Quantum maximum-distance-separable (MDS) code

Description

An \(((n,q^k,d))\) code constructed out of \(q\)-dimensional qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound \begin{align} 2(d-1) \leq n-k \end{align} becomes an equality.

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

The five-qubit code and \([[n,n-2,2]]\) codes, where \(n\) is even, are the only examples of MDS qubit codes.

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Cousins

References

[1]
G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011). DOI
[2]
Lingfei Jin and Chaoping Xing, “A Construction of New Quantum MDS Codes”. 1311.3009
[3]
M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004). DOI; quant-ph/0312164
[4]
R. Li and Z. Xu, “Construction of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:msub><mml:mrow /><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>quantum codes for odd prime power<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>”, Physical Review A 82, (2010). DOI; 0906.2509
[5]
Xianmang He, Liqing Xu, and Hao Chen, “New $q$-ary Quantum MDS Codes with Distances Bigger than $\frac{q}{2}$”. 1507.08355
[6]
Liangdong Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”. 1803.07927
[7]
X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014). DOI
[8]
B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015). DOI
[9]
X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013). DOI
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Zoo code information

Internal code ID: quantum_mds

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Zoo Code ID: quantum_mds

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
BibTeX:
@incollection{eczoo_quantum_mds, title={Quantum maximum-distance-separable (MDS) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/quantum_mds} }
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Permanent link:
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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/tree/main/codes/quantum/properties/quantum_mds.yml.