Simplex spherical code
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. For example, the spherical simplex code in \(n=3\) makes up the vertices of a tetrahedron. In general, the code makes up the vertices of an \(n\)-simplex. See [1; Sec. 7.7] for a parameterization.
Simplex spherical codes are the only tight spherical 2-designs [1; Tab. 9.3].
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].
Parents
- Polytope code — Simplex spherical codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a triangle (tetrahedron, 5-cell, \(n\)-simplex).
- Spherical sharp configuration
- Permutation spherical code
Cousins
- Binary antipodal code — A binary simplex code (also known as a shortened Hadamard code) to a \((2^m,2^m+1)\) simplex signal set under the antipodal mapping [3; Sec. 6.5.2]. This set is equivalent to the simplex code since all such codes are unique up to equivalence [1; pg. 18]. Simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.
- Icosahedron code — Vertices of a dodecahedron can be split up into vertices of five tetrahedra, which are simplex spherical codes for \(n=3\) [4].
- Hadamard code — The shortened Hadamard code maps to a \((2^m,2^m+1)\) simplex spherical code under the antipodal mapping [3; Sec. 6.5.2][1; pg. 18].
- Simplex code — Binary simplex codes map to simplex spherical codes under the antipodal mapping [3; 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].
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]
- Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
- [4]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
Page edit log
- Victor V. Albert (2022-11-15) — most recent
Cite as:
“Simplex spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/simplex_spherical