Frobenius code[1]

Description

Let \(C\) be a quantum cyclic code on \(n\) prime-dimensional qudits. \(C\) is a Frobenius code if there exists a positive integer \(t\) such that \(n\) divides \(p^t +1\).

Protection

Protects against Pauli noise.

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. All such codes with even \(t\) can be directly constructed.

Parents

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.), 2021. https://errorcorrectionzoo.org/c/frobenius
BibTeX:
@incollection{eczoo_frobenius, title={Frobenius code}, booktitle={The Error Correction Zoo}, year={2021}, 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.), 2021. https://errorcorrectionzoo.org/c/frobenius

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