# 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.

## Protection

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

## Realizations

Quantum mechanical SIC-POVMs [5].

## Parents

- Polytope code
- Cameron-Goethals-Seidel (CGS) isotropic subspace code — The CGS isotropic subspace code for \(q=2\) reduces to the Hessian polytope.

## Cousins

- \(E_6\) root lattice code — The 27 Hessian polyhedron codewords are intimately related to the \(E_6\) Lie group [6].
- Rectified Hessian polyhedron code — The (rectified) Hessian polyhedron is an analogue of a (octahedron) tetrahedron in 3D complex space [4; pg. 127].

## 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

## Page edit log

- Victor V. Albert (2022-11-16) — most recent

## Cite as:

“Hessian polyhedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hessian_polyhedron