Universally optimal code[1]
Description
A code that produces a minimum over all codes of its cardinality for a large family of potential functions. Such codes exist for the conventional \(q\)-ary and real spaces (see children below), but can also be formulated for more exotic spaces such as Lie groups, projective spaces, and real Grassmanians [2,3].
Notes
See notes on universally optimal codes [4].
Parent
Children
- Universally optimal sphere packing
- Sharp configuration — All sharp configurations are universally optimal [5,6], but not all universally optimal codes are sharp configurations.
- Universally optimal \(q\)-ary code
- Universally optimal spherical code
References
- [1]
- G. A. Kabatiansky, V. I. Levenshtein, “On Bounds for Packings on a Sphere and in Space”, Probl. Peredachi Inf., 14:1 (1978), 3–25; Problems Inform. Transmission, 14:1 (1978), 1–17
- [2]
- H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
- [3]
- A. Glazyrin, “Moments of isotropic measures and optimal projective codes”, (2020) arXiv:1904.11159
- [4]
- H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
- [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]
- H. Cohn and Y. Zhao, “Energy-Minimizing Error-Correcting Codes”, IEEE Transactions on Information Theory 60, 7442 (2014) arXiv:1212.1913 DOI
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 code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/univ_opt