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.

## Parent

- Bounded-energy code — Constant-energy codes are bounded-energy codes constrained to lie on a sphere.

## Child

## Cousins

- Constant-excitation (CE) code
- Quantum spherical code (QSC) — QSCs are quantum analogues of spherical and constant-energy codes because they store information in quantum superpositions of points on a sphere in quantum phase space.

## 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.

## Page edit log

- Victor V. Albert (2022-11-15) — most recent

## Cite as:

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