Alternative names: Cross polytope code, Hyperoctahedron code, Orthoplex code, Co-cube code.
Description
Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. The code makes up the vertices of an \(n\)-orthoplex (a.k.a. hyperoctahedron or cross polytope).
For \(n=3\), biorthogonal spherical codewords make up the vertices of an octahedron. For \(n=4\), codewords make up the vertices of a 16-cell, or, equivalently, the Möbius-Kantor complex polygon. A quaternion realization of the vertices yields the quaternion group \(Q\).
The set of permutations of \((0,0,\cdots,0,1)\) forms an orthogonal set and yields the biorthogonal code when combined with the set of permutations of \((0,0,\cdots,0,-1)\).
Protection
Biorthogonal spherical codes saturate the absolute bound for antipodal codes and the third Rankin bound [1]. Biorthogonal codes are unique up to equivalence [1; pg. 19], which follows from saturating the Boroczky bound [2]. The octahedron is the optimal antipodal configuration of 6 points in 3D space [3].Cousins
- 24-cell code— Vertices of a 24-cell can be split up into vertices of three 16-cells, which are biorthogonal spherical codes for \(n=4\) [4]. The vertices of a 24-cell are a union of the vertices of a tesseract and a 16-cell [5; Exam. 2.6].
- Dual polytope code— Orthoplexes and hypercubes are dual to each other.
- Hypercube code— Orthoplexes and hypercubes are dual to each other. The weighted union of the vertices of a hypercube and an orthoplex form a weighted spherical 5-design in dimensions \(\geq 3\) [6; Sec. 8.6, Ex. 5-2][5; Exam. 2.6].
- Binary antipodal code— Each first-order RM\((1,m)\) code maps to a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping [7][8; Sec. 6.4][1; pg. 19]. In other words, first-order RM (biorthogonal spherical) codes form orthoplexes in Hamming (Euclidean) space.
- Antiprism code— The antiprism reduces to the octahedron for \(q=3\).
- \([2^m,m+1,2^{m-1}]\) First-order RM code— Each first-order RM code maps to a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping [7][8; Sec. 6.4][1; pg. 19]. In other words, first-order RM (biorthogonal spherical) codes form orthoplexes in Hamming (Euclidean) space.
- Rhombic dodecahedron code— The vertices of a rhombic dodecahedron are a union of the vertices of a cube and an octahedron.
- Kerdock spherical code— Kerdock spherical codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) orthoplexes [9].
- Polyphase code— Biorthogonal spherical codes for dimension \(n=p\) with \(p\) an odd prime admit a polyphase realization [1; Sec. 7.7].
- PPM c-q code— Each signal of a PPM c-q code defines a different dimension, so PPM c-q codewords are c-q spherical codes lying on the vertices of a cross polytope.
Member of code lists
Primary Hierarchy
Parents
Biorthogonal spherical codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a square (octahedron, 16-cell, \(n\)-orthoplex).
Spherical sharp configurationSpherical design Sharp configuration Universally optimal ECC \(t\)-design
Biorthogonal spherical codes are the only tight spherical 3-designs [1; Tab. 9.3]. The weighted union of the vertices of a hypercube and an orthoplex form a weighted spherical 5-design in dimensions \(\geq 3\) [6; Sec. 8.6, Ex. 5-2][5; Exam. 2.6][5; Exam. 2.6].
Biorthogonal codewords form the minimal shell of the \(\mathbb{Z}^n\) hypercubic lattice.
Biorthogonal spherical code
Children
Each signal of a PPM code defines a different dimension, so PPM codewords are spherical codes lying on the vertices of a cross polytope.
The QPSK is equivalent to the biorthogonal spherical code for \(n=2\).
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]
- J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing Lines, Planes, etc.: Packings in Grassmannian Space”, (2002) arXiv:math/0208004
- [4]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [5]
- S. Borodachov, P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Energy bounds for weighted spherical codes and designs via linear programming”, (2024) arXiv:2403.07457
- [6]
- Stroud, Arthur H. Approximate calculation of multiple integrals. Prentice Hall, 1971.
- [7]
- G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels”, IEEE Transactions on Information Theory 44, 2384 (1998) DOI
- [8]
- Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
- [9]
- H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
Page edit log
- Victor V. Albert (2022-11-15) — most recent
Cite as:
“Biorthogonal spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/biorthogonal_spherical