## 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

## Parents

- Polytope code — Biorthogonal spherical codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a square (octahedron, 16-cell, \(n\)-orthoplex).
- Spherical sharp configuration
- Spherical design — Biorthogonal spherical codes are the only tight spherical 3-designs [1; Tab. 9.3].
- Lattice-shell code — Biorthogonal codewords form the minimal shell of the \(\mathbb{Z}^n\) hypercubic lattice.
- Permutation spherical code

## Child

- Quadrature PSK (QPSK) code — The QPSK is equivalent to the biorthogonal spherical code for \(n=2\).

## 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 \(n=4\) [3].
- \(\mathbb{Z}^n\) hypercubic lattice code — Biorthogonal spherical codewords form the minimal shell of the \(\mathbb{Z}^n\) hypercubic lattice.
- 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 [4][5; Sec. 6.4][1; pg. 19]. In other words, first-order RM (biorthogonal spherical) codes form orthoplexes in Hamming (Euclidean) space.
- \([2^m,m+1,2^{m-1}]\) First-order RM code — Each first-order RM code maps to a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping [4][5; Sec. 6.4][1; pg. 19]. In other words, first-order RM (biorthogonal spherical) codes form orthoplexes in Hamming (Euclidean) space.
- Hypercube code — Orthoplexes and hypercubes are dual to each other.
- Polyphase code — Biorthogonal spherical codes for dimension \(n=p\) 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]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [4]
- G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels”, IEEE Transactions on Information Theory 44, 2384 (1998) DOI
- [5]
- Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.

## 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