## Description

A spherical code that is a spherical design of strength \(2m-1\) for some \(m\) and that has \(m\) distances between distinct points. All known spherical sharp configrations are either obtained from the Leech or \(E_8\) lattice, certain regular polytopes, or are CGS isotropic subspace spherical codes [5; Table 1].

## Parents

- Universally optimal spherical code — All sharp configurations are universally optimal [4], but not all universally optimal spherical codes are sharp configurations. The one known exception is the 600-cell.
- Sharp configuration
- Spherical design — Spherical sharp configurations are spherical designs of strength \(2m-1\) for some \(m\).

## Children

- \(3_{21}\) polytope code — The \(3_{21}\) polytope code is a sharp configuration [4,6].
- Icosahedron code — The icosahedron is a sharp configuration [4,7].
- Biorthogonal spherical code
- Simplex spherical code
- Polygon code
- Witting polytope code — The Witting polytope code is a sharp configuration [4,6].
- Cameron-Goethals-Seidel (CGS) isotropic subspace code — CGS isotropic subspace codes are the only known spherical sharp configrations not derived from regular polytopes or lattices [4].

## Cousins

- \(\Lambda_{24}\) Leech lattice code — Several spherical sharp configrations are derived from the \(\Lambda_{24}\) Leech lattice [4].
- \(E_8\) Gosset lattice code — Several spherical sharp configrations are derived from the \(E_8\) Gosset lattice code [4].
- \(\Lambda_{24}\) Leech lattice-shell code — The smallest-shell \((24,196560,1)\) code is a spherical sharp configuration [4,8]. The \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes are also sharp configurations [4,9–11][5; Table 1].
- Smith \(40\)-point code — The Smith spherical code is conjectured to be a global minimum of completely monotonic potential functions [12].

## References

- [1]
- V. I. Levenshtein, "On choosing polynomials to obtain bounds in packing problems." Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius. 1978.
- [2]
- V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303
- [3]
- V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
- [4]
- 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
- [5]
- H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
- [6]
- A. V. KOLUSHOV and V. A. YUDIN, “On Korkin-Zolotarev’s construction”, Discrete Mathematics and Applications 4, (1994) DOI
- [7]
- Andreev, Nikolay N. "An extremal property of the icosahedron." East J. Approx 2.4 (1996): 459-462.
- [8]
- Andreev, N. N. Location of points on a sphere with minimal energy. (Russian) Tr. Mat. Inst. Steklova 219 (1997), Teor. Priblizh. Garmon. Anal., 27–31; translation in Proc. Steklov Inst. Math. 1997, no. 4(219), 20–24
- [9]
- E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
- [10]
- R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI
- [11]
- H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
- [12]
- B. Ballinger et al., “Experimental Study of Energy-Minimizing Point Configurations on Spheres”, Experimental Mathematics 18, 257 (2009) arXiv:math/0611451 DOI

## Page edit log

- Victor V. Albert (2023-03-05) — most recent
- Alexander Barg (2023-03-05)
- Victor V. Albert (2023-02-23)

## Cite as:

“Spherical sharp configuration”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/sharp_config