\([[5,1,3]]_q\) Galois-qudit code[1] 

Description

True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(GF(q)\) over \(GF(p)\).

Notes

This code is described in a talk by Gottesman.

Parents

Child

  • Five-qubit perfect code — The \([[5,1,3]]_q\) Galois-qudit code for \(q=2\) reduces to the five-qubit perfect code.

Cousin

  • Tenfold 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 [2].

References

[1]
D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[2]
D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
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Zoo Code ID: galois_5_1_3

Cite as:
\([[5,1,3]]_q\) Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_5_1_3
BibTeX:
@incollection{eczoo_galois_5_1_3, title={\([[5,1,3]]_q\) Galois-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_5_1_3} }
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Permanent link:
https://errorcorrectionzoo.org/c/galois_5_1_3

Cite as:

\([[5,1,3]]_q\) Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_5_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/small/galois_5_1_3.yml.