    November  2017, 11(4): 693-703. doi: 10.3934/amc.2017050

## On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$

 1 School of Mathematical Sciences, Huaiyin Normal University, Huaian 223300, China 2 School of Mathematics and Statistics, & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China 3 School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China

Received  October 2015 Revised  March 2016 Published  November 2017

For two odd integers $l,k$ with $0<l<k$ and $\gcd(l,k)=1$, let $m=2k$ and $d=\frac{2^{lk}+1}{2^l+1}$. In this paper, we study the cross-correlation between a binary $m$-sequence $(s_t)$ of length $2^m-1$ and its $d$-decimated sequences $(s_{dt+u}), 0≤q u<\frac{2^k+1}{3}.$ It is shown that the maximum magnitude of cross-correlation values is $2^{\frac{m}{2}+1}+1.$ Moreover, a new sequence family with maximum correlation magnitude $2^{\frac{m}{2}+1}+1$ and family size $2^{\frac{m}{2}}$ is proposed.

Citation: Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050
##### References:
  S. T. Choi, J. Y. Kim and J. S. No, On the cross-correlation of a $p$-ary m-sequence and its decimated sequence by $d=\frac{p^n+1}{p^k+1}+\frac{p^n-1}{2}$, arXiv: 1205.5959v1, 2012 Google Scholar  S. T. Choi, T. Lim, J. S. No and H. Chung, On the cross-correlation of a $p$-ary m-sequence of period $p^{2m}-1$ and its decimated sequence by $\frac{(p^{m}+1)^{2}}{2(p+1)}$, IEEE Trans. Inf. Theory, 58 (2012), 1873-1879.  doi: 10.1109/TIT.2011.2177573.  Google Scholar  H. Dobbertin, T. Helleseth, P. V. Kumar and H. Martinsen, Ternary $m$-sequences with three-valued cross-correlation function: New decimations of Welch and Niho type, IEEE Trans. Inf. Theory, 47 (2001), 1473-1481.  doi: 10.1109/18.923728.  Google Scholar  H. Dobbertin, P. Felke, T. Helleseth and P. Rosendahl, Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums, IEEE Trans. Inf. Theory, 52 (2006), 613-627.  doi: 10.1109/TIT.2005.862094.  Google Scholar  R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156.  doi: 10.1109/TIT.1968.1054106. Google Scholar  G. Gong, New designs for signal sets with low cross correlation, balance property, and large linear span: GF(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867.  doi: 10.1109/TIT.2002.804044.  Google Scholar  T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.  Google Scholar  T. Helleseth and P. V. Kumar, Sequences with low correlation, Handbook of Coding Theory, Vol. Ⅰ, Ⅱ, 1765-1853, North-Holland, Amsterdam, 1998. Google Scholar  Z. Hu, X. Li, D. Mills, E. Muller, W. Williems, Y. Yang and Z. Zhang, On the cross-correlation of sequences with the decimation factor $d=\frac{p^n+1}{p+1}-\frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 12 (2001), 255-263. Google Scholar  J. Y. Kim, S. T. Choi, J. S. No and H. Chung, A new family of $p$-ary sequences of period $(p^n-1)/2$ with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 3825-3829.  doi: 10.1109/TIT.2011.2133730.  Google Scholar  H. Liang and Y. Tang, The cross correlation distribution of a $p$-ary $m$-sequence of period $p^m-1$ and its decimated sequences by $(p^k+1)(p^m+1)/4$, Finite Fields Appl., 31 (2015), 137-161.  doi: 10.1016/j.ffa.2014.10.005.  Google Scholar  R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. Google Scholar  S. C. Liu and J. F. Komo, Nonbinary Kasami sequences over $GF(p)$, IEEE Trans. Inf. Theory, 38 (1992), 1409-1412.  doi: 10.1109/18.144728.  Google Scholar  J. Luo, Binary sequences with three-valued cross correlations of different lengths, IEEE Transactions on Information Theory, 62 (2016), 7532-7537, arXiv: 1503.05650v1. doi: 10.1109/TIT.2016.2620432. Google Scholar  J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation, in Proceeding of IWSDA'11, 2001, 44-47. doi: 10.1109/IWSDA.2011.6159435. Google Scholar  M. Moisio, A note on evaluations of some exponential sums, Acta Arith., 93 (2000), 117-119. Google Scholar  G. J. Ness and T. Helleseth, Cross correlation of $m$-sequences with different lengths, IEEE Trans. Inf. Theory, 52 (2006), 1637-1648.  doi: 10.1109/TIT.2006.871049.  Google Scholar  G. J. Ness and T. Helleseth, A new three-valued cross correlation between $m$-sequences of different lengths, IEEE Trans. Inf. Theory, 52 (2006), 4695-4701.  doi: 10.1109/TIT.2006.881715.  Google Scholar  G. J. Ness and T. Helleseth, Characterization of $m$-sequences of length $2^{2k}-1$ and $2^k-1$ with three-valued cross correlation, IEEE Trans. Inf. Theory, 53 (2007), 2236-2245.  doi: 10.1109/TIT.2007.896881.  Google Scholar  J. S. No and P. V. Kumar, A new family of binary pseudorandom sequences having optimal periodic correlation properties and larger linear span, IEEE Trans. Inf. Theory, 35 (1989), 371-379.  doi: 10.1109/18.32131. Google Scholar  E. Y. Seo, Y. S. Kim, J. S. No and D. J. Shin, Cross-correlation distribution of p-ary m-sequence of period $p^{4k}-1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$, IEEE Trans. Inf. Theory, 54 (2008), 3140-3149.  doi: 10.1109/TIT.2008.924694.  Google Scholar  Y. Sun, Z. Wang, H. Li and T. Yan, The cross-correlation distribution of a $p$-ary m-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$, Adv. Math. Commun., 7 (2013), 409-424.  doi: 10.3934/amc.2013.7.409.  Google Scholar  Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d=\frac{p^n+1}{p+1}-\frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329-342.  doi: 10.1007/s00200-010-0128-y.  Google Scholar

