# American Institute of Mathematical Sciences

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

 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.
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:
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##### 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, (1998).   Google Scholar [3] L. E. Dickson, Cyclotomy, higher congruences, and Waring's problem,, Amer. J. Math., 57 (1935), 391.  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.  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.  doi: 10.1109/18.904555.  Google Scholar [6] C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Cryptography,, World Scientific, (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.   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, (2009), 2603.   Google Scholar [9] D. Jungnickel and A. Pott, Difference sets: an introduction,, in Difference Sets, (1999), 259.   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.   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.   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.   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.  doi: 10.1109/TIT.2012.2210025.  Google Scholar [14] T. Storer, Cyclotomy and Difference Sets,, Markham, (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.  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.   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.   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.   Google Scholar
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