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Constant-energy spherical code

Root code for the Spherical Kingdom

Description

Code whose codewords are points on a real or complex sphere whose radius squared is called the energy. Typically, only angular distances between points are relevant for code performance, so one can normalize codewords of a constant-energy code to obtain up a spherical code, i.e., a constant energy code with energy one.

Protection

Constant-energy codes are sphere packings constrained to lie on a sphere, meaning that they can be used to transmit information through the AGWN channel. For a given dimension \(n\), number of codewords \(M\), and average energy \(P\), the constant-energy Gaussian channel coding problem asks to find a set of \(M\) codewords of energy \(nP\) that minimizes the error probability during transmission; see [1; Ch. 3].

Notes

See [2; Ch. 7] for more details.

Cousins

  • Real projective space code— Real projective space can be obtained from the sphere by identifying antipodal points, i.e., \(\mathbb{R}P^N = S^N/\mathbb{Z}_2\). As such, real projective space codes are in one-to-one correspondence with antipodal spherical codes, with each antipodal pair of spherical codewords corresponding to one line in projective space.
  • Quantum spherical code (QSC)— QSCs are quantum counterparts of spherical and constant-energy codes because they store information in quantum superpositions of points on a sphere in quantum phase space.

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Parents
Bounded-energy codeECC
Constant-energy codes are bounded-energy codes constrained to lie on a sphere.
Two-point homogeneous-space codeECC
Compact two-point homogeneous spaces \(G/H\) reduce to real spheres for \(G = SO(D+1)\) and \(H = SO(D)\) and to complex spheres for \(G = SU(D+1)\) and \(H = SU(D)\) [3; Table 1].
Constant-energy spherical code
Children
Spherical codePolytope Polyhedron Lattice-shell Spherical design

References

[1]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[2]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[3]
C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin, “Invariant Semidefinite Programs”, International Series in Operations Research & Management Science 219 (2011) arXiv:1007.2905 DOI
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Zoo Code ID: points_into_spheres

Cite as:
“Constant-energy spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/points_into_spheres
BibTeX:
@incollection{eczoo_points_into_spheres, title={Constant-energy spherical code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/points_into_spheres} }
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Cite as:

“Constant-energy spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/points_into_spheres

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/points_into_spheres.yml.