# Rectified Hessian polyhedron code

## 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. The code forms a spherical 5-design [1]. See [2; pg. 127][3; 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.

## Cousin

- Hessian polyhedron code — The (rectified) Hessian polyhedron is an analogue of a (octahedron) tetrahedron in 3D complex space [2; pg. 127].

## References

- [1]
- 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
- [2]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [3]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) 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