Description
A spherical code that (weakly) minimizes all completely monotonic potentials on the sphere for its cardinality. See [7][6; Sec. 12.4] for further discussion.
Parents
Children
- 600-cell code — The 600-cell is universally optimal, but it is not a spherical sharp configuration [5].
- Spherical sharp configuration — All sharp configurations are universally optimal [5], but not all universally optimal spherical codes are sharp configurations. The one known exception is the 600-cell.
Cousins
- 24-cell code — The 24-cell code is not universally optimal [8], but comes quite close [6; Exam. 12.4.29].
- Kerdock spherical code — Kerdock spherical codes are almost universally optimal [9].
References
- [1]
- V. A. Yudin, “Minimum potential energy of a point system of charges”, Diskr. Mat., 4:2 (1992), 115–121; Discrete Math. Appl., 3:1 (1993), 75–81
- [2]
- A. Askikhmin, A. Barg, and S. Litsyn, “Estimates of the distance distribution of codes and designs”, IEEE Transactions on Information Theory 47, 1050 (2001) DOI
- [3]
- A. V. Kolushov, V. A. Yudin, А. В. Колущов, and В. А. Удин, “Extremal dispositions of points on the sphere”, Analysis Mathematica 23, 25 (1997) DOI
- [4]
- E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere”, The Mathematical Intelligencer 19, 5 (1997) DOI
- [5]
- 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
- [6]
- P. Boyvalenkov, D. Danev, "Linear programming bounds." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [7]
- J. S. Brauchart and P. J. Grabner, “Distributing many points on spheres: Minimal energy and designs”, Journal of Complexity 31, 293 (2015) arXiv:1407.8282 DOI
- [8]
- H. Cohn, J. H. Conway, N. D. Elkies, and A. Kumar, “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
- [9]
- P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff, and M. M. Stoyanova, “Universal upper and lower bounds on energy of spherical designs”, (2015) arXiv:1509.07837
Page edit log
- Victor V. Albert (2023-03-05) — most recent
- Alexander Barg (2023-03-05)
- Victor V. Albert (2023-02-28)
Cite as:
“Universally optimal spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/univ_opt_spherical