Torus-layer spherical code (TLSC)[1] 

Description

Code whose codewords are elements of a foliation of the \(2n-1\)-dimensional hypersphere \(S^{2n-1}\) using flat tori \(S^1\times S^1\cdots\times S^1\). Related constructions include the spherical codes by Hopf foliations (SCHF) [2].

Decoding

Efficiently decodable [1].

Notes

See [3; Sec. 5.2] for an exposition.

Parent

  • Slepian group-orbit code — Polyphase codewords can be implemented by acting on the all-ones initial vector by diagonal orthogonal matrices whose entries are the codeword components [4; Ch. 8]. TLSC codes are generalizations of polyphase codes to other initial vectors and are examples of abelian Slepian-group codes.

Child

References

[1]
C. Torezzan, S. I. R. Costa, and V. A. Vaishampayan, “Spherical codes on torus layers”, 2009 IEEE International Symposium on Information Theory (2009) DOI
[2]
H. K. Miyamoto, H. N. Sa Earp, and S. I. R. Costa, “Constructive spherical codes in 2\({}^{\text{k}}\) dimensions”, 2019 IEEE International Symposium on Information Theory (ISIT) (2019) DOI
[3]
S. I. R. Costa et al., Lattices Applied to Coding for Reliable and Secure Communications (Springer International Publishing, 2017) DOI
[4]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
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Zoo Code ID: tlsc

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

“Torus-layer spherical code (TLSC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/tlsc

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/spherical/numerical/tlsc.yml.