Tenfold code[1; Prop. V.1] 

Description

A \([[10,1,4]]_{G}\) group code for finite Abelian \(G\). The code is defined using a graph that is closely related to the \([[5,1,3]]\) code.

Parents

Cousins

  • \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code — The \([[10,1,4]]_{G}\) Abelian group code for \(G=\mathbb{Z}_q\) is defined using a graph that is closely related to the \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code [1].
  • \([[5,1,3]]_q\) Galois-qudit code — The \([[10,1,4]]_{G}\) Abelian group code for \(G=GF(q)\) is defined using a graph that is closely related to the \([[5,1,3]]_{q}\) Galois-qudit code [1].
  • Five-qubit perfect code — The \([[10,1,4]]_{G}\) Abelian group code for \(G=\mathbb{Z}_2\) is defined using a graph that is closely related to the \([[5,1,3]]\) five-qudit code [1]. The former code can be obtained by converting the latter into a code that is oblivious to collective \(Z\)-type rotations [2; Exam. 6].

References

[1]
D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
[2]
J. Hu et al., “Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers”, IEEE Transactions on Information Theory 68, 1795 (2022) arXiv:2011.00197 DOI
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Zoo Code ID: group_10_1_4

Cite as:
“Tenfold code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_10_1_4
BibTeX:
@incollection{eczoo_group_10_1_4, title={Tenfold code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_10_1_4} }
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Cite as:

“Tenfold code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/group_10_1_4

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/small/group_10_1_4.yml.