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Subsystem Galois-qudit code

Alternative names: Gauge Galois-qudit code.
Root code for the Galois-qudit Kingdom

Description

Subsystem QECC encoding into a \(q^n\)-dimensional Hilbert space consisting of \(n\) Galois qudits.

Cousin

  • Galois-qudit code— Subsystem Galois-qudit codes reduce to (subspace) Galois-qudit codes when there is no gauge subsystem.

Member of code lists

Primary Hierarchy

Quantum Domain
Galois-qudit Kingdom
Parents
Subsystem group-based quantum codeSubsystem QECC Quantum
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [1]; see Sec. 5.3 of Ref. [2]. Interpreted this way, subsystem Galois-qudit codes are subsystem group quantum codes whose physical spaces are constructed using Galois fields \(\mathbb{F}_q\) as groups. More general versions of such qudits can be valued in a Galois ring [3], over which there also exists a Fourier transform [4].
Subsystem Galois-qudit code
Children
Subsystem qubit code
Subsystem Galois-qudit quantum codes for \(q=2\) correspond to subsystem qubit codes.
Subsystem Galois-qudit stabilizer code

References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[2]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
[3]
R. Zucchini, “Calibrated hypergraph states: II calibrated hypergraph state construction and applications”, (2025) arXiv:2501.18968
[4]
Y. Zhang, “Quantum Fourier Transform Over Galois Rings”, (2009) arXiv:0904.2560
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Zoo Code ID: subsystem_galois_into_galois

Cite as:
“Subsystem Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/subsystem_galois_into_galois
BibTeX:
@incollection{eczoo_subsystem_galois_into_galois, title={Subsystem Galois-qudit code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/subsystem_galois_into_galois} }
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Permanent link:
https://errorcorrectionzoo.org/c/subsystem_galois_into_galois

Cite as:

“Subsystem Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/subsystem_galois_into_galois

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