Description
Encodes states (codewords) into coordinates in the hyperbolic plane \(\mathbb{H}^2\).Protection
Designed to communicate information over channels for which a Lorentzian metric is appropriate [3]. Linear programming bounds exist for hyperbolic surfaces [4].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 [5].
- Quadrature-amplitude modulation (QAM) code— Hyperbolic QAM constellations may yield improved performance over Euclidean ones [6].
- Quantum spherical code (QSC)— Quantum versions [7] of hyperbolic sphere packings have been developed for hyperbolic space \(SO(n,1)/O(n)\), which is the negative-curvature analogue of the sphere.
Member of code lists
Primary Hierarchy
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) 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]
- M. F. Bourque and B. Petri, “Linear programming bounds for hyperbolic surfaces”, (2023) arXiv:2302.02540
- [5]
- 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.
- [6]
- 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.
- [7]
- Y. Wang, Y. Xu, and Z.-W. Liu, “Encoded quantum gates by geometric rotation on tessellations”, (2024) arXiv:2410.18713
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