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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].

Member of code lists

Primary Hierarchy

Classical Domain
Error-correcting code (ECC)
Block code
Parents
Block code
Universally optimal code
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 codeMDS GRS
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
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Zoo Code ID: univ_opt

Cite as:
“Universally optimal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/univ_opt
BibTeX:
@incollection{eczoo_univ_opt, title={Universally optimal code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/univ_opt} }
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Permanent link:
https://errorcorrectionzoo.org/c/univ_opt

Cite as:

“Universally optimal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/univ_opt

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/properties/block/universally_optimal/univ_opt.yml.