# American Institute of Mathematical Sciences

May  2016, 10(2): 321-331. doi: 10.3934/amc.2016008

## On $\omega$-cyclic-conjugated-perfect quaternary GDJ sequences

 1 Department of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 2 Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1 3 Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, Sichuan 610031

Received  May 2014 Published  April 2016

A sequence is called perfect if its autocorrelation function is a delta function. In this paper, we give a new definition of autocorrelation function: $\omega$-cyclic-conjugated autocorrelation. As a result, we present several classes of $\omega$-cyclic-conjugated-perfect quaternary Golay sequences, where $\omega=\pm 1$. We also considered such perfect property for $4^q$-QAM Golay sequences, $q\ge 2$ being an integer.
Citation: Yang Yang, Guang Gong, Xiaohu Tang. On $\omega$-cyclic-conjugated-perfect quaternary GDJ sequences. Advances in Mathematics of Communications, 2016, 10 (2) : 321-331. doi: 10.3934/amc.2016008
##### References:
 [1] R. Appuswamy and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE Trans. Inf. Theory, 52 (2006), 3817-3826. doi: 10.1109/TIT.2006.878171.  Google Scholar [2] S. Boztaş and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010, (2010), 1300-1304. Google Scholar [3] C. Y. Chang, Y. Li and J. Hirata, New 64-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 56 (2009), 2479-2485. doi: 10.1109/TIT.2010.2043871.  Google Scholar [4] C. V. Chong, R. Venkataramani and V. Tarokh, A new construction of 16-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 49 (2003), 2953-2959. doi: 10.1109/TIT.2003.818418.  Google Scholar [5] D. C. Chu, Polyphase codes with good periodic correlation properties, IEEE Trans. Inf. Theory, 18 (1972), 531-532. Google Scholar [6] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes, IEEE Trans. Inf. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.  Google Scholar [7] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, John Wiley & Sons, London, 1996. Google Scholar [8] R. Frank, S. Zadoff and R. Heimiller, Phase shift pulse codes with good periodic correlation properties, IRE Trans. Inf. Theory, 8 (1962), 381-382. Google Scholar [9] M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., 39 (1949), 437-444. Google Scholar [10] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.  Google Scholar [11] S. W. Golomb and G. Gong, Signal Designs with Good Correlation: For Wireless Communication, Cryptography and Radar Applications, Cambridge Univeristy Press, Cambridge, 2005. doi: 10.1017/CBO9780511546907.  Google Scholar [12] G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay sequences, in ISIT 2012, (2012), 1024-1028. Google Scholar [13] G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay sequences and their applications, IEEE Trans. Commun., 61 (2013), 3967-3978. Google Scholar [14] T. Hoholdt and J. Justesen, Ternary sequences with perfect periodic auto-correlation, IEEE Trans. Inf. Theory, IT-29 (1983), 597-600. doi: 10.1109/TIT.1983.1056707.  