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, i.e., the 72 roots of \(E_6\) after normalization to the unit sphere. See [1; pg. 127][2; pg. 126] for realizations of the 72 codewords.Cousins
- Hessian polyhedron code— The Hessian and rectified Hessian polyhedra are analogues of the tetrahedron and octahedron 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 from two tetrahedra.
- Dual polytope code— The rectified and double Hessian polyhedra are dual to each other, analogous to the octahedron and cube.
- \(E_6\) root lattice— The Voronoi cell of the \(E_6\) root lattice is the dual of the Gosset \(1_{22}\) polytope [2; Ch. 21, pg. 465].
Member of code lists
Primary Hierarchy
Parents
Rectified Hessian polyhedron codewords form the minimal shell of the \(E_6\) lattice.
The rectified Hessian polyhedron code forms a spherical 5-design [3].
Rectified Hessian polyhedron code
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