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

Yang  Yang; Guang Gong; Xiaohu  Tang

doi:10.3934/amc.2016008

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

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

Yang Yang ^1, , Guang Gong ^2, and Xiaohu Tang ^3,

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

Abstract
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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.

Keywords: quaternary sequences, quadrature amplitude modulation (QAM), Golay sequence, $\omega$-cyclic-conjugated autocorrelation., $\omega$-cyclic autocorrelation, Golay-Davis-Jedweb (GDJ) sequences.

Mathematics Subject Classification: Primary: 94A05, 60G3.

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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