Hessian polyhedron code

Description

Also known as the Schlafli configuration. Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. The code can be obtained from the Schlafli graph [1; Ch. 9]. The points correspond to the 27 lines on a smooth cubic surface in the complex projective plane [2]. The code forms a tight spherical 4-design [3; Exam. 7.3]. See [1; Exam. 1.2.5] ([4; pg. 119]) for a real (complex) realization of the 27 codewords.

Figure I: Projection of the coordinates of the Hessian polytope.

Protection

The Hessian polytope code has degree \(d=2\) and saturates the absolute bound [1].

Realizations

Quantum mechanical SIC-POVMs [5].

Parents

Cousins

References

[1]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[2]
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
[3]
A. Roy and S. Suda, “Complex spherical designs and codes”, (2011) arXiv:1104.4692
[4]
H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
[5]
B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”, (2019) arXiv:1911.05809
[6]
L. Manivel, “Configurations of lines and models of Lie algebras”, (2005) arXiv:math/0507118
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Zoo Code ID: hessian_polyhedron

Cite as:
“Hessian polyhedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hessian_polyhedron
BibTeX:
@incollection{eczoo_hessian_polyhedron, title={Hessian polyhedron code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hessian_polyhedron} }
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“Hessian polyhedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hessian_polyhedron

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/spherical/polytope/hessian_polyhedron/hessian_polyhedron.yml.