American Institute of Mathematical Sciences

Advanced
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

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.

[2]

S. Boztaş and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010, (2010), 1300-1304.

[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.

[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.

[5]

D. C. Chu, Polyphase codes with good periodic correlation properties, IEEE Trans. Inf. Theory, 18 (1972), 531-532.

[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.

[7]

P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, John Wiley & Sons, London, 1996.

[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.

[9]

M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., 39 (1949), 437-444.

[10]

M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.

[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.

[12]

G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay sequences, in ISIT 2012, (2012), 1024-1028.

[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.

[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.

[15]

V. P. Ipatov, Periodic Discrete Signals with Optimal Correlation Properties, Radio i svyaz, 1992.

[16]

E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004, Springer, 2004, 197-207.

[17]

C. E. Lee, Perfect $q$-ary sequences from multiplicative characters over $GF(p)$, Electr. Letters, 28 (1992), 833-834.

[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.

[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.

[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.

[21]

Y. Li and W. B. Chu, More Golay sequences, IEEE Trans. Inf. Theory, 51 (2005), 1141-1145. doi: 10.1109/TIT.2004.842775.

[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.

[23]

H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electr. Syst., 31 (1995), 495-498.

[24]

A. Milewski, Periodic sequences with optimal properties for channel estimation and fast start-up equalization, IBM J. Res. Devel., 27 (1983), 425-431.

[25]

M. J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Crypt., 53 (2009), 1-15. doi: 10.1007/s10623-009-9301-3.

[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.

[27]

A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861. doi: 10.1016/j.disc.2006.06.048.

[28]

M. B. Pursley, A Introduction to Digital Communications, Pearson Prentice Hall, 2005.

[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.

[30]

D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68 (1980), 593-619.

[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.

[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.

[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.

[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.

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.

[2]

S. Boztaş and P. Udaya, Nonbinary sequences with perfect and nearly perfect autocorrelation, in ISIT 2010, (2010), 1300-1304.

[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.

[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.

[5]

D. C. Chu, Polyphase codes with good periodic correlation properties, IEEE Trans. Inf. Theory, 18 (1972), 531-532.

[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.

[7]

P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, John Wiley & Sons, London, 1996.

[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.

[9]

M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., 39 (1949), 437-444.

[10]

M. J. E. Golay, Complementary series, IRE Trans. Inf. Theory, 7 (1961), 82-87.

[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.

[12]

G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay sequences, in ISIT 2012, (2012), 1024-1028.

[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.

[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.

[15]

V. P. Ipatov, Periodic Discrete Signals with Optimal Correlation Properties, Radio i svyaz, 1992.

[16]

E. I. Krengel, Almost-perfect and odd-perfect ternary sequences, in SETA 2004, Springer, 2004, 197-207.

[17]

C. E. Lee, Perfect $q$-ary sequences from multiplicative characters over $GF(p)$, Electr. Letters, 28 (1992), 833-834.

[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.

[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.

[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.

[21]

Y. Li and W. B. Chu, More Golay sequences, IEEE Trans. Inf. Theory, 51 (2005), 1141-1145. doi: 10.1109/TIT.2004.842775.

[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.

[23]

H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electr. Syst., 31 (1995), 495-498.

[24]

A. Milewski, Periodic sequences with optimal properties for channel estimation and fast start-up equalization, IBM J. Res. Devel., 27 (1983), 425-431.

[25]

M. J. Mossinghoff, Wieferich pairs and Barker sequences, Des. Codes Crypt., 53 (2009), 1-15. doi: 10.1007/s10623-009-9301-3.

[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.

[27]

A. Pott, Difference triangles and negaperiodic autocorrelation functions, Discrete Math., 308 (2008), 2854-2861. doi: 10.1016/j.disc.2006.06.048.

[28]

M. B. Pursley, A Introduction to Digital Communications, Pearson Prentice Hall, 2005.

[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.

[30]

D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68 (1980), 593-619.

[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.

[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.

[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.

[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.

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