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
- Slepian group-orbit code
- Spherical design code — The orbit of any point under the real Clifford subgroup is a spherical 7-design [4], and some are 11-designs [5].
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
- Barnes-Wall (BW) lattice code — The automorphism group of BW lattices is a subgroup of index 2 of a real Clifford group [1,2] (see [3] for an explanation).
- Disphenoidal 288-cell code — The disphenoidal 288-cell code is a group-orbit code with the group being the real Clifford group in \(4\) dimensions.
- Clifford subgroup-orbit QSC — Clifford group-orbit QSCs are quantum analogues of real Clifford subgroup-orbit codes.
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
Page edit log
- Victor V. Albert (2022-11-22) — most recent
Cite as:
“Real-Clifford subgroup-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/sidelnikov