# 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. Google Scholar [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. doi: 10.1109/18.796380. Google Scholar [3] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'', Research Studies Press, (1996). Google Scholar [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. doi: 10.1109/18.746823. Google Scholar [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. Google Scholar [6] M. J. E. Golay, Multislit spectroscopy,, J. Opt. Soc. Amer., 39 (1949), 437. doi: 10.1364/JOSA.39.000437. Google Scholar [7] M. J. E. Golay, Complementary series,, IRE Trans., 7 (1961), 82. Google Scholar [8] M. J. E. Golay, Note on complementary series,, Proc. IRE, 50 (1962). Google Scholar [9] S. W. Golomb and G. Gong, "Signal Designs with Good Correlation: For Wireless Communication, Cryptography and Radar Applications,'', Cambridge Univeristy Press, (2005). doi: 10.1017/CBO9780511546907. Google Scholar [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), 303. doi: 10.1109/CIS.2009.227. Google Scholar [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. doi: 10.1109/TCOMM.2002.807628. Google Scholar [12] N. Levanon, "Radar Principles,'', Wiley Interscience, (1988). Google Scholar [13] H. D. Lüke, Binary odd periodic complementary sequences,, IEEE Trans. Inf. Theory, 43 (1997), 365. doi: 10.1109/18.567768. Google Scholar [14] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences,, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495. doi: 10.1109/7.366335. Google Scholar [15] M. G. Parker, K. G. Paterson and C. Tellambura, Golay complementary sequences,, in, (2002). Google Scholar [16] K. G. Paterson, Generalized Reed-Muller codes and power control for OFDM modulation,, IEEE. Trans. Inf. Theory, 46 (2000), 104. doi: 10.1109/18.817512. Google Scholar [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. Google Scholar [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. Google Scholar [19] M. B. Pursley, "An Introduction to Digital Communications,'', Pearson Prentice Hall, (2005). Google Scholar [20] D. V. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79. Google Scholar [21] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences,, Proc. IEEE, 68 (1980), 593. doi: 10.1109/PROC.1980.11697. Google Scholar [22] H. D. Schotten, New optimum ternary complementary sets and almost quadriphase, perfect sequences,, in, (1995), 1106. Google Scholar [23] R. Sivaswamy, Self-clutter cancellation and ambiguity properties of subcomplementary sequences,, IEEE Trans. Aerosp. Electron. Sysr., AES-18 (1982), 163. doi: 10.1109/TAES.1982.309223. Google Scholar [24] C. C. Tseng and C. L. Liu, Complementary sets of sequences,, IEEE Trans. Inf. Theory, 18 (1972), 644. doi: 10.1109/TIT.1972.1054860. Google Scholar [25] H. Wen, F. Hu and F. Jin, Design of odd periodic complementary binary signal set,, in, 2 (2004), 590. Google Scholar

