# American Institute of Mathematical Sciences

May  2013, 7(2): 113-125. doi: 10.3934/amc.2013.7.113

## Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions

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

Received  April 2012 Published  May 2013

A pair of two sequences is called the even periodic (odd periodic) complementary sequence pair if the sum of their even periodic (odd periodic) correlation function is a delta function. The well-known Golay aperiodic complementary sequence pair (Golay pair) is a special case of even periodic (odd periodic) complementary sequence pair. In this paper, we presented several classes of even periodic and odd periodic complementary pairs based on the generalized Boolean functions, but which do not form Gloay pairs. The proposed sequences could be used to design signal sets, which have been applied in direct sequence code division multiple (DS-CDMA) cellular communication systems.
Citation: Yang Yang, Xiaohu Tang, Guang Gong. Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 113-125. doi: 10.3934/amc.2013.7.113
##### References:
 [1] L. Bömer and M. Antweiler, Periodic complementary binary sequences, IEEE. Trans. Inf. Theory, 35 (1990), 1487-1494. [2] 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. [3] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'' Research Studies Press, John Wiley & Sons Ltd, London, 1996. [4] K. Q. Feng, P. J.-S. Shiue and Q. Xiang, On aperiodic and periodic complementary binary sequences, IEEE Trans. Inf. Theory, 45 (1999), 296-303. doi: 10.1109/18.746823. [5] H. Ganapathy, D. A. Pados and G. N. Karystinos, New bounds and optimal binary signature sets-Part I: Periodic total squared correlation, IEEE Trans. Inf. Theory, 59 (2011), 1123-1132. [6] M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., 39 (1949), 437-444. doi: 10.1364/JOSA.39.000437. [7] M. J. E. Golay, Complementary series, IRE Trans., 7 (1961), 82-87. [8] M. J. E. Golay, Note on complementary series, Proc. IRE, 50 (1962), 84. [9] 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. [10] H. L. Jin, G. D. Liang, Z. H. Liu and C. Q. Xu, The necessary condition of families of odd periodic perfect complementary sequence pairs, in "2009 International Conference on Computational Intelligence and Security,'' (2009), 303-307. doi: 10.1109/CIS.2009.227. [11] G. N. Karystinos and D. A. Pados, New bounds on the total squared correlation and optimal design of DS-CDMA binary signature sets, IEEE Trans. Commun., 51 (2003), 48-51. doi: 10.1109/TCOMM.2002.807628. [12] N. Levanon, "Radar Principles,'' Wiley Interscience, New York, 1988. [13] H. D. Lüke, Binary odd periodic complementary sequences, IEEE Trans. Inf. Theory, 43 (1997), 365-367. doi: 10.1109/18.567768. [14] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495-498. doi: 10.1109/7.366335. [15] M. G. Parker, K. G. Paterson and C. Tellambura, Golay complementary sequences, in "Wiley Encyclopedia of Telecommunications'' (ed. J.G. Proakis), Wiley Interscience, New York, 2002. [16] 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. [17] M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part I: System analysis, IEEE Trans. Inf. Theory, 25 (1977), 795-799. [18] M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part II: Code sequence analysis, IEEE Trans. Inf. Theory, 25 (1977), 800-803. [19] M. B. Pursley, "An Introduction to Digital Communications,'' Pearson Prentice Hall, U.S., 2005. [20] D. V. Sarwate, Meeting the Welch bound with equality, in "Sequences and Their Applications: Proceedings of SETA'98'' (eds. C. Ding, T. Helleseth and H. Niederreiter), Springer-Verlag, London, (1999), 79-102. [21] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68 (1980), 593-619. doi: 10.1109/PROC.1980.11697. [22] H. D. Schotten, New optimum ternary complementary sets and almost quadriphase, perfect sequences, in "Int. Conference on Neural Networks and Signal Processing (ICNNSP'95),'' Nanjing, China, (1995), 1106-1109. [23] R. Sivaswamy, Self-clutter cancellation and ambiguity properties of subcomplementary sequences, IEEE Trans. Aerosp. Electron. Sysr., AES-18 (1982), 163-180. doi: 10.1109/TAES.1982.309223. [24] C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inf. Theory, 18 (1972), 644-651. doi: 10.1109/TIT.1972.1054860. [25] H. Wen, F. Hu and F. Jin, Design of odd periodic complementary binary signal set, in "Ninth IEEE Symposium on Computers and Communications 2004,'' 2 (2004), 590-593.

