Description
True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Solomon (GRS) codes via either the Hermitian construction [2–4] or the Galois-qudit CSS construction [1,5].
Rate
Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound [6].
Parent
- True Galois-qudit stabilizer code — Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.
Child
Cousins
- Generalized RS (GRS) code
- Quantum maximum-distance-separable (MDS) code — Some Galois-qudit GRS codes are quantum MDS [2].
- Galois-qudit CSS code — Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.
- Hermitian Galois-qudit code — Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.
- Concatenated quantum code — Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound [6].
- Random stabilizer code — Concatenations of quantum GRS codes and random stabilizer codes can achieve the quantum GV bound [6].
- Holographic tensor-network code — Galois-qudit GRS codes can be used to construct holographic p-adic (i.e., tree-tensor-network) codes on Bruhat-Tits trees and buildings and on Drinfeld symmetric spaces [7,8].
- Perfect-tensor code — GRS codes can yield perfect tensors via a generalized Hermitian construction [7,8].
- EA Galois-qudit stabilizer code — Galois-qudit GRS codes can be used to construct EA Galois-qudit stabilizer codes [9,10].
References
- [1]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [2]
- L. Jin and C. Xing, “A Construction of New Quantum MDS Codes”, (2020) arXiv:1311.3009
- [3]
- X. Liu, L. Yu, and H. Liu, “New quantum codes from Hermitian dual-containing codes”, International Journal of Quantum Information 17, 1950006 (2019) DOI
- [4]
- L. Jin et al., “Application of Classical Hermitian Self-Orthogonal MDS Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 56, 4735 (2010) DOI
- [5]
- Z. Li, L.-J. Xing, and X.-M. Wang, “Quantum generalized Reed-Solomon codes: Unified framework for quantum maximum-distance-separable codes”, Physical Review A 77, (2008) arXiv:0812.4514 DOI
- [6]
- Y. Ouyang, “Concatenated Quantum Codes Can Attain the Quantum Gilbert–Varshamov Bound”, IEEE Transactions on Information Theory 60, 3117 (2014) arXiv:1004.1127 DOI
- [7]
- M. Marcolli, “Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces”, (2018) arXiv:1801.09623
- [8]
- M. Heydeman et al., “Nonarchimedean Holographic Entropy from Networks of Perfect Tensors”, (2018) arXiv:1812.04057
- [9]
- K. Guenda, S. Jitman, and T. A. Gulliver, “Constructions of Good Entanglement-Assisted Quantum Error Correcting Codes”, (2016) arXiv:1606.00134
- [10]
- P. J. Nadkarni and S. S. Garani, “Entanglement-assisted Reed–Solomon codes over qudits: theory and architecture”, Quantum Information Processing 20, (2021) DOI
Page edit log
- Victor V. Albert (2022-07-22) — most recent
Cite as:
“Galois-qudit GRS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_grs