Google Scholar [15] V. P. Ipatov, Periodic Discrete Signals with Optimal Correlation Properties, Radio i svyaz, 1992. Google Scholar [16] E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004, Springer, 2004, 197-207. Google Scholar [17] C. E. Lee, Perfect $q$-ary sequences from multiplicative characters over $GF(p)$, Electr. Letters, 28 (1992), 833-834. Google Scholar [18] H. Lee and S. W. Golomb, A new construction of 64-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 52 (2006), 1663-1670. doi: 10.1109/TIT.2006.871616.  Google Scholar [19] Y. Li, Commnents on "A new construction of 16-QAM Golay complementary sequences'' and extension for 64-QAM Golay sequences, IEEE Trans. Inf. Theory, 54 (2008), 3246-3251. doi: 10.1109/TIT.2008.924735.  Google Scholar [20] Y. Li, A construction of general QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 56 (2010), 5765-5771. doi: 10.1109/TIT.2010.2070151.  Google Scholar [21] Y. Li and W. B. Chu, More Golay sequences, IEEE Trans. Inf. Theory, 51 (2005), 1141-1145. doi: 10.1109/TIT.2004.842775.  Google Scholar [22] Z. L. Liu, Y. Li and Y. L. Guan, New constructions of general QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 59 (2013), 7684-7692. doi: 10.1109/TIT.2013.2278178.  Google Scholar [23] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electr. Syst., 31 (1995), 495-498. Google Scholar [24] A. Milewski, Periodic sequences with optimal properties for channel estimation and fast start-up equalization, IBM J. Res. Devel., 27 (1983), 425-431. Google Scholar [25] M. J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Crypt., 53 (2009), 1-15. doi: 10.1007/s10623-009-9301-3.  Google Scholar [26] K. G. Paterson, Generalized Reed-Muller codes and power control for OFDM modulation, IEEE. Trans. Inf. Theory, 46 (2000), 104-120. doi: 10.1109/18.817512.  Google Scholar [27] A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861. doi: 10.1016/j.disc.2006.06.048.  Google Scholar [28] M. B. Pursley, A Introduction to Digital Communications, Pearson Prentice Hall, 2005. Google Scholar [29] A. Rathinakumar and A. K. Chaturvedi, Complete mutually orthogonal Golay complementary sets from Reed-Muller codes, IEEE. Trans. Inf. Theory, 54 (2008), 1339-1346. doi: 10.1109/TIT.2007.915980.  Google Scholar [30] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68 (1980), 593-619. Google Scholar [31] H. D. Schotten and H. D. Lüke, New perfect and $\omega$-cyclic-perfect Sequences, in Proc. Int. Symp. Inf. Theory Appl., 1996, 82-85. Google Scholar [32] J. R. Seberry, B. J. Wysocki and T. A. Wysocki, On a use of Golay sequences for asynchronous DS CDMA applications, in Advanced Signal Processing for Communication Systems, Springer, 2002, 183-196. Google Scholar [33] X. H. Tang, P. Z. Fan and J. Lindner, Multiple binary ZCZ sequence sets with good cross-correlation property based on complementary sequence sets, IEEE Trans. Inf. Theory, 56 (2010), 4038-4045. doi: 10.1109/TIT.2010.2050796.  Google Scholar [34] Y. Yang, F. Huo and G. Gong, Large zero odd periodic autocorrelation zone of Golay sequences and QAM Golay sequences, in ISIT 2012, 2012, 1024-1028. Google Scholar

