Description
Spherical code obtained by taking vectors from a fixed shell of the \(E_6\) lattice and normalizing them to lie on the unit sphere.
The minimal shell of the lattice yields the \((6,72,1)\) code, whose codewords form the vertices of the \(1_{22}\) polytope and the rectified Hessian polyhedron.
Cousins
- \(E_6\) root lattice— The \(E_6\) lattice-shell code is obtained from a shell of the \(E_6\) lattice.
- Sharp configuration— The 36 antipodal pairs of the smallest \(E_6\) lattice shell form a sharp configuration in \(\mathbb{R}P^5\) [1].
- \(t\)-design— The 36 antipodal pairs of the smallest \(E_6\) lattice shell form a 2-design in \(\mathbb{R}P^5\) [1].
- Real projective space code— The 36 antipodal pairs of the smallest \(E_6\) lattice shell form a sharp configuration and a 2-design in \(\mathbb{R}P^5\) [1].
- Hessian polyhedron code— Double Hessian polyhedron codewords form the minimal lattice-shell code of the \(E_6^{\perp}\) lattice [2].
Member of code lists
Primary Hierarchy
Parents
\(E_6\) lattice-shell code
Children
Rectified Hessian polyhedron codewords form the minimal shell of the \(E_6\) lattice.
References
- [1]
- 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
- [2]
- 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 (2026-06-08) — most recent
- Victor V. Albert (2022-11-29)
Cite as:
“\(E_6\) lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/esix_shell, arXiv:2606.11484