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.
Parents
Children
- \([[13,1,5]]\) cyclic code
- Five-qubit perfect code — The \([[5,1,3]]\) code is the smallest qubit Frobenius code [1; Table I].
References
- [1]
- S. Dutta and P. P. Kurur, “Quantum Cyclic Code of length dividing \(p^{t}+1\)”, (2011) arXiv:1011.5814
Page edit log
- En-Jui Kuo (2024-03-18) — most recent
- Victor V. Albert (2024-03-18)
- Nolan Coble (2021-12-03)
Cite as:
“Frobenius code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/frobenius