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Quantum maximum-distance-separable (MDS) code[13]

Description

A type of block quantum code whose parameters satisfy the quantum Singleton bound with equality.

An \(((n,K,d))\) code constructed out of \(q\)-dimensional qudits is a quantum MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound [13], \begin{align} K \leq q^{n-2(d-1)} \tag*{(1)}\end{align} becomes an equality for such codes. When \(K = q^k\) for some integer \(k\), the above reduces to \(2(d-1) \leq n-k\). Such codes are pure [3]; see also [4] mentioned in Ref. [5]. The length \(n\) of a quantum MSD code with distance \(d \geq 3\) is bounded by the qudit dimension, \(n \leq q^2 + d - 2\) [5].

Protection

Given \(n\) and \(k\), MDS codes have the highest distance possible of all codes and so have the best possible error correction properties.

Notes

See Ref. [6] for an overview of quantum MDS codes.Tables of quantum MDS codes [5].

Cousins

Primary Hierarchy

Parents
Quantum maximum-distance-separable (MDS) code
Children
The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [8].
The five-qubit code is one of the two qubit quantum MDS codes.
The three-qutrit code is the smallest nontrivial quantum MDS code.

References

[1]
E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters 84, 2525 (2000) arXiv:quant-ph/9604034 DOI
[2]
N. J. Cerf and R. Cleve, “Information-theoretic interpretation of quantum error-correcting codes”, Physical Review A 56, 1721 (1997) arXiv:quant-ph/9702031 DOI
[3]
E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[4]
A. Winter, private communication (2019).
[5]
F. Huber and M. Grassl, “Quantum Codes of Maximal Distance and Highly Entangled Subspaces”, Quantum 4, 284 (2020) arXiv:1907.07733 DOI
[6]
A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
[7]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
[8]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
[9]
R. Li and Z. Xu, “Construction of[[n,n−4,3]]qquantum codes for odd prime powerq”, Physical Review A 82, (2010) arXiv:0906.2509 DOI
[10]
X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
[11]
L. Lu, W. Ma, R. Li, Y. Ma, and L. Guo, “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
[12]
X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
[13]
B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
[14]
X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
[15]
M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) 1104 (2015) arXiv:1502.05267 DOI
[16]
S. Ball, “Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes”, Designs, Codes and Cryptography 91, 1685 (2022) DOI
[17]
H. Liu and S. Liu, “A class of constacyclic codes are generalized Reed–Solomon codes”, Designs, Codes and Cryptography 91, 4143 (2023) DOI
[18]
H. Q. Dinh, T. Bag, A. K. Upadhyay, R. Bandi, and R. Tansuchat, “A class of skew cyclic codes and application in quantum codes construction”, Discrete Mathematics 344, 112189 (2021) DOI
[19]
P. K. Sarvepalli, A. Klappenecker, and M. Rötteler, “Asymmetric quantum codes: constructions, bounds and performance”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 1645 (2009) DOI
[20]
M. F. EZERMAN, S. JITMAN, H. M. KIAH, and S. LING, “PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION”, International Journal of Quantum Information 11, 1350027 (2013) arXiv:1006.1694 DOI
[21]
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
[22]
Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
[23]
D. Alsina and M. Razavi, “Absolutely maximally entangled states, quantum-maximum-distance-separable codes, and quantum repeaters”, Physical Review A 103, (2021) arXiv:1907.11253 DOI
[24]
N. Alon, J. Edmonds, and M. Luby, “Linear time erasure codes with nearly optimal recovery”, Proceedings of IEEE 36th Annual Foundations of Computer Science DOI
[25]
N. Alon and M. Luby, “A linear time erasure-resilient code with nearly optimal recovery”, IEEE Transactions on Information Theory 42, 1732 (1996) DOI
[26]
T. Bergamaschi, L. Golowich, and S. Gunn, “Approaching the Quantum Singleton Bound with Approximate Error Correction”, (2022) arXiv:2212.09935
[27]
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
[28]
L. Jin and C. Xing, “A Construction of New Quantum MDS Codes”, (2020) arXiv:1311.3009
[29]
S. A. Aly and A. Klappenecker, “Subsystem Code Constructions”, (2008) arXiv:0712.4321
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Cite as:

“Quantum maximum-distance-separable (MDS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_mds

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/quantum_mds.yml.