## Description

Also known as a binary signal constellation. An \((n,K,4d/n)\) spherical code obtained from a binary \((n,K,d)\) code via a component-wise antipodal mapping (also known as a Euclidean-space image or \(Y_2\) construction) \(0\to +1\) and \(1 \to -1\) [1; Example 1.2.1].

## Parent

## Children

- Binary PSK (BPSK) code — A single-bit binary code yields a spherical \((n,2,4)\) spherical code under the antipodal mapping, which is equivalent to the BPSK code for dimension \(n=2\).
- Kerdock spherical code

## Cousins

- Binary PSK (BPSK) code — A binary antipodal code can be thought of as a concatenation of a binary outer code with a BPSK inner code.
- Binary code — Binary antipodal codes are spherical codes obtained from binary codes.
- Slepian group-orbit code — Any length-\(n\) binary linear code can be used to define a diagonal subgroup of \(n\)-dimensional rotation matrices with \(\pm 1\) on the diagonals via the antipodal mapping \(0\to+1\) and \(1\to-1\). The orbit of this subgroup yields the corresponding Slepian group-orbit code; see [1; Thm. 8.5.2].
- Biorthogonal spherical code — An RM\((1,m)\) code maps to a \((2^m,2^{m+1})\) biorthogonal signal set under the antipodal mapping [2][3; Sec. 6.4]. This set is equivalent to the biorthogonal code since all such codes are unique up to equivalence [1; pg. 19].
- Hypercube code — Binary antipodal codes are subcodes of a hypercube code since the hypercube code corresponds to the Hamming \(n\)-cube embedded into the unit \(n\)-sphere.
- Simplex spherical 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.

## References

- [1]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [2]
- G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels”, IEEE Transactions on Information Theory 44, 2384 (1998) DOI
- [3]
- 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:

“Binary antipodal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_antipodal