Description
An \(((n,K,d))\) code constructed out of \(q\)-dimensional qubits or Galois qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound is satisfied.
In other words, the quantum Singleton \begin{align} K \leq q^{n-2(d-1)} \tag*{(1)}\end{align} becomes an equality. When \(K = q^k\) for some integer \(k\), the above reduces to \(2(d-1) \leq n-k\). Such codes are pure [3]. The length \(n\) of a quantum MSD code with distance \(d \geq 3\) is bounded by the qudit dimension, \(n \leq q^2 + d - 2\) [4].
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. [5] for an overview of quantum MDS codes.
Parents
Children
- \([[2m,2m-2,2]]\) error-detecting code — The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes [6].
- 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.
- \([[6,2,3]]_{q}\) code
- \([[7,3,3]]_{q}\) 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 [7].
- Hermitian Galois-qudit code — Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [6,8–10], in particular from cyclic [7], constacyclic [10–12] and negacyclic [13] codes.
- Constacyclic code — Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [6,8–10], in particular from cyclic [7], constacyclic [10–12], and negacyclic [13] codes.
- Generalized RS (GRS) code — Some MDS codes are constructed from cyclic and constacyclic codes [14] which are GRS codes [15,16].
- Skew-cyclic CSS code — Some MDS codes are constructed from cyclic and constacyclic codes using the Galois-qudit CSS construction [17].
- Asymmetric quantum code — An asymmetric Singleton bound and linear programming bounds for asymmetric CSS codes have been formulated [18]. Asymmetric MDS codes have been characterized [19].
- EA MDS code
- 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 [20].
- Galois-qudit RS code — A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
- Galois-qudit quantum RM code — There exists a quantum RM code \([[q, q − 2ν − 2, ν + 2]]_q\) for \( 0\leq v \leq \frac{(q-2)}{2}\) and \([[q^2,q^2-2v-2,v+2]]_q\) for \(0\leq v \leq q-2\). Both these codes satisfy the quantum Singleton bound.
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]
- F. Huber and M. Grassl, “Quantum Codes of Maximal Distance and Highly Entangled Subspaces”, Quantum 4, 284 (2020) arXiv:1907.07733 DOI
- [5]
- A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
- [6]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [7]
- G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
- [8]
- 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
- [9]
- X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
- [10]
- L. Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
- [11]
- X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
- [12]
- B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
- [13]
- X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
- [14]
- M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) (2015) arXiv:1502.05267 DOI
- [15]
- S. Ball, “Grassl–Rötteler cyclic and consta-cyclic MDS codes are generalised Reed–Solomon codes”, Designs, Codes and Cryptography 91, 1685 (2022) DOI
- [16]
- H. Liu and S. Liu, “A class of constacyclic codes are generalized Reed–Solomon codes”, Designs, Codes and Cryptography 91, 4143 (2023) DOI
- [17]
- H. Q. Dinh et al., “A class of skew cyclic codes and application in quantum codes construction”, Discrete Mathematics 344, 112189 (2021) DOI
- [18]
- 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
- [19]
- M. F. EZERMAN et al., “PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION”, International Journal of Quantum Information 11, 1350027 (2013) arXiv:1006.1694 DOI
- [20]
- 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