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February  2015, 9(1): 117-128. doi: 10.3934/amc.2015.9.117

Two new classes of binary sequence pairs with three-level cross-correlation

Xiaohui Liu 1, , Jinhua Wang 1, and  Dianhua Wu 2,

1. 

School of Sciences, Nantong University, Nantong, Jiangsu 226007, China, China
2. 

Department of Mathematics, Guangxi Normal University, Guilin, Guangxi 541004, China

Received  June 2014 Revised  August 2014 Published  February 2015

A pair of binary sequences is generalized from the concept of a two-level autocorrelation function of single binary sequence. In this paper, we describe two classes of binary sequence pairs of period $N=2q$, where $q=4f+1$ is an odd prime and $f$ is an even integer. Those classes of binary sequence pairs are based on cyclic almost difference set pairs. They have optimal three-level cross-correlation, and either balanced or almost balanced.
Keywords: almost difference set pair, Binary sequence pair, cyclotomy., three-level cross-correlation, optimal correlation value.
Mathematics Subject Classification: Primary: 05B10; Secondary: 94A5.
Citation: Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117
References:
[1]

L. D. Baumert, Cyclic Difference Sets, Springer-Verlag, 1971.  Google Scholar

[2]

T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, North-Holland/Elsevier, Amsterdam, 1998.  Google Scholar

[3]

L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math., 57 (1935), 391-424. doi: 10.2307/2371217.  Google Scholar

[4]

C. Ding, T. Helleseth and K. Y. Lam, Several classes of binary sequences with three-level autocorrelation, IEEE Trans. Inf. Theory, 45 (1999), 2606-2612. doi: 10.1109/18.796414.  Google Scholar

[5]

C. Ding, T. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inf. Theory, 47 (2001), 428-433. doi: 10.1109/18.904555.  Google Scholar

[6]

C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Cryptography, World Scientific, Singapore, 1996. doi: 10.1142/9789812779380.  Google Scholar

[7]

H. L. Jin and C. Q. Xu, The study of methods for constructing a family of pseudorandom binary sequence pairs based on the cyclotomic class (in Chinese), Acta Electr. Sin., 38 (2010), 1608-1611. Google Scholar

[8]

S. Y. Jin and H. Y. Song, Note on a pair of binary sequences with ideal two-level crosscorrelation, in Proc. ISIT2009, Seoul, 2009, 2603-2607. Google Scholar

[9]

D. Jungnickel and A. Pott, Difference sets: an introduction, in Difference Sets, Sequences and Their Correlation Properties (eds. A. Pott, P.V. Kumar, T. Helleseth and D. Jungnickel), Kluwer Academic Publishers, 1999, 259-295.  Google Scholar

[10]

J. Z. Li and P. H. Ke, Study on the almost difference set pairs and almost perfect autocorrelation binary sequence pairs (in Chinese), J. Wuyi University, 27 (2008), 10-14. Google Scholar

[11]

K. Liu and C. Q. Xu, On binary sequence pairs with two-level periodic cross-correlation function, IEICE Trans. Funda., E93-A (2010), 2278-2285. Google Scholar

[12]

F. Mao, T. Jiang, C. L. Zhao and Z. Zhou, Study of pseudorandom binary sequence pairs (in Chinese), J. Commun., 26 (2005), 94-98. Google Scholar

[13]

X. P. Peng, C. Q. Xu and K. T. Arasu, New families of binary sequence pairs with two-level and three-level correlation, IEEE Trans. Inf. Theory, 58 (2012), 2968-2978. doi: 10.1109/TIT.2012.2210025.  Google Scholar

[14]

T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.  Google Scholar

[15]

T. W. Sze, S. Chanson, C. Ding, T. Helleseth and M. G.Parker, Logarithm authentication codes, Infor. Comput., 148 (2003), 93-108. doi: 10.1016/S0890-5401(03)00053-1.  Google Scholar

[16]

