Description
Family of \((n=2^{2r},n^2,2-2/\sqrt{n})\) spherical codes for \(r \geq 2\), obtained from Kerdock codes via the antipodal mapping [4; pg. 157]. These codes are optimal for their parameters for \(2\leq r\leq 5\), they are unique for \(r\in\{2,3\}\), and they form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [5].
Parents
- Binary antipodal code
- Spherical design — Kerdock codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [5].
Cousins
- Kerdock code — Kerdock spherical codes can be obtained from Kerdock codes using the antipodal mapping [4; pg. 157].
- Universally optimal spherical code — Kerdock spherical codes are almost universally optimal [6].
References
- [1]
- König, Hermann. "Isometric imbeddings of Euclidean spaces into finite dimensional lp-spaces." Banach Center Publications 34.1 (1995): 79-87. <http://eudml.org/doc/251336>.
- [2]
- P. J. CAMERON and J. J. SEIDEL, “QUADRATIC FORMS OVER GF(2)”, Geometry and Combinatorics 290 (1991) DOI
- [3]
- Levenshtein, V. I. "Bounds on the maximal cardinality of a code with bounded modulus of the inner product." Soviet Math. Dokl. Vol. 25. No. 2. 1982.
- [4]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [5]
- H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
- [6]
- P. G. Boyvalenkov et al., “Universal upper and lower bounds on energy of spherical designs”, (2015) arXiv:1509.07837
Page edit log
- Victor V. Albert (2023-11-23) — most recent
Cite as:
“Kerdock spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/kerdock_spherical