Also known as Schläfli configuration.

## Description

Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron (a.k.a. diplo-Schläfli) code. The code can be obtained from the Schläfli graph [3; Ch. 9]. The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in the complex projective plane [4–7].

See [8][3; Exam. 1.2.5] ([9; 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 [3].

## Realizations

Quantum mechanical SIC-POVMs [10].

## Parents

- Polytope code
- Cameron-Goethals-Seidel (CGS) isotropic subspace code — The CGS isotropic subspace code for \(q=2\) reduces to the Hessian polytope.
- Spherical design — The Hessian polytope code forms a tight spherical 4-design [11; Exam. 7.3].

## Cousins

- \(E_6\) root lattice code — The 27 Hessian polyhedron codewords are intimately related to the \(E_6\) Lie group [12].
- \(E_6\) lattice-shell code — Double Hessian polyhedron codewords form the minimal lattice-shell code of the \(E_6^{\perp}\) lattice [13].
- Rectified Hessian polyhedron code — The (rectified) Hessian polyhedron is an analogue of a (octahedron) tetrahedron in 3D complex space, while the double Hessian polyhedron is the analogue of a cube [9; pg. 127]. The rectified and double Hessian polyhedra are dual to each other, just like the octahedron and cube. Moreover, the double Hessian consists of two Hessians, just like the cube can be constructed with two tetrahedra.
- Hessian QSC — The Hessian QSC is the quantum generalization of the classical Hessian polyhedron code.

## References

- [1]
- Gosset, Thorold. "On the regular and semi-regular figures in space of n dimensions." Messenger of Mathematics 29 (1900): 43-48.
- [2]
- Schoute, P. H. "On the relation between the vertices of a definite six-dimensional polytope and the lines of a cubic surface." Proc. Roy. Acad. Amsterdam. Vol. 13. 1910.
- [3]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [4]
- P. du Val, “On the Directrices of a Set of Points in a Plane”, Proceedings of the London Mathematical Society s2-35, 23 (1933) DOI
- [5]
- Arnold, V. I. (1999). Symplectization, complexification and mathematical trinities. The Arnoldfest, 23-37.
- [6]
- 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
- [7]
- Y.-H. He and J. McKay, “Sporadic and Exceptional”, (2015) arXiv:1505.06742
- [8]
- J. S. Frame, “The classes and representations of the groups of 27 lines and 28 bitangents”, Annali di Matematica Pura ed Applicata 32, 83 (1951) DOI
- [9]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [10]
- B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”, (2019) arXiv:1911.05809
- [11]
- A. Roy and S. Suda, “Complex spherical designs and codes”, (2011) arXiv:1104.4692
- [12]
- L. Manivel, “Configurations of lines and models of Lie algebras”, (2005) arXiv:math/0507118
- [13]
- J. H. Conway and N. J. A. Sloane, “The Cell Structures of Certain Lattices”, Miscellanea Mathematica 71 (1991) DOI

## 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