Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30309
On Quantum BCH Codes and Its Duals

Authors: MANISH GUPTA, J. S. Bhullar

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

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1061956

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1337

References:


[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