An Upper Bound for Signal Transmission Error Probability in Hyperbolic Spaces

We introduce and discuss the concept of Gaussian probability density function (pdf) for the n-dimensional hyperbolic space which has been proposed as an environment for coding and decoding signals. An upper bound for the error probability of signal transmission associated with the hyperbolic distance is established. The pdf and the upper bound were developed using Poincare models for the hyperbolic spaces.

[1]  G. David Forney,et al.  Geometrically uniform codes , 1991, IEEE Trans. Inf. Theory.

[2]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[3]  Sueli I. Rodrigues Costa,et al.  Graphs, tessellations, and perfect codes on flat tori , 2004, IEEE Transactions on Information Theory.

[4]  Clarice Dias de Albuquerque,et al.  Topological quantum codes on compact surfaces with genus g≥2 , 2009 .

[5]  Sueli I. Rodrigues Costa,et al.  Signal constellations in the hyperbolic plane: A proposal for new communication systems , 2006, J. Frankl. Inst..

[6]  Sueli I. Rodrigues Costa,et al.  Fisher information matrix and hyperbolic geometry , 2005, IEEE Information Theory Workshop, 2005..

[7]  C. Atkinson.  Rao's distance measure , 1981 .

[8]  John G. Ratcliffe,et al.  Geometry of Discrete Groups , 2019, Foundations of Hyperbolic Manifolds.

[9]  Tom Carroll,et al.  Extremal Problems for Conditioned Brownian Motion and the Hyperbolic Metric , 2000 .

[10]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[11]  Sueli I. Rodrigues Costa,et al.  "AWGN"-signal transmission in hyperbolic spaces , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[12]  Sergio Benedetto,et al.  Principles of Digital Transmission: With Wireless Applications , 1999 .

[13]  H. Bombin,et al.  Homological error correction: Classical and quantum codes , 2006, quant-ph/0605094.

[14]  A. Kitaev.  Fault tolerant quantum computation by anyons , 1997, quant-ph/9707021.