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Frobenius code[1]

Description

A cyclic prime-qudit stabilizer code whose length \(n\) divides \(p^t + 1\) for some positive integer \(t\).

Decoding

Adapted from the Berlekamp decoding algorithm for classical BCH codes. There exists a polynomial time quantum algorithm to correct errors of weight at most \(\tau\), where \(\delta=2\tau+1\) is the BCH distance of the code [1].

Notes

Frobenius codes that are also stabilizer codes have been completely classified. No such codes exist when \(t\) is odd.

Member of code lists

Primary Hierarchy

Quantum Domain
Modular-qudit Kingdom
Modular-qudit codeQECC Quantum
Modular-qudit USt code
Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum
Parents
Modular-qudit stabilizer code
Cyclic quantum codeQECC Quantum
Frobenius code
Children
Five-qubit perfect code
The \([[5,1,3]]\) code is the smallest qubit Frobenius code [1; Table I].
\([[13,1,5]]\) cyclic code

References

[1]
S. Dutta and P. P. Kurur, “Quantum Cyclic Code of length dividing \(p^{t}+1\)”, (2011) arXiv:1011.5814
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Zoo Code ID: frobenius

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

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/frobenius.yml.