# \(3_{21}\) polytope code[1]

Also known as Hess polytope code, Hesse 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 [2–4]. Antipodal pairs of points correspond to the 28 bitangent lines of a general quartic plane curve [5–8].

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

- Polytope code
- Spherical sharp configuration — The \(3_{21}\) polytope code is a sharp configuration [7,9].
- Spherical design — The \(3_{21}\) polytope code forms a 5-design [2][3; Ch. 14].

## Cousins

- Witting polytope code — \(3_{21}\) polytope codewords form the kissing configuration of the Witting polytope code; see Ref. [7].
- \(E_7\) lattice-shell code — \(3_{21}\) polytope codewords form the minimal lattice-shell code of the \(E_7^{\perp}\) lattice [10].

## 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

## Page edit log

- Victor V. Albert (2022-11-28) — most recent

## Cite as:

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