# American Institute of Mathematical Sciences

May  2012, 6(2): 237-247. doi: 10.3934/amc.2012.6.237

## Quaternary periodic complementary/Z-complementary sequence sets based on interleaving technique and Gray mapping

 1 College of Communication Engineering, Chongqing University, Chongqing 400044, China, and Chongqing Key Laboratory of Emergency Communication, Chongqing Communication Institute, Chongqing 400035, China 2 College of Communication Engineering, Chongqing University, Chongqing 400044, China 3 Chongqing Key Laboratory of Emergency Communication, Chongqing Communication Institute, Chongqing 400035, China, China

Received  June 2011 Revised  January 2012 Published  April 2012

A family of quaternary periodic complementary sequence (PCS) or Z-complementary sequence (PZCS) sets is presented. By combining an interleaving technique and the inverse Gray mapping, the proposed method transforms the known binary PCS/PZCS sets with odd length of sub-sequences into quaternary PCS/PZCS sets, but the length of new sub-sequences is twice as long as that of the original sub-sequences, which is a drawback of this proposed method. However, the shortcoming that the method proposed by J. W. Jang, et al. is merely fit for even length of sub-sequences is overcome. As a consequence, the union of our and J. W. Jang, et al.'s methods allows us to construct quaternary PCS/PZCS sets from binary PCS/PZCS sets with sub-sequences of arbitrary length.
Citation: Fanxin Zeng, Xiaoping Zeng, Zhenyu Zhang, Guixin Xuan. Quaternary periodic complementary/Z-complementary sequence sets based on interleaving technique and Gray mapping. Advances in Mathematics of Communications, 2012, 6 (2) : 237-247. doi: 10.3934/amc.2012.6.237
##### References:
 [1] R. Appuswamy and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE Trans. Inform. Theory, 52 (2006), 3817-3826. doi: 10.1109/TIT.2006.878171.  Google Scholar [2] L. Bömer and M. Antweiler, Periodic complementary binary sequences, IEEE Trans. Inform. Theory, 36 (1990), 1487-1497. doi: 10.1109/18.59954.  Google Scholar [3] H. H. Chen, D. Hank and M. E. Magañz, Design of next-generation CDMA using orthogonal complementary codes and offset stacked spreading, IEEE Wireless Commun., 2007 (2007), 61-69. doi: 10.1109/MWC.2007.386614.  Google Scholar [4] J. H. Chung and K. Yang, New design of quaternary low-correlation zone sequences sets and quaternary Hadamard matrices, IEEE Trans. Inform. Theory, 54 (2008), 3733-3737. doi: 10.1109/TIT.2008.926406.  Google Scholar [5] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.  Google Scholar [6] D. Ž. Đoković, Note on periodic complementary sets of binary sequences, Des. Codes Cryptogr., 13 (1998), 251-256. doi: 10.1023/A:1008245823233.  Google Scholar [7] D. Ž. Đoković, Periodic complementary sets of binary sequences, Int. Math. Forum, 4 (2009), 717-725.  Google Scholar [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applictions,'' John Wiley & Sons Inc., 1996. Google Scholar [9] P. Z. Fan, W. N. Yuan and Y. F. Tu, Z-complementary binary sequences, IEEE Signal Proc. Letters, 14 (2007), 509-512. doi: 10.1109/LSP.2007.891834.  Google Scholar [10] K. Feng, P. J. S. Shyue and Q. Xiang, On the aperiodic and periodic complementary binary sequences, IEEE Trans. Inform. Theory, 45 (1999), 296-303. doi: 10.1109/18.746823.  Google Scholar [11] R. L. Frank, Polyphase complementary codes, IEEE Trans. Inform. Theory, IT-26 (1980), 641-647. doi: 10.1109/TIT.1980.1056272.  Google Scholar [12] M. J. E. Golay, Complementary series, IEEE Trans. Inform. Theory, IT-7 (1960), 82-87.  Google Scholar [13] S. W. Golomb and G. Gong, "Signal Design for Good Correlation: for Wireless Communications, Cryptography and Radar Applications,'' Cambridge University Press, 2005. doi: 10.1017/CBO9780511546907.  Google Scholar [14] J. W. Jang, Y. S. Kim, S. H. Kim and D. W. Lim, New construction methods of quaternary periodic complementary sequence sets, Adv. Math. Commun., 4 (2010), 61-68. doi: 10.3934/amc.2010.4.61.  Google Scholar [15] S. M. Krone and D. V. Sarwate, Quadriphase sequences for spread spectrum multiple-access communication, IEEE Trans. Inform. Theory, IT-30 (1984), 520-529. doi: 10.1109/TIT.1984.1056913.  Google Scholar [16] M. G. Parker, C. Tellambura and K. G. Paterson, Golay complementary sequences, in "Wiley Encyclopedia of Telecommunicatins'' (ed. J.G. Proakis), Wiley, 2003. doi: 10.1002/0471219282.eot367.  Google Scholar [17] R. Sivaswamy, Multiphase complementary codes, IEEE Trans. Inform. Theory, IT-24 (1978), 545-552. Google Scholar [18] N. Suehiro and M. Hatori, N-shift cross-orthogonal sequences, IEEE Trans. Inform. Theory, 34 (1988), 143-146. doi: 10.1109/18.2615.  Google Scholar [19] S. M. Tseng and M. R. Bell, Asynchronous multicarrier DS-CDMA using mutually orthogonal complementary sets of sequences, IEEE Trans. Commun., 48 (2000), 53-59. doi: 10.1109/26.818873.  Google Scholar [20] C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inform. Theory, IT-18 (1972), 644-652. doi: 10.1109/TIT.1972.1054860.  Google Scholar [21] Y. F. Tu, P. Z. Fan, L. Hao and X. H. Tang, A simple method for generating optimal Z-periodic complementary Sequence set based on phase shift, IEEE Signal Proc. Letters, 17 (2010), 891-893. doi: 10.1109/LSP.2010.2068288.  Google Scholar [22] Z. Y. Zhang, F. X. Zeng and G. X. Xuan, A class of complementary sequences with multi-width zero cross-correlation zone, IEICE Trans. Fundam., E93-A (2010), 1508-1517. Google Scholar

