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Witting polytope code

Alternative names: \(4_{21}\) polytope code, Gosset polytope code.

Description

Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope, the \(4_{21}\) 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 of the \(4_{21}\) polytope code correspond to the 120 tritangent planes of a canonical sextic curve in \(\mathbb{C}P^3\) [4–7].

A representation of the codewords consists of the 112 vectors obtained from permutations of \((0,0,0,0,0,0,\pm 2,\pm 2)\) together with the 128 vectors \((\pm 1)^{\times 8}\) for which the number of minus signs is even. After normalization, these are precisely the 240 minimal vectors of the \(E_8\) lattice. See [8; pg. 132] for a complex representation.

Recursively taking its kissing configurations yields the \((7,56,1/3)\), \((6,27,1/4)\), \((5,16,1/5)\), \((4,10,1/6)\), and \((3,6,1/7)\) spherical codes [1].

Protection

Code yields an optimal solution to the kissing problem in 8D [9,10], saturates the Levenshtein bound [11], and is unique up to equivalence [1]. It carries a 4-class association scheme, and the inner products with respect to any codeword are \(\pm 1\), \(\pm 1/2\), and \(0\) with multiplicities \(1\), \(56\), and \(126\), respectively [1].

Notes

The Witting polytope yields 40 states of a qudit of dimension 4 and a non-probabilistic version of Bell’s theorem [12–15].

Cousins

  • Complex projective space code— Antipodal pairs of points of the Witting polytope code correspond to the 120 tritangent planes of a canonical sextic curve in \(\mathbb{C}P^3\) [4–7].
  • Sharp configuration— The 120 antipodal pairs of the Witting polytope code form a sharp configuration in \(\mathbb{R}P^7\) [6].
  • \(t\)-design— Antipodal pairs of points of the Witting polytope code form a 3-design in \(\mathbb{R}P^7\) [6].
  • Real projective space code— The 120 antipodal pairs of the Witting polytope code form a sharp configuration and a 3-design in \(\mathbb{R}P^7\) [6].
  • \(E_8\) Gosset lattice— The Voronoi cell of the \(E_8\) Gosset lattice is the dual of the Gosset \(4_{21}\) polytope [2; Ch. 21, pg. 464].
  • 600-cell code— The 120 vertices of the 600-cell are the unit icosians, and these icosian units, together with their multiples by \((1-\sqrt{5})/2\), form the 240 minimal vectors of a version of the \(E_8\) lattice, i.e., the Witting polytope [2; Ch. 8, pg. 210].
  • Hessian polyhedron code— The Hessian polyhedron code forms the next recursive kissing configuration after the \(3_{21}\) polytope code in the \(E_8\) lattice-shell/Witting polytope sequence [1]. The Schläfli graph is a subgraph of the graph formed by the vertices of the Witting polytope [16; Sec. 3.11].
  • \(3_{21}\) polytope code— \(3_{21}\) polytope codewords form the first recursive kissing configuration of the Witting polytope code [1,6][2; Ch. 9, pg. 264]. The Gosset graph is a subgraph of the graph formed by the vertices of the Witting polytope [16; Sec. 3.11].
  • \(2_{41}\) polytope code— Vertices of the \(2_{41}\) and \(4_{21}\) polytopes minimize each other’s potential functions [17].
  • Clifford subgroup-orbit QSC— Logical constellations of the Clifford subgroup-orbit code for \(r=2\) form vertices of Witting polytopes.

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy spherical codeECC
Spherical code
Slepian group-orbit codeECC
Polytope code
Dual polytope code
Self-dual polytope code
Parents
Self-dual polytope code
The Witting polytope is self-dual as a complex polytope. The \(4_{21}\) polytope is not self-dual [18].
\(E_8\) Gosset lattice-shell codeLattice-shell ECC
The minimal shell of the lattice yields the \((8,240,1)\) code, whose codewords form the vertices of the \(4_{21}\) polytope.
Spherical sharp configurationSpherical design Sharp configuration Universally optimal ECC \(t\)-design
The Witting polytope code is a sharp configuration [6,19].
Spherical designECC \(t\)-design
The Witting polytope code forms a tight spherical 7-design [1][2; Ch. 14].
Real-Clifford subgroup-orbit codeSpherical design ECC \(t\)-design
The Witting polytope code is equivalent to the real Clifford subgroup-orbit code for \(n=8\).
Witting polytope code

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]
V. I. Arnold (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]
L. Schläfli, Theorie der vielfachen Kontinuität, vol. 38 (Zürcher & Furrer, 1901)
[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]
R. Penrose, “On Bell non-locality without probabilities: some curious geometry.” Quantum Reflections (2000): 1-27
[13]
J. Zimba and R. Penrose, “On bell non-locality without probabilities: More curious geometry”, Studies in History and Philosophy of Science Part A 24, 697 (1993) DOI
[14]
J. E. Massad and P. K. Aravind, “The Penrose dodecahedron revisited”, American Journal of Physics 67, 631 (1999) DOI
[15]
M. Waegell and P. K. Aravind, “The Penrose dodecahedron and the Witting polytope are identical in CP3”, Physics Letters A 381, 1853 (2017) arXiv:1701.06512 DOI
[16]
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer Berlin Heidelberg, 1989) DOI
[17]
S. Borodachov, “Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials”, Aequationes mathematicae 98, 509 (2024) DOI
[18]
H. S. M. Coxeter, Regular Polytopes (Courier Corporation, 1973)
[19]
A. V. KOLUSHOV and V. A. YUDIN, “On Korkin-Zolotarev’s construction”, Discrete Mathematics and Applications 4, (1994) 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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“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/8d/witting_polytope.yml.