bounds for hyperbolic surfaces”, (2023) arXiv:2302.02540
- [7]
- M. Wackenhuth, “Bounds on hyperbolic sphere packings: On a conjecture by Cohn and Zhao”, (2024) arXiv:2411.07139
- [8]
- M. Wackenhuth, “Linear programming bounds in homogeneous spaces, I: Optimal packing density”, (2025) arXiv:2505.23572
- [9]
- Silva, E. B., and R. Palazzo Jr. “M-PSK signal constellations in hyperbolic space achieving better performance than the M-PSK signal constellations in Euclidean space.” 1999 IEEE Information Theory Workshop, Metsovo, Greece. 1999.
- [10]
- Silva, E. B., R. Palazzo Jr, and M. Firer. “Performance analysis of QAM-like constellations in hyperbolic space.” 2000 International Symposium on Information Theory and its Applications, Honolulu, USA. 2000.
Page edit log
- Victor V. Albert (2022-11-02) — most recent
- Victor V. Albert (2022-02-16)
Cite as:
“Hyperbolic sphere packing”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hyperbolic