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 \tag*{(1)}\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.
Parents
Children
- \([[2m,2m-2,2]]\) error-detecting code — The \([[2m,2m-2,2]]\) error-detecting code is one of the two qubit quantum MDS [1].
- Five-qubit perfect code — The five-qubit code is one of the two qubit quantum MDS codes.
- Three-qutrit code — The three-qutrit code is the smallest nontrivial quantum MDS code.
Cousins
- Maximum distance separable (MDS) code
- Cyclic linear \(q\)-ary code — Quantum MDS codes can be constructed from \(q\)-ary cyclic codes using the Hermitian construction [2].
- Quantum Reed-Solomon code — A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
- True Galois-qudit stabilizer code — Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [1][3][4][5], in particular from cyclic [2], constacyclic [6][7] and negacyclic [8] codes.
- Singleton-bound approaching AQECC — Singleton-bound approaching AQECCs saturate the quantum Singleton bound.
- Galois-qudit GRS code — Some Galois-qudit GRS codes are quantum MDS [9].
- Galois-qudit RS code — A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
References
- [1]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [2]
- G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
- [3]
- 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
- [4]
- X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
- [5]
- L. Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
- [6]
- X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
- [7]
- B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
- [8]
- X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
- [9]
- L. Jin and C. Xing, “A Construction of New Quantum MDS Codes”, (2020) arXiv:1311.3009
Page edit log
- Victor V. Albert (2022-07-22) — most recent
- Victor V. Albert (2022-01-10)
- Qingfeng (Kee) Wang (2021-12-20)
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