Abstract
Jacket matrices motivated by the center weight Hadamard matrices have played an important role in signal processing, communications, image compression, cryptography, etc. In this paper, we suggest a design approach for the Pauli block jacket matrix achieved by substituting some Pauli matrices for all elements of common matrices. Since, the well-known Pauli matrices have been widely utilized for quantum information processing, the large-order Pauli block jacket matrix that contains commutative row operations are investigated in detail. After that some special Abelian groups are elegantly generated from any independent rows of the yielded Pauli block jacket matrix. Finally, we show how the Pauli block jacket matrix can simplify the coding theory of quantum error-correction. The quantum codes we provide do not require the dual-containing constraint necessary for the standard quantum error-correction codes, thus allowing us to construct quantum codes of the large codeword length. The proposed codes can be constructed structurally by using the stabilizer formalism of Abelian groups whose generators are selected from the row operations of the Pauli block jacket matrix, and hence have advantages of being fast constructed with the asymptotically good behaviors.
Article PDF
Similar content being viewed by others
References
Ahmed N., Rao K.R.: Orthogonal Transforms for Digital Signal Processing. Springer, Berlin (1975)
Lee M.H., Kaveh M.: Fast Hadamard transform based on a simple matrix factorization. IEEE Trans. Acoust. Speech Signal Process. 34(6), 1666–1667 (1986)
Lee M.H.: The center weighted Hardamard transform. IEEE Trans. Circuits Syst. CAS-36, 1247–1249 (1989)
Rao K.Y., Hershey J.E.: Hadamard Matrix Analysis and Synthesis. Kluwer, Norwell (1997)
Lee M.H.: A new reverse jacket transform and its fast algorithm. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 47(1), 39–47 (2000)
Lee M.H., Rajan B.S., Park J.Y.: A generalized reverse jacket transform. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 48(7), 684–691 (2001)
Chen Z., Lee M.H., Zeng G.: Fast cocyclic jacket transform. IEEE Trans. Signal Process. 56(5), 2143–2148 (2008)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error Correcting Codes. Elsevier, Amsterdam (1988)
Zeng G., Lee M.H.: A generalized reverse block jacket transform. IEEE Trans. Circuits Syst. I 55(6), 1589–1600 (2008)
Song, W., Lee, M.H.: Orthogonal space-time block codes design using jacket transform for MIMO transmission system. In: IEEE International Conference on Communications (ICC 2008), Beijing, China (2008)
Lee M.H., Hou J.: Fast block inverse jacket transform. IEEE Signal Process. Lett. 13(8), 461–464 (2006)
Lee M.H., Finlayson K.: A Simple Element Inverse Jacket Transform Coding. IEEE ITW, New Zealand (2005)
Lee M.H., Zeng G.: Family of fast jacket transform algorithms. Elect. Lett. 43(11), 651 (2007)
Hottinen A., Tirkkonen O., Wichman R.: Multi-Antenna Transceiver Techniques for 3G and Beyond. Wiley, New York (2003)
Nielsen M.A., Chuang I.L.: Quantum computation and quantum information. Cambrige University Press, Cambridge (2002)
Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error-correction via codes over GF(4). IEEE Trans. Inform. Theory 44, 1369–1387 (1998)
Steane A.M.: Enlagement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inform. Theory 45(7), 2492–2495 (1999)
MacKay D.J.C., Mitchison G.J., McFadden P.L.: Sparse-graph codes for quantum error correction. IEEE Trans. Inform. Theory 50, 2315–2330 (2004)
Matsumoto R.: Improvement of Ashikhmin–Litsyn–Tsfasman bound for quantum codes. IEEE Trans. Inform. Theory 48, 2122–2125 (2002)
Djordjevic I.B.: Quantum LDPC codes from balanced incomplete block designs. IEEE Commun. Lett. 12(5), 389–391 (2008)
Acknowledgments
The authors are grateful to the anonymous referees for their detailed suggestions. This work was supported by Natural Science Foundation of Hunan Province (Nos. 07JJ3128, 2008RS4016), Postdoctoral Science Foundation of China (Nos. 20070420184, 200801341), and in part by Joint Project KOSEF/NSFC Korea Research Foundation KRF-2007-521-D00330, Chonbuk National University, Korea.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article can be found at http://dx.doi.org/10.1007/s11128-009-0160-7
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Guo, Y., Peng, J. & Lee, M.H. Fast quantum codes based on Pauli block jacket matrices. Quantum Inf Process 8, 361–378 (2009). https://doi.org/10.1007/s11128-009-0113-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-009-0113-1