Binary antipodal code 

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

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.
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Zoo Code ID: binary_antipodal

Cite as:
“Binary antipodal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_antipodal
BibTeX:
@incollection{eczoo_binary_antipodal, title={Binary antipodal code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/binary_antipodal} }
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“Binary antipodal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_antipodal

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/q-ary/binary_antipodal.yml.