\(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code[1,2] 

Description

Member of a family of \(((2^m,2^{2^m−5m+1},8))\) CSS-like union stabilizer codes constructed using the classical Goethals and Preparata codes.

They can be viewed as union stabilizer codes constructed from the codespace of a \([[2^m,2^m-7m+3,8]]\) code (which itself is obtained from Steane enlargement) and the coset representatives used to obtain the Goethals and Preparata codes [3; Thm. 10.3].

The Goethals and Preparata codes can each be used to obtain families of union stabilizer codes with distance 8 and 6, respectively [1]. A construction using the \(\mathbb{Z}_4\) versions of these codes and the Gray map yields qubit code families with similar parameters [4].

Parent

Cousins

  • Goethals code — The \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code is constructed using the classical Goethals and Preparata codes [1,2]. A construction using the \(\mathbb{Z}_4\) versions of the Goethals and Preparata codes and the Gray map yields qubit code families with similar parameters [4].
  • Preparata code — The \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code is constructed using the classical Goethals and Preparata codes [1,2]. A construction using the \(\mathbb{Z}_4\) versions of the Goethals and Preparata codes and the Gray map yields qubit code families with similar parameters [4].
  • Gray code — A construction using the \(\mathbb{Z}_4\) versions of the Goethals and Preparata codes and the Gray map yields qubit code families with similar parameters as the \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code [4].

References

[1]
M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop (2008) arXiv:0801.2144 DOI
[2]
M. Grassl and M. Rotteler, “Quantum Goethals-Preparata codes”, 2008 IEEE International Symposium on Information Theory (2008) arXiv:0801.2150 DOI
[3]
M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
[4]
S. Ling and P. Sole. 2008. Nonadditive quantum codes from Z4-codes. https://hal.archives-ouvertes.fr/hal-00338309/fr/.
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Zoo Code ID: quantum_goethals_preparata

Cite as:
“\(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_goethals_preparata
BibTeX:
@incollection{eczoo_quantum_goethals_preparata, title={\(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_goethals_preparata} }
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Cite as:

“\(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_goethals_preparata

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/union_stabilizer/quantum_goethals_preparata.yml.