Also known as \(4_{21}\) real polytope code.
Description
Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope, the \(4_{21}\) real polytope, and the minimal lattice-shell code of the \(E_8\) lattice. The code is optimal and unique up to equivalence [1–3]. Antipodal pairs of points correspond to the 120 tritangent planes of a canonic sextic curve [4–7].
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 [8; pg. 132] for a complex representation.
Protection
Code yields an optimal solution to the kissing problem in 4D [9,10] and saturates the Levenshtein bound [11].
Parents
- Polytope code
- \(E_8\) Gosset lattice-shell code — Witting polytope codewords form the minimal shell of the \(E_8\) lattice.
- Spherical sharp configuration — The Witting polytope code is a sharp configuration [6,12].
- Spherical design — The Witting polytope code forms a tight spherical 7-design [1][2; Ch. 14].
- Real-Clifford subgroup-orbit code — The Witting polytope code is equivalent to the real Clifford subgroup-orbit code for \(n=8\).
Cousins
- \(3_{21}\) polytope code — \(3_{21}\) polytope codewords form the kissing configuration of the Witting polytope code; see Ref. [6].
- 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. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
- [4]
- P. du Val, “On the Directrices of a Set of Points in a Plane”, Proceedings of the London Mathematical Society s2-35, 23 (1933) DOI
- [5]
- Arnold, V. I. (1999). Symplectization, complexification and mathematical trinities. The Arnoldfest, 23-37.
- [6]
- 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
- [7]
- Y.-H. He and J. McKay, “Sporadic and Exceptional”, (2015) arXiv:1505.06742
- [8]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [9]
- Schläfli, L. (1901). Theorie der vielfachen Kontinuität (Vol. 38). Zürcher & Furrer.
- [10]
- O. R. Musin, “The kissing number in four dimensions”, (2006) arXiv:math/0309430
- [11]
- 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
- [12]
- A. V. KOLUSHOV and V. A. YUDIN, “On Korkin-Zolotarev’s construction”, Discrete Mathematics and Applications 4, (1994) DOI
Page edit log
- Victor V. Albert (2022-11-28) — most recent
Cite as:
“Witting polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/witting_polytope