\([[5,1,3]]_q\) Galois-qudit code[1]
Description
True stabilizer code that generalizes the five-qubit perfect code to Galois qudits of prime-power dimension \(q=p^m\). It has \(4(m-1)\) stabilizer generators expressed as \(X_{\gamma} Z_{\gamma} Z_{-\gamma} X_{-\gamma} I\) and its cyclic permutations, with \(\gamma\) iterating over basis elements of \(GF(q)\) over \(GF(p)\).
Notes
This code is described in a talk by Gottesman.
Parents
- True Galois-qudit stabilizer code
- Graph quantum code — The \([[5,1,3]]_q\) code is equivalent via a single-Galois-qudit Clifford circuit to a graph quantum code for the group \(G=GF(q)\) [2].
- Cyclic quantum code
- Small-distance block quantum code
Child
- Five-qubit perfect code — The \([[5,1,3]]_q\) Galois-qudit code for \(q=2\) reduces to the five-qubit perfect code.
Cousin
- \([[10,1,4]]_{G}\) tenfold code — The \([[10,1,4]]_{G}\) Abelian group code for \(G=GF(q)\) is defined using a graph that is closely related to the \([[5,1,3]]_{q}\) Galois-qudit code [2].
References
- [1]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [2]
- D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
Page edit log
- Sarah Meng Li (2022-02-21) — most recent
- Victor V. Albert (2022-02-21)
- Victor V. Albert (2023-01-14)
Cite as:
“\([[5,1,3]]_q\) Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_5_1_3