Witting polytope code 

Description

Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope and the \(4_{21}\) real polytope. This code is optimal and unique up to equivalence [1,2]. Codewords form the minimal lattice-shell code of the \(E_8\) lattice.

A representation of the codewords consists of all 112 permutations of the four vectors \((0,0,0,0,0,0,\pm 2,\pm 2)\) along with the 128 vectors \((\pm 1)^{\times 8}\) where the number of minus signs is even. See [3; pg. 132] for a complex representation.

Protection

Code yields an optimal solution to the kissing problem in 4D [4,5] and saturates the Levenshtein bound [6].

Parents

Cousin

  • Clifford subgroup-orbit QSC — Logical constellations of the Clifford subgroup-orbit code for \(r=2\) form vertices of Witting polytopes.

References

[1]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[2]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[3]
H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
[4]
Schläfli, L. (1901). Theorie der vielfachen Kontinuität (Vol. 38). Zürcher & Furrer.
[5]
O. R. Musin, “The kissing number in four dimensions”, (2006) arXiv:math/0309430
[6]
A. M. Odlyzko and N. J. A. Sloane, “New bounds on the number of unit spheres that can touch a unit sphere in n dimensions”, Journal of Combinatorial Theory, Series A 26, 210 (1979) DOI
[7]
A. V. KOLUSHOV and V. A. YUDIN, “On Korkin-Zolotarev’s construction”, Discrete Mathematics and Applications 4, (1994) DOI
[8]
H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
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Zoo Code ID: witting_polytope

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

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/witting_polytope.yml.