Cameron-Goethals-Seidel (CGS) isotropic subspace code[1]
Description
Member of a \((q(q^2-q+1),(q+1)(q^3+1),2-2/q^2)\) family of spherical codes for any prime-power \(q\). Constructed from generalized quadrangles, which in this case correspond to sets of totally isotropic points and lines in the projective space \(PG_{5}(q)\) [2; Ex. 9.4.5]. There exist multiple distinct spherical codes using this construction for \(q>3\) [3].
Protection
CGS isotropic subspace codes saturate the Levenshtein bound [2; pg. 64].
Parents
- Spherical design code
- Spherical sharp configuration — CGS isotropic subspace codes are the only known spherical sharp configrations not derived from regular polytopes or lattices [3].
Child
- Hessian polyhedron code — The CGS isotropic subspace code for \(q=2\) reduces to the Hessian polytope.
Cousin
- Projective geometry code — CSG isotropic subspace codes are constructed from incidence matrices of \(PG_5(q)\) [2; Ex. 9.4.5].
References
- [1]
- P. J. Cameron, J. M. Goethals, and J. J. Seidel, “Strongly regular graphs having strongly regular subconstituents”, Journal of Algebra 55, 257 (1978) DOI
- [2]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [3]
- 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
Page edit log
- Victor V. Albert (2023-02-23) — most recent
Cite as:
“Cameron-Goethals-Seidel (CGS) isotropic subspace code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/cgs_spherical