show all references

##### References:
 [1] L. Bömer and M. Antweiler, Periodic complementary binary sequences,, IEEE. Trans. Inf. Theory, 35 (1990), 1487. Google Scholar [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. doi: 10.1109/18.796380. Google Scholar [3] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'', Research Studies Press, (1996). Google Scholar [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. doi: 10.1109/18.746823. Google Scholar [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. Google Scholar [6] M. J. E. Golay, Multislit spectroscopy,, J. Opt. Soc. Amer., 39 (1949), 437. doi: 10.1364/JOSA.39.000437. Google Scholar [7] M. J. E. Golay, Complementary series,, IRE Trans., 7 (1961), 82. Google Scholar [8] M. J. E. Golay, Note on complementary series,, Proc. IRE, 50 (1962). Google Scholar [9] S. W. Golomb and G. Gong, "Signal Designs with Good Correlation: For Wireless Communication, Cryptography and Radar Applications,'', Cambridge Univeristy Press, (2005). doi: 10.1017/CBO9780511546907. Google Scholar [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), 303. doi: 10.1109/CIS.2009.227. Google Scholar [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. doi: 10.1109/TCOMM.2002.807628. Google Scholar [12] N. Levanon, "Radar Principles,'', Wiley Interscience, (1988). Google Scholar [13] H. D. Lüke, Binary odd periodic complementary sequences,, IEEE Trans. Inf. Theory, 43 (1997), 365. doi: 10.1109/18.567768. Google Scholar [14] H. D. Lüke and H. D. Schotten, Odd-perfect almost binary correlation sequences,, IEEE Trans. Aerosp. Electron. Syst., 31 (1995), 495. doi: 10.1109/7.366335. Google Scholar [15] M. G. Parker, K. G. Paterson and C. Tellambura, Golay complementary sequences,, in, (2002). Google Scholar [16] K. G. Paterson, Generalized Reed-Muller codes and power control for OFDM modulation,, IEEE. Trans. Inf. Theory, 46 (2000), 104. doi: 10.1109/18.817512. Google Scholar [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. Google Scholar [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. Google Scholar [19] M. B. Pursley, "An Introduction to Digital Communications,'', Pearson Prentice Hall, (2005). Google Scholar [20] D. V. Sarwate, Meeting the Welch bound with equality,, in, (1999), 79. Google Scholar [21] D. V. Sarwate and M. B. Pursley, Crosscorrelation properties of pseudorandom and related sequences,, Proc. IEEE, 68 (1980), 593. doi: 10.1109/PROC.1980.11697. Google Scholar [22] H. D. Schotten, New optimum ternary complementary sets and almost quadriphase, perfect sequences,, in, (1995), 1106. Google Scholar [23] R. Sivaswamy, Self-clutter cancellation and ambiguity properties of subcomplementary sequences,, IEEE Trans. Aerosp. Electron. Sysr., AES-18 (1982), 163. doi: 10.1109/TAES.1982.309223. Google Scholar [24] C. C. Tseng and C. L. Liu, Complementary sets of sequences,, IEEE Trans. Inf. Theory, 18 (1972), 644. doi: 10.1109/TIT.1972.1054860. Google Scholar [25] H. Wen, F. Hu and F. Jin, Design of odd periodic complementary binary signal set,, in, 2 (2004), 590. Google Scholar
 [1] Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim, Dae-Woon Lim. New construction methods of quaternary periodic complementary sequence sets. Advances in Mathematics of Communications, 2010, 4 (1) : 61-68. doi: 10.3934/amc.2010.4.61 [2] 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 [3] Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9 [4] Frank Fiedler. Small Golay sequences. Advances in Mathematics of Communications, 2013, 7 (4) : 379-407. doi: 10.3934/amc.2013.7.379 [5] Ugo Locatelli, Letizia Stefanelli. Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1155-1187. doi: 10.3934/dcdsb.2015.20.1155 [6] Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028 [7] Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control & Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042 [8] Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks & Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551 [9] Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935 [10] Leszek Gasiński, Nikolaos S. Papageorgiou. A pair of positive solutions for $(p,q)$-equations with combined nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 203-215. doi: 10.3934/cpaa.2014.13.203 [11] Bernard Bonnard, Olivier Cots, Jérémy Rouot, Thibaut Verron. Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019029 [12] Gaofei Wu, Yuqing Zhang, Xuefeng Liu. New complementary sets of length $2^m$ and size 4. Advances in Mathematics of Communications, 2016, 10 (4) : 825-845. doi: 10.3934/amc.2016043 [13] Mitali Sarkar, Young Hae Lee. Optimum pricing strategy for complementary products with reservation price in a supply chain model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1553-1586. doi: 10.3934/jimo.2017007 [14] Claude Carlet, Sylvain Guilley. Complementary dual codes for counter-measures to side-channel attacks. Advances in Mathematics of Communications, 2016, 10 (1) : 131-150. doi: 10.3934/amc.2016.10.131 [15] Finley Freibert. The classification of complementary information set codes of lengths $14$ and $16$. Advances in Mathematics of Communications, 2013, 7 (3) : 267-278. doi: 10.3934/amc.2013.7.267 [16] Lisha Wang, Huaming Song, Ding Zhang, Hui Yang. Pricing decisions for complementary products in a fuzzy dual-channel supply chain. Journal of Industrial & Management Optimization, 2019, 15 (1) : 343-364. doi: 10.3934/jimo.2018046 [17] Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915 [18] Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004 [19] Anna Gierzkiewicz, Klaudiusz Wójcik. Lefschetz sequences and detecting periodic points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 81-100. doi: 10.3934/dcds.2012.32.81 [20] A. Gasull, Víctor Mañosa, Xavier Xarles. Rational periodic sequences for the Lyness recurrence. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 587-604. doi: 10.3934/dcds.2012.32.587

2018 Impact Factor: 0.879

## Metrics

• PDF downloads (7)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]