show all references

##### References:
 [1] R. Appuswamy and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE Trans. Inf. Theory, 52 (2006), 3817-3826. doi: 10.1109/TIT.2006.878171.  Google Scholar [2] S. Boztaş and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010, (2010), 1300-1304. Google Scholar [3] C. Y. Chang, Y. Li and J. Hirata, New 64-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 56 (2009), 2479-2485. doi: 10.1109/TIT.2010.2043871.  Google Scholar [4] C. V. Chong, R. Venkataramani and V. Tarokh, A new construction of 16-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 49 (2003), 2953-2959. doi: 10.1109/TIT.2003.818418.  Google Scholar [5] D. C. Chu, Polyphase codes with good periodic correlation properties, IEEE Trans. Inf. Theory, 18 (1972), 531-532. Google Scholar [6] J. A. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences and Reed-Muller codes, IEEE Trans. Inf. Theory, 45 (1999), 2397-2417. doi: 10.1109/18.796380.  Google Scholar [7] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, John Wiley & Sons, London, 1996. Google Scholar [8] R. Frank, S. Zadoff and R. Heimiller, Phase shift pulse codes with good periodic correlation properties, IRE Trans. Inf. Theory, 8 (1962), 381-382. Google Scholar [9] M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., 39 (1949), 437-444. Google Scholar [10] M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.  Google Scholar [11] S. W. Golomb and G. Gong, Signal Designs with Good Correlation: For Wireless Communication, Cryptography and Radar Applications, Cambridge Univeristy Press, Cambridge, 2005. doi: 10.1017/CBO9780511546907.  Google Scholar [12] G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay sequences, in ISIT 2012, (2012), 1024-1028. Google Scholar [13] G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay sequences and their applications, IEEE Trans. Commun., 61 (2013), 3967-3978. Google Scholar [14] T. Hoholdt and J. Justesen, Ternary sequences with perfect periodic auto-correlation, IEEE Trans. Inf. Theory, IT-29 (1983), 597-600. doi: 10.1109/TIT.1983.1056707.  Google Scholar [15] V. P. Ipatov, Periodic Discrete Signals with Optimal Correlation Properties, Radio i svyaz, 1992. Google Scholar [16] E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004, Springer, 2004, 197-207. Google Scholar [17] C. E. Lee, Perfect $q$-ary sequences from multiplicative characters over $GF(p)$, Electr. Letters, 28 (1992), 833-834. Google Scholar [18] H. Lee and S. W. Golomb, A new construction of 64-QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 52 (2006), 1663-1670. doi: 10.1109/TIT.2006.871616.  Google Scholar [19] Y. Li, Commnents on "A new construction of 16-QAM Golay complementary sequences'' and extension for 64-QAM Golay sequences, IEEE Trans. Inf. Theory, 54 (2008), 3246-3251. doi: 10.1109/TIT.2008.924735.  Google Scholar [20] Y. Li, A construction of general QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 56 (2010), 5765-5771. doi: 10.1109/TIT.2010.2070151.  Google Scholar [21] Y. Li and W. B. Chu, More Golay sequences, IEEE Trans. Inf. Theory, 51 (2005), 1141-1145. doi: 10.1109/TIT.2004.842775.  Google Scholar [22] Z. L. Liu, Y. Li and Y. L. Guan, New constructions of general QAM Golay complementary sequences, IEEE Trans. Inf. Theory, 59 (2013), 7684-7692. doi: 10.1109/TIT.2013.2278178.  Google Scholar [23] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electr. Syst., 31 (1995), 495-498. Google Scholar [24] A. Milewski, Periodic sequences with optimal properties for channel estimation and fast start-up equalization, IBM J. Res. Devel., 27 (1983), 425-431. Google Scholar [25] M. J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Crypt., 53 (2009), 1-15. doi: 10.1007/s10623-009-9301-3.  Google Scholar [26] K. G. Paterson, Generalized Reed-Muller codes and power control for OFDM modulation, IEEE. Trans. Inf. Theory, 46 (2000), 104-120. doi: 10.1109/18.817512.  Google Scholar [27] A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861. doi: 10.1016/j.disc.2006.06.048.  Google Scholar [28] M. B. Pursley, A Introduction to Digital Communications, Pearson Prentice Hall, 2005. Google Scholar [29] A. Rathinakumar and A. K. Chaturvedi, Complete mutually orthogonal Golay complementary sets from Reed-Muller codes, IEEE. Trans. Inf. Theory, 54 (2008), 1339-1346. doi: 10.1109/TIT.2007.915980.  Google Scholar [30] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68 (1980), 593-619. Google Scholar [31] H. D. Schotten and H. D. Lüke, New perfect and $\omega$-cyclic-perfect Sequences, in Proc. Int. Symp. Inf. Theory Appl., 1996, 82-85. Google Scholar [32] J. R. Seberry, B. J. Wysocki and T. A. Wysocki, On a use of Golay sequences for asynchronous DS CDMA applications, in Advanced Signal Processing for Communication Systems, Springer, 2002, 183-196. Google Scholar [33] X. H. Tang, P. Z. Fan and J. Lindner, Multiple binary ZCZ sequence sets with good cross-correlation property based on complementary sequence sets, IEEE Trans. Inf. Theory, 56 (2010), 4038-4045. doi: 10.1109/TIT.2010.2050796.  Google Scholar [34] Y. Yang, F. Huo and G. Gong, Large zero odd periodic autocorrelation zone of Golay sequences and QAM Golay sequences, in ISIT 2012, 2012, 1024-1028. Google Scholar
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