n k t 2 t + 1 d 8 1 1 3 3 10 1 2 5 4 20 1 3 7 7 22 1 4 9 7–8 32 1 5 11 11 34 1 6 13 11–12 48 1 7 15 13–17 50 1 8 17 13–17 56 1 9 19 15–19 58 1 10 21 15–20 8 2 1 3 3 16 2 2 5 6 20 2 3 7 6–7 28 2 4 9 10 32 2 5 11 10–11 46 2 6 13 12–16 52 2 7 15 14–18 54 2 8 17 14–18 56 2 9 19 14–19 82 2 10 21 18–28 n k t 2 t + 1 d 12 3 1 3 4 16 3 2 5 5 24 3 3 7 7–8 30 3 4 9 9–10 40 3 5 11 10–13 48 3 6 13 11–16 52 3 7 15 13–17 54 3 8 17 13–18 72 3 9 19 15–24 82 3 10 21 18–27 12 4 1 3 4 20 4 2 5 6 24 4 3 7 6–8 32 4 4 9 8–10 40 4 5 11 10–13 50 4 6 13 12–16 52 4 7 15 12–17 70 4 8 17 15–23 80 4 9 19 16–26 96 4 10 21 18–31 n k t 2 t + 1 d 16 5 1 3 4–5 22 5 2 5 6–7 28 5 3 7 7–9 36 5 4 9 8–11 42 5 5 11 9–13 50 5 6 13 11–16 60 5 7 15 13–19 78 5 8 17 15–25 86 5 9 19 18–28 98 5 10 21 19–32 16 6 1 3 4 24 6 2 5 6–7 28 6 3 7 6–8 36 6 4 9 8–11 48 6 5 11 10–15 58 6 6 13 12–19 64 6 7 15 14–21 84 6 8 17 17–27 92 6 9 19 18–29 104 6 10 21 19–33 TABLE II COMPARISON OF OUR [[ n, k ]] t -CODES AND THE BOUNDS ON THE MINIMUM DISTANCE d OF A STABILIZER CODE [[ n, k, d ]] . ACKNOWLEDGMENTS We thank Daniel Gottesman and Peter Shor for helpful discussions. RD is partly supported by QCIS, University of Technology, Sydney, and the NSF of China (Grant Nos. 60736011 and 60702080). BZ is supported by NSERC and QuantumWorks. Centre for Quantum Technologies is a Re- search Centre of Excellence funded by Ministry of Education and National Research Foundation of Singapore. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario thought the Ministry of Research & Innovation. REFERENCES [12] M. Grassl, P. W. Shor, G. Smith, J. A. Smolin, and B. Zeng. Generalized Concatenated Quantum Codes. Physical Review A 79(5):050306(R), 2009. [13] M. Grassl, P. W. Shor, and B. Zeng. Generalized Concatenation for Quantum Codes. In Proceedings of the 2009 IEEE International Symposium on Information Theory , pp. 95-3-957, 2009. [14] D. Hu, W. Tang, M. Zhao, Q. Chen, S. Yu, and C. H. Oh. Graph- ical nonbinary quantum error-correcting codes. Physical Review A , 78(1):012306, 2008. [15] L. Ioffe and M. Mezard. Asymmetric quantum error-correcting codes. Physical Review A , 75(3):032345, 2007. [16] E. Knill and R. Laflamme. Theory of quantum error-correc ting codes. Physical Review A , 55(2):900–911, 1997. [17] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek. Perfectquantum error correcting code. Physical Review Letters , 77(1):198–201, 1996. [18] R. Lang and P. W. Shor. Nonadditive quantum error correcting codes adapted to the amplitude damping channel. arXiv:0712.2586 , 2007. [19] D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Yamamoto. Ap- proximate quantum error correction can lead to better codes. Physical Review A , 56(4):2567–2573, 1997. [20] S. Y. Looi, L. Yu, V. Gheorghiu, and R. B. Griffiths. Quant um error correcting codes using qudit graph states. Physical Review A , 78(4):042303, 2008. [21] J. L. Massey. Zero Error. Lecture at Information Theory Winter School 2007 , La Colle sur Loup, France, 2007. Available on-line at http://itwinterschool07.eurecom.fr/Tutorials/Massey Zero error.pdf [22] M. Nielsen and I. Chuang. Quantum computation and quantum infor- mation . Cambridge University Press, Cambridge, England, 2000. [23] P. K. Sarvepalli, M. R¨otteler, and A. Klappenecker. Asymmetric quantum LDPC codes. In Proceedings of the 2008 IEEE International Symposium on Information Theory , pp. 305–309, 2008. [24] P. K. Sarvepalli, A. Klappenecker, and M. R¨otteler. Asymmetric quantum codes: constructions, bounds, and performance, Proceedings of the Royal Society London, Series A , 465(2105):1645–1672, 2009. [25] P. W. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A , 52(4):2493–2496, 1995. [26] P. W. Shor, G. Smith, J. A. Smolin, and B. Zeng. High performance single-error-correcting quantum codes for the amplitude damping chan- nel. arXiv:0907.5149 , 2009. [27] A. Steane. Multiple particle interference and quantum error correction. Proceedings of the Royal Society London, Series A , 452(1954):2551– 2577, 1996. [11] M. Grassl, T. Beth, and T. Pellizzari. Codes for the quantum erasure channel. Physical Review A , 56(1):33–38, 1997. [10] M. Grassl. Tables of quantum error-correcting codes, available on-line at http://www.codetables.de [9] D. Gottesman. Stabilizer Codes and Quantum Error Correction . PhD thesis, California Institute of Technology, Pasadena, USA, 1997. [8] A. S. Fletcher, P. W. Shor, and M. Z. Win. Channel-adapted quantum error correction for the amplitude damping channel. IEEE Transactions on Information Theory , 54(12):5705–5718, 2008. [7] Z. W. E. Evans, A. M. Stephens, J. H. Cole, and L. C. L. Hollen- berg. Error correction optimisation in the presence of x/z asymmetry. arXiv:0709.3875 , 2007. [6] I. L. Chuang, D. W. Leung, and Y. Yamamoto. Bosonic quantum codes for amplitude damping. Physical Review A , 56(2):1114–1125, 1997. [5] A. R. Calderbank and P. W. Shor. Good quantum error-correcting codes exist. Physical Review A , 54(2):1098–1106, 1996. [4] W. Bosma, J. J. Cannon, and C. Playoust. The Magma Algebra System I: The User Language. Journal of Symbolic Computation , 24(3–4):235– 265, 1997. [3] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters. Mixed state entanglement and quantum error correction. Physical Review A , 54(5):3824–3851, 1996. [2] D. Bacon. Operator quantum error-correcting subsystems for self- correcting quantum memories. Physical Review A , 73(1):012340, 2006. [1] P. Aliferis and J. Preskill. Fault-tolerant quantum computation against biased noise. Physical Review A 78(5):052331, 2008.