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##### References:
 [1] L. Bömer and M. Antweiler, Periodic complementary binary sequences, IEEE. Trans. Inf. Theory, 35 (1990), 1487-1494. [2] 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. [3] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'' Research Studies Press, John Wiley & Sons Ltd, London, 1996. [4] K. Q. Feng, P. J.-S. Shiue and Q. Xiang, On aperiodic and periodic complementary binary sequences, IEEE Trans. Inf. Theory, 45 (1999), 296-303. doi: 10.1109/18.746823. [5] H. Ganapathy, D. A. Pados and G. N. Karystinos, New bounds and optimal binary signature sets-Part I: Periodic total squared correlation, IEEE Trans. Inf. Theory, 59 (2011), 1123-1132. [6] M. J. E. Golay, Multislit spectroscopy, J. Opt. Soc. Amer., 39 (1949), 437-444. doi: 10.1364/JOSA.39.000437. [7] M. J. E. Golay, Complementary series, IRE Trans., 7 (1961), 82-87. [8] M. J. E. Golay, Note on complementary series, Proc. IRE, 50 (1962), 84. [9] 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. [10] H. L. Jin, G. D. Liang, Z. H. Liu and C. Q. Xu, The necessary condition of families of odd periodic perfect complementary sequence pairs, in "2009 International Conference on Computational Intelligence and Security,'' (2009), 303-307. doi: 10.1109/CIS.2009.227. [11] G. N. Karystinos and D. A. Pados, New bounds on the total squared correlation and optimal design of DS-CDMA binary signature sets, IEEE Trans. Commun., 51 (2003), 48-51. doi: 10.1109/TCOMM.2002.807628. [12] N. Levanon, "Radar Principles,'' Wiley Interscience, New York, 1988. [13] H. D. Lüke, Binary odd periodic complementary sequences, IEEE Trans. Inf. Theory, 43 (1997), 365-367. doi: 10.1109/18.567768. [14] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495-498. doi: 10.1109/7.366335. [15] M. G. Parker, K. G. Paterson and C. Tellambura, Golay complementary sequences, in "Wiley Encyclopedia of Telecommunications'' (ed. J.G. Proakis), Wiley Interscience, New York, 2002. [16] 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. [17] M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part I: System analysis, IEEE Trans. Inf. Theory, 25 (1977), 795-799. [18] M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication--Part II: Code sequence analysis, IEEE Trans. Inf. Theory, 25 (1977), 800-803. [19] M. B. Pursley, "An Introduction to Digital Communications,'' Pearson Prentice Hall, U.S., 2005. [20] D. V. Sarwate, Meeting the Welch bound with equality, in "Sequences and Their Applications: Proceedings of SETA'98'' (eds. C. Ding, T. Helleseth and H. Niederreiter), Springer-Verlag, London, (1999), 79-102. [21] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences, Proc. IEEE, 68 (1980), 593-619. doi: 10.1109/PROC.1980.11697. [22] H. D. Schotten, New optimum ternary complementary sets and almost quadriphase, perfect sequences, in "Int. Conference on Neural Networks and Signal Processing (ICNNSP'95),'' Nanjing, China, (1995), 1106-1109. [23] R. Sivaswamy, Self-clutter cancellation and ambiguity properties of subcomplementary sequences, IEEE Trans. Aerosp. Electron. Sysr., AES-18 (1982), 163-180. doi: 10.1109/TAES.1982.309223. [24] C. C. Tseng and C. L. Liu, Complementary sets of sequences, IEEE Trans. Inf. Theory, 18 (1972), 644-651. doi: 10.1109/TIT.1972.1054860. [25] H. Wen, F. Hu and F. Jin, Design of odd periodic complementary binary signal set, in "Ninth IEEE Symposium on Computers and Communications 2004,'' 2 (2004), 590-593.
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