Description
Spherical \((n,n+1,2+2/n)\) code whose codewords are all permutations of the \(n+1\)-dimensional vector \((1,1,\cdots,1,-n)\), up to normalization, forming an \(n\)-simplex. Codewords are all equidistant and their components add up to zero. Simplex spherical codewords in 2 (3, 4) dimensions form the vertices of a triangle (tetrahedron, 5-cell) In general, the code makes up the vertices of an \(n\)-simplex. The union of a simplex and its antipodal simplex forms the vertices of a bi-simplex, which has \(2(n+1)\) vertices.
See [1; Sec. 7.7] for a parameterization.
Protection
Simplex spherical codes saturate the absolute bound, the Levenshtein bound and, for \(2 < \rho \leq 4\), the first two Rankin bounds [1]. All simplex codes are unique up to equivalence [1; pg. 18], which follows from saturating the Boroczky bound [2].Notes
See the corresponding Bendwavy database entry for the 5-cell [3].Cousins
- Dodecahedron code— Vertices of a dodecahedron can be split up into vertices of five tetrahedra, which are simplex spherical codes for \(n=3\) [4].
- Binary antipodal code— Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the antipodal mapping [5; Sec. 6.5.2][1; pg. 18]. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.
- Antiprism code— The antiprism reduces to the tetrahedron for \(q=2\).
- Simplex integer-based code— Codewords of simplex integer-based codes are restricted to lie in a discrete simplex.
- \([2^m-1,m,2^{m-1}]\) simplex code— Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the antipodal mapping [5; Sec. 6.5.2][1; pg. 18]. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.
- Polyphase code— Simplex spherical codes for dimension \(n=(p-1)/2\) with \(p\) an odd prime admit a polyphase realization [1; Sec. 7.7].
- Petersen spherical code— Codewords of the Petersen spherical code correspond to midpoints of the \(5\)-cell [6].
- Pauli tessellation QSC— Each codeword of the Pauli tessellation QSC is a quantum superposition of vertices of a tetrahedron with \(\pm 1\) coefficients.
- \([[2^r-1,1,3]]\) simplex code— Each \([[2^r-1,1,3]]\) simplex code is a color code whose qubits are placed on the vertices, edges, and faces of an \((r-1)\)-simplex [7,8].
Primary Hierarchy
Parents
The simplex is self-dual.
Spherical sharp configurationSpherical design Sharp configuration Universally optimal ECC \(t\)-design
Simplex spherical codes are the only tight spherical 2-designs [1; Tab. 9.3]. The bi-simplex is a spherical 3-design since antipodal codes have zero averages over odd-degree polynomials.
Simplex spherical code
References
- [1]
- T. Ericson and V. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
- [2]
- K. Boroczky, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hung. 32 (1978), 243–261
- [3]
- R. Klitzing. “Pen.” Polytopes & their Incidence Matrices. bendwavy.org/klitzing/incmats/pen.htm
- [4]
- H. S. M. Coxeter, Regular Polytopes (Courier Corporation, 1973)
- [5]
- G. D. Forney (2003). 6.451 Principles of Digital Communication II, Spring 2003
- [6]
- C. Bachoc and F. Vallentin, “Optimality and uniqueness of the (4,10,1/6) spherical code”, Journal of Combinatorial Theory, Series A 116, 195 (2009) arXiv:0708.3947 DOI
- [7]
- H. Bombin, “Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes”, (2015) arXiv:1311.0879
- [8]
- B. J. Brown, N. H. Nickerson, and D. E. Browne, “Fault-tolerant error correction with the gauge color code”, Nature Communications 7, (2016) arXiv:1503.08217 DOI
Page edit log
- Shubham P. Jain (2026-05-05) — most recent
- Victor V. Albert (2022-11-15)
Cite as:
“Simplex spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/simplex_spherical