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##### References:
 [1] R. Appuswamy and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE Trans. Inform. Theory, 52 (2006), 3817-3826. doi: 10.1109/TIT.2006.878171.  Google Scholar [2] L. Bömer and M. Antweiler, Periodic complementary binary sequences, IEEE Trans. Inform. Theory, 36 (1990), 1487-1497. doi: 10.1109/18.59954.  Google Scholar [3] H. H. Chen, D. Hank and M. E. Magañz, Design of next-generation CDMA using orthogonal complementary codes and offset stacked spreading, IEEE Wireless Commun., 2007 (2007), 61-69. doi: 10.1109/MWC.2007.386614.  Google Scholar [4] J. H. Chung and K. Yang, New design of quaternary low-correlation zone sequences sets and quaternary Hadamard matrices, IEEE Trans. Inform. Theory, 54 (2008), 3733-3737. doi: 10.1109/TIT.2008.926406.  Google Scholar [5] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.  Google Scholar [6] D. Ž. Đoković, Note on periodic complementary sets of binary sequences, Des. Codes Cryptogr., 13 (1998), 251-256. doi: 10.1023/A:1008245823233.  Google Scholar [7] D. Ž. Đoković, Periodic complementary sets of binary sequences, Int. Math. Forum, 4 (2009), 717-725.  Google Scholar [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applictions,'' John Wiley & Sons Inc., 1996. Google Scholar [9] P. Z. Fan, W. N. Yuan and Y. F. Tu, Z-complementary binary sequences, IEEE Signal Proc. Letters, 14 (2007), 509-512. doi: 10.1109/LSP.2007.891834.  Google Scholar [10] K. Feng, P. J. S. Shyue and Q. Xiang, On the aperiodic and periodic complementary binary sequences, IEEE Trans. Inform. Theory, 45 (1999), 296-303. doi: 10.1109/18.746823.  Google Scholar [11] R. L. Frank, Polyphase complementary codes, IEEE Trans. Inform. Theory, IT-26 (1980), 641-647. doi: 10.1109/TIT.1980.1056272.  Google Scholar [12] M. J. E. Golay, Complementary series, IEEE Trans. Inform. Theory, IT-7 (1960), 82-87.  Google Scholar [13] S. W. Golomb and G. Gong, "Signal Design for Good Correlation: for Wireless Communications, Cryptography and Radar Applications,'' Cambridge University Press, 2005. doi: 10.1017/CBO9780511546907.  Google Scholar [14] J. W. Jang, Y. S. Kim, S. H. Kim and D. W. Lim, New construction methods of quaternary periodic complementary sequence sets, Adv. Math. Commun., 4 (2010), 61-68. doi: 10.3934/amc.2010.4.61.  Google Scholar [15] S. M. Krone and D. V. Sarwate, Quadriphase sequences for spread spectrum multiple-access communication, IEEE Trans. Inform. Theory, IT-30 (1984), 520-529. doi: 10.1109/TIT.1984.1056913.  Google Scholar [16] M. G. Parker, C. Tellambura and K. G. Paterson, Golay complementary sequences, in "Wiley Encyclopedia of Telecommunicatins'' (ed. J.G. Proakis), Wiley, 2003. doi: 10.1002/0471219282.eot367.  Google Scholar [17] R. Sivaswamy, Multiphase complementary codes, IEEE Trans. Inform. Theory, IT-24 (1978), 545-552. Google Scholar [18] N. Suehiro and M. Hatori, N-shift cross-orthogonal sequences, IEEE Trans. Inform. Theory, 34 (1988), 143-146. doi: 10.1109/18.2615.  Google Scholar [19] S. M. Tseng and M. R. Bell, Asynchronous multicarrier DS-CDMA using mutually orthogonal complementary sets of sequences, IEEE Trans. Commun., 48 (2000), 53-59. doi: 10.1109/26.818873.  Google Scholar [20] C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inform. Theory, IT-18 (1972), 644-652. doi: 10.1109/TIT.1972.1054860.  Google Scholar [21] Y. F. Tu, P. Z. Fan, L. Hao and X. H. Tang, A simple method for generating optimal Z-periodic complementary Sequence set based on phase shift, IEEE Signal Proc. Letters, 17 (2010), 891-893. doi: 10.1109/LSP.2010.2068288.  Google Scholar [22] Z. Y. Zhang, F. X. Zeng and G. X. Xuan, A class of complementary sequences with multi-width zero cross-correlation zone, IEICE Trans. Fundam., E93-A (2010), 1508-1517. Google Scholar
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