\(((5+2r,3\times 2^{2r+1},2))\) Rains code

\(((5+2r,3\times 2^{2r+1},2))\) Rains code[1]

Member of a family of \(((5+2r,3\times 2^{2r+1},2))\) CWS codes; see also [3,4][2; Exam. 8].

Smolin-Smith-Wehner (SSW) code— The SSW code outperforms the Rains codes in terms of code parameters at odd \(n > 11\) [5,6].

Parents

The \(((5+2r,3\times 2^{2r+1},2))\) qubit code family is a CWS family whose graph state is the union of the ring and Bell-pair graphs [5,6].

\(((5+2r,3\times 2^{2r+1},2))\) Rains code

Children

The \(((5,6,2))\) code is the smallest nontrivial Rains code [7] (see also [8][2; Exam. 8]).

[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]: 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
[3]: K. Feng and C. Xing, “A new construction of quantum error-correcting codes”, Transactions of the American Mathematical Society 360, 2007 (2007) DOI
[4]: A. Rigby, J. C. Olivier, and P. Jarvis, “Heuristic construction of codeword stabilized codes”, Physical Review A 100, (2019) arXiv:1907.04537 DOI
[5]: 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
[6]: Cross, Andrew William. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. Diss. Massachusetts Institute of Technology, 2008.
[7]: E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
[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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“\(((5+2r,3\times 2^{2r+1},2))\) Rains code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/rains