Alternative names: \(2_{21}\) polytope code, Schläfli configuration.
Description
Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) polytope. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron (a.k.a. diplo-Schläfli) code. The code can be obtained from the Schläfli graph [3; Ch. 9]. The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in \(\mathbb{C}P^3\) [4–8].
See [9][3; Exam. 1.2.5][10; pg. 119] for a real (complex) realization of the 27 codewords.
Realizations
Quantum mechanical SIC-POVMs [11].Notes
See the corresponding Bendwavy database entries for the Hessian complex polyhedron [12] and its real-space embedding as the \(2_{21}\) polytope [13].Cousins
- \(E_6\) root lattice— The 27 Hessian polyhedron codewords are intimately related to the \(E_6\) Lie group [14].
- \(E_6\) lattice-shell code— Double Hessian polyhedron codewords form the minimal lattice-shell code of the \(E_6^{\perp}\) lattice [15].
- \(3_{21}\) polytope code— The Hessian polyhedron code forms the next recursive kissing configuration after the \(3_{21}\) polytope code in the \(E_8\) lattice-shell/Witting polytope sequence [16].
- Complex projective space code— The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in \(\mathbb{C}P^3\) [5–8].
- Witting polytope code— The Hessian polyhedron code forms the next recursive kissing configuration after the \(3_{21}\) polytope code in the \(E_8\) lattice-shell/Witting polytope sequence [16]. The Schläfli graph is a subgraph of the graph formed by the vertices of the Witting polytope [17; Sec. 3.11].
- Rectified 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 [10; 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.
- Hessian QSC— Each codeword of the Hessian QSC is a quantum superposition of vertices of a Hessian complex polyhedron.
Primary Hierarchy
Parents
The \(2_{21}\) polytope is self-dual [4].
Cameron-Goethals-Seidel (CGS) isotropic subspace codeSpherical design Sharp configuration Universally optimal ECC \(t\)-design
The CGS isotropic subspace code for \(q=2\) reduces to the Hessian polytope.
The Hessian polytope code forms a tight spherical 4-design [18; Exam. 7.3]. The double Hessian polytope code forms a spherical 5-design [19].
Hessian polyhedron code
References
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- [2]
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- [3]
- T. Ericson and V. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
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- H. S. M. Coxeter, Regular Complex Polytopes (Cambridge University Press, 1991)
- [11]
- B. C. Stacey, “Sporadic SICs and Exceptional Lie Algebras”, (2019) arXiv:1911.05809
- [12]
- R. Klitzing. “3-3-3-3-3.” Complex Polytopes. bendwavy.org/klitzing/complex/3-3-3-3-3.htm
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- R. Klitzing. “Jak.” Polytopes & their Incidence Matrices. bendwavy.org/klitzing/incmats/jak.htm
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- L. Manivel, “Configurations of lines and models of Lie algebras”, (2005) arXiv:math/0507118
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- J. H. Conway and N. J. A. Sloane, “The Cell Structures of Certain Lattices”, Miscellanea Mathematica 71 (1991) DOI
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- E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
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- A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer Berlin Heidelberg, 1989) DOI
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- A. Roy and S. Suda, “Complex spherical designs and codes”, (2011) arXiv:1104.4692
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- B. Venkov. Réseaux et designs sphériques. In Réseaux euclidiens, designs sphériques et formes modulaires, volume 37 of Monogr. Enseign. Math., pages 10–86. Enseignement Math., Geneva, 2001
Page edit log
- Shubham P. Jain (2026-05-05) — most recent
- Victor V. Albert (2022-11-16)
Cite as:
“Hessian polyhedron code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hessian_polyhedron