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\(((2m+1,3 \times 2^{2m-3},2))\) Rains code[1]

Description

Member of a family of pure odd-length distance-two CWS codes with parameters \(((2m+1,3 \times 2^{2m-3},2))\) for all \(m \geq 2\), constructed recursively from the \(((5,6,2))\) code [4,5][2; Lem. 5 and Thm. 4][3; Exam. 8].

Cousin

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Codeword stabilized (CWS) codeQECC Quantum
Parents
Codeword stabilized (CWS) code
The \(((2m+1,3 \times 2^{2m-3},2))\) qubit code family is a CWS family whose graph state is the union of the ring and Bell-pair graphs [6,7].
Small-distance block quantum codeQECC Quantum
\(((2m+1,3 \times 2^{2m-3},2))\) Rains code
Children
\(((5,6,2))\) qubit code
The \(((5,6,2))\) code is the smallest nontrivial Rains code [2] (see also [8][3; Exam. 8]).

References

[1]
E. M. Rains, R. H. Hardin, P. W. Shor, and N. J. A. Sloane, “A Nonadditive Quantum Code”, Physical Review Letters 79, 953 (1997) arXiv:quant-ph/9703002 DOI
[2]
E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[3]
V. Aggarwal and A. R. Calderbank, “Boolean Functions, Projection Operators, and Quantum Error Correcting Codes”, IEEE Transactions on Information Theory 54, 1700 (2008) arXiv:cs/0610159 DOI
[4]
K. Feng and C. Xing, “A new construction of quantum error-correcting codes”, Transactions of the American Mathematical Society 360, 2007 (2007) DOI
[5]
A. Rigby, J. C. Olivier, and P. Jarvis, “Heuristic construction of codeword stabilized codes”, Physical Review A 100, (2019) arXiv:1907.04537 DOI
[6]
A. Cross, G. Smith, J. A. Smolin, and B. Zeng, “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
[7]
A. W. Cross. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. PhD thesis, Massachusetts Institute of Technology, 2008
[8]
D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh, “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
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Zoo Code ID: rains

Cite as:
“\(((2m+1,3 \times 2^{2m-3},2))\) Rains code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rains
BibTeX:
@incollection{eczoo_rains, title={\(((2m+1,3 \times 2^{2m-3},2))\) Rains code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/rains} }
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Cite as:

“\(((2m+1,3 \times 2^{2m-3},2))\) Rains code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rains

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/union/cws/rains.yml.