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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. See [1; pg. 127][2; pg. 126] for realizations of the 72 codewords.

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.

Member of code lists

Primary Hierarchy

Classical Domain
Spherical Kingdom
Constant-energy codeECC
Spherical code
Slepian group-orbit codeECC
Polytope code
Parents
Polytope code
\(E_6\) lattice-shell codeLattice-shell ECC
Rectified Hessian polyhedron codewords form the minimal shell of the \(E_6\) lattice.
Spherical design\(t\)-design ECC
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
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Zoo Code ID: rect_hessian_polyhedron

Cite as:
“Rectified Hessian polyhedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/rect_hessian_polyhedron
BibTeX:
@incollection{eczoo_rect_hessian_polyhedron, title={Rectified Hessian polyhedron code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rect_hessian_polyhedron} }
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Permanent link:
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/polytope/6-8d/rect_hessian_polyhedron.yml.