Description
Galois-qudit \([[n,1]]_q\) pure CSS code constructed from a dual-containing QR code via the Galois-qudit CSS construction. For \(q\) not divisible by \(n\), its distance satisfies \(d^2-d+1 \geq n\) when \(n \equiv 3\) modulo 4 [2; Thm. 40] and \(d \geq \sqrt{n}\) when \(n\equiv 1\) modulo 4 [2; Thm. 41].
Parents
- Galois-qudit CSS code
- Quantum duadic code — Quantum QR codes are quantum duadic codes since QR codes are duadic codes.
Children
- \([[7,1,3]]\) Steane code — The Steane code is a qubit quantum QR code [3,4].
- Quantum Golay code — The Golay code is a qubit quantum QR code [3,4].
- \([[11,1,5]]_3\) qutrit Golay code — The qutrit Golay code is a qutrit quantum QR code since the ternary Golay code is a QR code.
Cousins
- \(q\)-ary quadratic-residue (QR) code
- Quantum data-syndrome (QDS) code — A family of qubit quantum QR codes can be made into QDS codes [4; Thm. 14].
- Quantum maximum-distance-separable (MDS) code — Almost all quantum QR codes for prime-dimensional qudits are quantum MDS [1; Corr. 11].
References
- [1]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [2]
- A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [3]
- C.-Y. Lai and C.-C. Lu, “A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices”, IEEE Transactions on Information Theory 57, 7163 (2011) arXiv:0712.0103 DOI
- [4]
- A. Ashikhmin, C.-Y. Lai, and T. A. Brun, “Quantum Data-Syndrome Codes”, IEEE Journal on Selected Areas in Communications 38, 449 (2020) arXiv:1907.01393 DOI
Page edit log
- Victor V. Albert (2024-05-05) — most recent
Cite as:
“Quantum quadratic-residue (QR) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_quad_residue