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 \([[5,1,3]]\) code and \([[n,n-2,2]]\) codes, where \(n\) is even, are the only examples of MDS qubit codes.

Parent

Cousins

Zoo code information

Internal code ID: quantum_mds

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on github.com (edit & pull request)

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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} }
Permanent link:
https://errorcorrectionzoo.org/c/quantum_mds

References

[1]
Hualu Liu and Xiusheng Liu, “Constructions of quantum MDS codes”. 2002.06040

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.