Classical Bose-Chaudhuri-Hocquenghem (BCH) codes C that contain their dual codes can be used to construct quantum stabilizer codes this chapter studies the properties of such codes. It had been shown that a BCH code of length n which contains its dual code satisfies the bound on weight of any non-zero codeword in C and converse is also true. One impressive difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum errorcorrecting codes have been derived as binary stabilizer codes. We were able to shed more light on the structure of dual containing BCH codes. These results make it possible to determine the parameters of quantum BCH codes in terms of weight of non-zero dual codeword.
<p>[1] P. W. Shor, “Scheme for reducing decoherence in quantum computer
memory,” Phys. Rev. A, vol 52, pp. R2493-R2496, October 1995.
[2] A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev.
Lett., vol. 77, pp. 793-797, July 1996.
[3] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting
codes exist,” Phys. Rev. A, vol. 54, pp. 1098-1105, August 1996.
[4] A. Steane, “Multiple particle interference and quantum error correction,”
Proc. Roy. Soc. Lond. A, vol. 452, pp. 2551-2577, November 1996.
[5] E. Knill and R. Laflamme, “A theory of quantum error-correcting
codes,” Phys. Rev. A, vol. 55, pp. 900-911, February 1997.
[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wooters,
“Mixed state entanglement and quantum error correcting codes,” Phys.
Rev. A, vol. 54, pp. 3824-3851, November 1996.
[7] D. Gottesman, “Class of quantum error-correcting codes saturating the
quantum hamming bound,” Phys. Rev. A, vol. 54, pp. 1862-1868,
September 1996.
[8] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. Sloane, “Quantum
error correction and orthogonal geometry,” Phys. Rev. Lett., vol. 78, pp.
405-408, January, 1997.
[9] A. M. Steane, “Enlargement of Calderbank-Shor-Steane quantum
codes,” IEEE Trans. Inf. Theory, vol. 45, pp. 2492-2495, November
1999.
[10] A. Y. Kitaev, “Quantum error correction with imperfect gates,” in Proc.
3rd Int. Conf. of Quantum Communication and Measurement, New
York, May 1997, pp. 181-188.
[11] D. Aharonov and M. Ben-Or, “Fault-tolerant quantum computation with
constant error rate,” in Proc. 29th Ann. ACM Symp. on Theory of
Computing, New York, May 1997, pp.176-188.
[12] E. Knill and R. Laflamme, “Concatenated quantum codes,” quant
ph/9608012, August 1996.
[13] P. W. Shor, “Fault-tolerant quantum computation,” in Proc. 37th FOCS,
Los Alamitos, CA, March 1996, pp. 56-65.
[14] J. Preskill, “Reliable quantum computers,” Proc. R. Soc. Lond. A, pp.
454-385, August 1997.
[15] D. Gottesman, “A theory of fault-tolerant quantum computation,” Phys.
Rev. A, vol. 57, pp. 127-137, January 1998.
[16] A. M. Steane, “Efficient fault-tolerant quantum computing,” Nature, vol.
399, pp.124-126, May 1999.
[17] D. Gottesman, “Fault-tolerant quantum computation with local gates,” J.
Modern Optics, vol. 47, pp. 333-345, February 2000.
[18] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane,
“Quantum error correction via codes over GF (4),” IEEE Trans. Inform.
Theory, vol. 44, pp. 1369–1387, July 1998.
[19] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect quantum
error-correcting code,” Phys. Rev. Lett., vol. 77, pp. 198–201, July
1996.
[20] F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting
Codes. Amsterdam, the Netherlands: Elsevier, 1977.
[21] E. M. Rains, “Nonbinary quantum codes,” IEEE Trans. Inform. Theory,
vol. 45, pp. 1827–1832, Sept. 1999.
[22] A. M. Steane, “Simple quantum error correcting codes,” Phys. Rev.
Lett., vol. 77, pp. 793–797, 1996.
[23] G. Cohen, S. Encheva and S. Litsyn, “On Binary Construction of
Quantum Codes,” IEEE Trans. Inf. Theory, vol. 45, pp. 2495-2498,
November 1999.
[24] M. Grassl and T. Beth, “Quantum BCH codes,” in Proc. X. Int. Symp.
Theoret. Elec. Eng., Magdeburg, 1999, pp. 207–212.
[25] M. Grassl, T. Beth, and T. Pellizzari, “Codes for the quantum erasure
channel,” Phys. Rev. Lett. A, vol. 56, no. 1, pp. 33–38, 1997.
[26] Salah A. Aly, A. Klappenecker, and P. K. Sarvepalli, “On Quantum and
Classical BCH Codes,” IEEE Trans. Inf. Theory, vol. 53, pp. 1183-1188,
2007.
[27] M. Grassl, W. Geiselmann, and T. Beth, “Quantum reed-solomon
codes,” in Proc. AAECC Conf., 1999.
[28] A. Thangaraj, S. W. McLaughlin, “Quantum Codes form Cyclic Codes
over GF(4
<p>[1] P. W. Shor, “Scheme for reducing decoherence in quantum computer
memory,” Phys. Rev. A, vol 52, pp. R2493-R2496, October 1995.
