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Permutation spherical code[1,2]

Description

Slepian group-orbit code whose codewords are constructed from an arbitrary unit vector in two possible variants. Variant 1 consists of codewords that are permutations of the vector's coordinates, while Variant 2 consists of such permutations and all possible sign changes of the vector's components.

Rate

Number of codewords cannot increase exponentially with dimension \(n\) [3]

Decoding

Efficient maximum-likelihood decoder determining the Voronoi region of an error word.

Notes

See [4; Ch. 4] for more details and tables of optimal codes.

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy codeECC
Spherical code
Slepian group-orbit codeECC
Parents
Slepian group-orbit code
Permutations and sign changes can be implemented on vectors by orthogonal matrices, so permutation spherical codes are Slepian group-orbit codes.
Permutation spherical code
Children
Biorthogonal spherical code
Simplex spherical code

References

[1]
D. Slepian, “Permutation modulation”, Proceedings of the IEEE 53, 228 (1965) DOI
[2]
I. Ingemarsson, “Optimized permutation modulation”, IEEE Transactions on Information Theory 36, 1098 (1990) DOI
[3]
H. Landau, “How does a porcupine separate its quills?”, IEEE Transactions on Information Theory 17, 157 (1971) DOI
[4]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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Zoo Code ID: permutation_spherical

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

Cite as:

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

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