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
Page edit log
- Nolan Coble (2021-12-03) — most recent
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.