Y. Z. Wang and C. Q. Xu, Divisible difference set pairs and approach for the study of almost binary sequence pair (in Chinese), Acta Electr. Sin., 37 (2009), 692-695. Google Scholar

[17]

C. Q. Xu, Difference set pairs and approach for the study of perfect binary array pairs (in Chinese), Acta Electr. Sin., 29 (2001), 87-89. Google Scholar

[18]

X. Q. Zhao, W. C. He, Z. W. Wang and S. L. Jia, The theory of the perfect binary array pairs (in Chinese), Acta Electr. Sin., 27 (1999), 34-37. Google Scholar

show all references

References:
[1]

L. D. Baumert, Cyclic Difference Sets, Springer-Verlag, 1971.  Google Scholar

[2]

T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, North-Holland/Elsevier, Amsterdam, 1998.  Google Scholar

[3]

L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math., 57 (1935), 391-424. doi: 10.2307/2371217.  Google Scholar

[4]

C. Ding, T. Helleseth and K. Y. Lam, Several classes of binary sequences with three-level autocorrelation, IEEE Trans. Inf. Theory, 45 (1999), 2606-2612. doi: 10.1109/18.796414.  Google Scholar

[5]

C. Ding, T. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation, IEEE Trans. Inf. Theory, 47 (2001), 428-433. doi: 10.1109/18.904555.  Google Scholar

[6]

C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Cryptography, World Scientific, Singapore, 1996. doi: 10.1142/9789812779380.  Google Scholar

[7]

H. L. Jin and C. Q. Xu, The study of methods for constructing a family of pseudorandom binary sequence pairs based on the cyclotomic class (in Chinese), Acta Electr. Sin., 38 (2010), 1608-1611. Google Scholar

[8]

S. Y. Jin and H. Y. Song, Note on a pair of binary sequences with ideal two-level crosscorrelation, in Proc. ISIT2009, Seoul, 2009, 2603-2607. Google Scholar

[9]

D. Jungnickel and A. Pott, Difference sets: an introduction, in Difference Sets, Sequences and Their Correlation Properties (eds. A. Pott, P.V. Kumar, T. Helleseth and D. Jungnickel), Kluwer Academic Publishers, 1999, 259-295.  Google Scholar

[10]

J. Z. Li and P. H. Ke, Study on the almost difference set pairs and almost perfect autocorrelation binary sequence pairs (in Chinese), J. Wuyi University, 27 (2008), 10-14. Google Scholar

[11]

K. Liu and C. Q. Xu, On binary sequence pairs with two-level periodic cross-correlation function, IEICE Trans. Funda., E93-A (2010), 2278-2285. Google Scholar

[12]

F. Mao, T. Jiang, C. L. Zhao and Z. Zhou, Study of pseudorandom binary sequence pairs (in Chinese), J. Commun., 26 (2005), 94-98. Google Scholar

[13]

X. P. Peng, C. Q. Xu and K. T. Arasu, New families of binary sequence pairs with two-level and three-level correlation, IEEE Trans. Inf. Theory, 58 (2012), 2968-2978. doi: 10.1109/TIT.2012.2210025.  Google Scholar

[14]

T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.  Google Scholar

[15]

T. W. Sze, S. Chanson, C. Ding, T. Helleseth and M. G.Parker, Logarithm authentication codes, Infor. Comput., 148 (2003), 93-108. doi: 10.1016/S0890-5401(03)00053-1.  Google Scholar

[16]

Y. Z. Wang and C. Q. Xu, Divisible difference set pairs and approach for the study of almost binary sequence pair (in Chinese), Acta Electr. Sin., 37 (2009), 692-695. Google Scholar

[17]

C. Q. Xu, Difference set pairs and approach for the study of perfect binary array pairs (in Chinese), Acta Electr. Sin., 29 (2001), 87-89. Google Scholar

[18]

X. Q. Zhao, W. C. He, Z. W. Wang and S. L. Jia, The theory of the perfect binary array pairs (in Chinese), Acta Electr. Sin., 27 (1999), 34-37. Google Scholar

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