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

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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

Abstract
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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.
[2]	T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, North-Holland/Elsevier, Amsterdam, 1998.
[3]	L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math., 57 (1935), 391-424. doi: 10.2307/2371217.
[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.
[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.
[6]	C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Cryptography, World Scientific, Singapore, 1996. doi: 10.1142/9789812779380.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.
[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.
[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.
[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.
[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.

show all references

References:

[1]	L. D. Baumert, Cyclic Difference Sets, Springer-Verlag, 1971.
[2]	T. W. Cusick, C. Ding and A. Renvall, Stream Ciphers and Number Theory, North-Holland/Elsevier, Amsterdam, 1998.
[3]	L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem, Amer. J. Math., 57 (1935), 391-424. doi: 10.2307/2371217.
[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.
[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.
[6]	C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Cryptography, World Scientific, Singapore, 1996. doi: 10.1142/9789812779380.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.
[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.
[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.
[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.
[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.

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