[2] A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev.
Lett., vol. 77, pp. 793-797, July 1996.
[3] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting
codes exist,” Phys. Rev. A, vol. 54, pp. 1098-1105, August 1996.
[4] A. Steane, “Multiple particle interference and quantum error correction,”
Proc. Roy. Soc. Lond. A, vol. 452, pp. 2551-2577, November 1996.
[5] E. Knill and R. Laflamme, “A theory of quantum error-correcting
codes,” Phys. Rev. A, vol. 55, pp. 900-911, February 1997.
[6] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wooters,
“Mixed state entanglement and quantum error correcting codes,” Phys.
Rev. A, vol. 54, pp. 3824-3851, November 1996.
[7] D. Gottesman, “Class of quantum error-correcting codes saturating the
quantum hamming bound,” Phys. Rev. A, vol. 54, pp. 1862-1868,
September 1996.
[8] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. Sloane, “Quantum
error correction and orthogonal geometry,” Phys. Rev. Lett., vol. 78, pp.
405-408, January, 1997.
[9] A. M. Steane, “Enlargement of Calderbank-Shor-Steane quantum
codes,” IEEE Trans. Inf. Theory, vol. 45, pp. 2492-2495, November
1999.
[10] A. Y. Kitaev, “Quantum error correction with imperfect gates,” in Proc.
3rd Int. Conf. of Quantum Communication and Measurement, New
York, May 1997, pp. 181-188.
[11] D. Aharonov and M. Ben-Or, “Fault-tolerant quantum computation with
constant error rate,” in Proc. 29th Ann. ACM Symp. on Theory of
Computing, New York, May 1997, pp.176-188.
[12] E. Knill and R. Laflamme, “Concatenated quantum codes,” quant
ph/9608012, August 1996.
[13] P. W. Shor, “Fault-tolerant quantum computation,” in Proc. 37th FOCS,
Los Alamitos, CA, March 1996, pp. 56-65.
[14] J. Preskill, “Reliable quantum computers,” Proc. R. Soc. Lond. A, pp.
454-385, August 1997.
[15] D. Gottesman, “A theory of fault-tolerant quantum computation,” Phys.
Rev. A, vol. 57, pp. 127-137, January 1998.
[16] A. M. Steane, “Efficient fault-tolerant quantum computing,” Nature, vol.
399, pp.124-126, May 1999.
[17] D. Gottesman, “Fault-tolerant quantum computation with local gates,” J.
Modern Optics, vol. 47, pp. 333-345, February 2000.
[18] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane,
“Quantum error correction via codes over GF (4),” IEEE Trans. Inform.
Theory, vol. 44, pp. 1369–1387, July 1998.
[19] R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, “Perfect quantum
error-correcting code,” Phys. Rev. Lett., vol. 77, pp. 198–201, July
1996.
[20] F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting
Codes. Amsterdam, the Netherlands: Elsevier, 1977.
[21] E. M. Rains, “Nonbinary quantum codes,” IEEE Trans. Inform. Theory,
vol. 45, pp. 1827–1832, Sept. 1999.
[22] A. M. Steane, “Simple quantum error correcting codes,” Phys. Rev.
Lett., vol. 77, pp. 793–797, 1996.
[23] G. Cohen, S. Encheva and S. Litsyn, “On Binary Construction of
Quantum Codes,” IEEE Trans. Inf. Theory, vol. 45, pp. 2495-2498,
November 1999.
[24] M. Grassl and T. Beth, “Quantum BCH codes,” in Proc. X. Int. Symp.
Theoret. Elec. Eng., Magdeburg, 1999, pp. 207–212.
[25] M. Grassl, T. Beth, and T. Pellizzari, “Codes for the quantum erasure
channel,” Phys. Rev. Lett. A, vol. 56, no. 1, pp. 33–38, 1997.
[26] Salah A. Aly, A. Klappenecker, and P. K. Sarvepalli, “On Quantum and
Classical BCH Codes,” IEEE Trans. Inf. Theory, vol. 53, pp. 1183-1188,
2007.
[27] M. Grassl, W. Geiselmann, and T. Beth, “Quantum reed-solomon
codes,” in Proc. AAECC Conf., 1999.
[28] A. Thangaraj, S. W. McLaughlin, “Quantum Codes form Cyclic Codes
over GF(4
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55022", author = "J. S. Bhullar and Manish Gupta", title = "On Quantum BCH Codes and Its Duals", abstract = "Classical Bose-Chaudhuri-Hocquenghem (BCH) codes C that contain their dual codes can be used to construct quantum stabilizer codes this chapter studies the properties of such codes. It had been shown that a BCH code of length n which contains its dual code satisfies the bound on weight of any non-zero codeword in C and converse is also true. One impressive difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum errorcorrecting codes have been derived as binary stabilizer codes. We were able to shed more light on the structure of dual containing BCH codes. These results make it possible to determine the parameters of quantum BCH codes in terms of weight of non-zero dual codeword.
", keywords = "Quantum Codes, BCH Codes, Dual BCH Codes,
Designed Distance.", volume = "7", number = "6", pages = "979-4", }
