Real-Clifford subgroup-orbit code[1,2] 

Description

Slepian group-orbit code of dimension \(2^r\), approximate asympotic size \(2.38 \cdot 2^{r(r+1)/2+1}\), and distance \(1\). Code is constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [3], onto the vector \((1,0,0,\cdots,0)\). This group is the automorphism group of BW lattice, and the resulting codes coincide with the optimal spherical codes for dimensions \(\{4,8,16\}\).

Taking the orbit under the entire real Clifford group yields spherical codes twice the points and with distance \(2-\sqrt{2}\).

Parents

Children

  • \(BW_{32}\) lattice-shell code — The minimal \(BW_{32}\) lattice-shell code is equivalent to the real Clifford subgroup-orbit code for \(n=32\).
  • \(\Lambda_{16}\) lattice-shell code — The minimal \(\Lambda_{16}\) lattice-shell code is equivalent to the real Clifford subgroup-orbit code for \(n=16\).
  • 24-cell code — The 24-cell code is equivalent to the real Clifford subgroup-orbit code for \(n=4\).
  • Witting polytope code — The Witting polytope code is equivalent to the real Clifford subgroup-orbit code for \(n=8\).

Cousins

References

[1]
V. M. Sidelnikov, On a finite group of matrices and codes on the Euclidean sphere (in Russian), Probl. Peredach. Inform. 33 (1997), 35–54 (1997); English translation in Problems Inform. Transmission 33 (1997), 29–44
[2]
V. M. Sidelnikov, “On a finite group of matrices generating orbit codes on Euclidean sphere”, Proceedings of IEEE International Symposium on Information Theory DOI
[3]
G. Nebe, E. M. Rains, and N. J. A. Sloane, “The invariants of the Clifford groups”, (2000) arXiv:math/0001038
[4]
V. M. Sidelnikov, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
[5]
V. M. Sidelnikov, “Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial”, Algebra i Analiz, 11:4 (1999), 183–203; St. Petersburg Math. J., 11:4 (2000), 673–686
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Zoo Code ID: sidelnikov

Cite as:
“Real-Clifford subgroup-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/sidelnikov
BibTeX:
@incollection{eczoo_sidelnikov, title={Real-Clifford subgroup-orbit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/sidelnikov} }
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Cite as:

“Real-Clifford subgroup-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/sidelnikov

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