## Description

Spherical \((6,72,1)\) code whose codewords are the vertices of the rectified Hessian complex polyhedron and the \(1_{22}\) real polytope. Codewords form the minimal lattice-shell code of the \(E_6\) lattice. See [1; pg. 127][2; pg. 126] for realizations of the 72 codewords.

## Parents

- Polytope code
- \(E_6\) lattice-shell code — Rectified Hessian polyhedron codewords form the minimal shell of the \(E_6\) lattice.
- Spherical design — The rectified Hessian polyhedron code forms a spherical 5-design [3].

## Cousins

- 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 [1; 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.
- Dual polytope code — The rectified and double Hessian polyhedra are dual to each other, analogous to the octahedron and cube.

## References

- [1]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- P. de la Harpe and C. Pache, “Spherical designs and finite group representations (some results of E. Bannai)”, European Journal of Combinatorics 25, 213 (2004) DOI

## Page edit log

- Victor V. Albert (2022-11-29) — most recent

## Cite as:

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