\(3_{21}\) polytope code[1] 

Also known as Hess polytope code, 7-ic semi-regular figure code.

Description

Spherical \((7,56,1/3)\) code whose codewords are the vertices of the \(3_{21}\) real polytope (a.k.a. the Hess polytope). The vertices form the kissing configuration of the Witting polytope code. The code is optimal and unique up to equivalence [24]. Antipodal pairs of points correspond to the 28 bitangent lines of a general quartic plane curve [58].

A representation of the codewords consists of all seven permutations of the eight vectors \((\pm 1,0,\pm 1,\pm 1,0,0,0)\).

Parents

Cousins

References

[1]
Gosset, Thorold. "On the regular and semi-regular figures in space of n dimensions." Messenger of Mathematics 29 (1900): 43-48.
[2]
E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[3]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[4]
H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
[5]
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
[6]
Arnold, V. I. (1999). Symplectization, complexification and mathematical trinities. The Arnoldfest, 23-37.
[7]
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
[8]
Y.-H. He and J. McKay, “Sporadic and Exceptional”, (2015) arXiv:1505.06742
[9]
A. V. KOLUSHOV and V. A. YUDIN, “On Korkin-Zolotarev’s construction”, Discrete Mathematics and Applications 4, (1994) DOI
[10]
J. H. Conway and N. J. A. Sloane, “The Cell Structures of Certain Lattices”, Miscellanea Mathematica 71 (1991) DOI
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Zoo Code ID: hess_polytope

Cite as:
\(3_{21}\) polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hess_polytope
BibTeX:
@incollection{eczoo_hess_polytope, title={\(3_{21}\) polytope code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hess_polytope} }
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Cite as:

\(3_{21}\) polytope code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hess_polytope

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