# Biorthogonal spherical 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. Biorthogonal spherical codes are the only tight spherical 3-designs [1; Tab. 9.3].

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 Mobius-Kantor complex polygon. A quaternion realization of the vertices yields the quaternion group \(Q\). In general, the code makes up the vertices of an \(n\)-orthoplex.

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
- Permutation spherical code

## Child

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

## Cousins

- Binary antipodal code — An RM\((1,m)\) code maps to a \((2^m,2^{m+1})\) biorthogonal signal set under the antipodal mapping [3][4; Sec. 6.4]. This set is equivalent to the biorthogonal code since all such codes are unique up to equivalence [1; pg. 19].
- 24-cell code — Vertices of a 24-cell can be split up into vertices of three 16-cells, which are biorthogonal spherical codes for \(n=4\) [5].
- Hadamard code — The augmented Hadamard code maps to a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping [3][4; Sec. 6.4][1; pg. 19].
- Reed-Muller (RM) code — The RM\((1,m)\) maps to a \((2^m,2^{m+1})\) biorthogonal spherical code under the antipodal mapping [3][4; Sec. 6.4][1; pg. 19].
- 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]
- G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels”, IEEE Transactions on Information Theory 44, 2384 (1998) DOI
- [4]
- Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.
- [5]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.

## 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