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##### References:
  S. T. Choi, J. Y. Kim and J. S. No, On the cross-correlation of a $p$-ary m-sequence and its decimated sequence by $d=\frac{p^n+1}{p^k+1}+\frac{p^n-1}{2}$, arXiv: 1205.5959v1, 2012 Google Scholar  S. T. Choi, T. Lim, J. S. No and H. Chung, On the cross-correlation of a $p$-ary m-sequence of period $p^{2m}-1$ and its decimated sequence by $\frac{(p^{m}+1)^{2}}{2(p+1)}$, IEEE Trans. Inf. Theory, 58 (2012), 1873-1879.  doi: 10.1109/TIT.2011.2177573.  Google Scholar  H. Dobbertin, T. Helleseth, P. V. Kumar and H. Martinsen, Ternary $m$-sequences with three-valued cross-correlation function: New decimations of Welch and Niho type, IEEE Trans. Inf. Theory, 47 (2001), 1473-1481.  doi: 10.1109/18.923728.  Google Scholar  H. Dobbertin, P. Felke, T. Helleseth and P. Rosendahl, Niho type cross-correlation functions via Dickson polynomials and Kloosterman sums, IEEE Trans. Inf. Theory, 52 (2006), 613-627.  doi: 10.1109/TIT.2005.862094.  Google Scholar  R. Gold, Maximal recursive sequences with 3-valued recursive cross-correlation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156.  doi: 10.1109/TIT.1968.1054106. Google Scholar  G. Gong, New designs for signal sets with low cross correlation, balance property, and large linear span: GF(p) case, IEEE Trans. Inf. Theory, 48 (2002), 2847-2867.  doi: 10.1109/TIT.2002.804044.  Google Scholar  T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequences, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.  Google Scholar  T. Helleseth and P. V. Kumar, Sequences with low correlation, Handbook of Coding Theory, Vol. Ⅰ, Ⅱ, 1765-1853, North-Holland, Amsterdam, 1998. Google Scholar  Z. Hu, X. Li, D. Mills, E. Muller, W. Williems, Y. Yang and Z. Zhang, On the cross-correlation of sequences with the decimation factor $d=\frac{p^n+1}{p+1}-\frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 12 (2001), 255-263. Google Scholar  J. Y. Kim, S. T. Choi, J. S. No and H. Chung, A new family of $p$-ary sequences of period $(p^n-1)/2$ with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 3825-3829.  doi: 10.1109/TIT.2011.2133730.  Google Scholar  H. Liang and Y. Tang, The cross correlation distribution of a $p$-ary $m$-sequence of period $p^m-1$ and its decimated sequences by $(p^k+1)(p^m+1)/4$, Finite Fields Appl., 31 (2015), 137-161.  doi: 10.1016/j.ffa.2014.10.005.  Google Scholar  R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. Google Scholar  S. C. Liu and J. F. Komo, Nonbinary Kasami sequences over $GF(p)$, IEEE Trans. Inf. Theory, 38 (1992), 1409-1412.  doi: 10.1109/18.144728.  Google Scholar  J. Luo, Binary sequences with three-valued cross correlations of different lengths, IEEE Transactions on Information Theory, 62 (2016), 7532-7537, arXiv: 1503.05650v1. doi: 10.1109/TIT.2016.2620432. Google Scholar  J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation, in Proceeding of IWSDA'11, 2001, 44-47. doi: 10.1109/IWSDA.2011.6159435. Google Scholar  M. Moisio, A note on evaluations of some exponential sums, Acta Arith., 93 (2000), 117-119. Google Scholar  G. J. Ness and T. Helleseth, Cross correlation of $m$-sequences with different lengths, IEEE Trans. Inf. Theory, 52 (2006), 1637-1648.  doi: 10.1109/TIT.2006.871049.  Google Scholar  G. J. Ness and T. Helleseth, A new three-valued cross correlation between $m$-sequences of different lengths, IEEE Trans. Inf. Theory, 52 (2006), 4695-4701.  doi: 10.1109/TIT.2006.881715.  Google Scholar  G. J. Ness and T. Helleseth, Characterization of $m$-sequences of length $2^{2k}-1$ and $2^k-1$ with three-valued cross correlation, IEEE Trans. Inf. Theory, 53 (2007), 2236-2245.  doi: 10.1109/TIT.2007.896881.  Google Scholar  J. S. No and P. V. Kumar, A new family of binary pseudorandom sequences having optimal periodic correlation properties and larger linear span, IEEE Trans. Inf. Theory, 35 (1989), 371-379.  doi: 10.1109/18.32131. Google Scholar  E. Y. Seo, Y. S. Kim, J. S. No and D. J. Shin, Cross-correlation distribution of p-ary m-sequence of period $p^{4k}-1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$, IEEE Trans. Inf. Theory, 54 (2008), 3140-3149.  doi: 10.1109/TIT.2008.924694.  Google Scholar  Y. Sun, Z. Wang, H. Li and T. Yan, The cross-correlation distribution of a $p$-ary m-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$, Adv. Math. Commun., 7 (2013), 409-424.  doi: 10.3934/amc.2013.7.409.  Google Scholar  Y. Xia, X. Zeng and L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d=\frac{p^n+1}{p+1}-\frac{p^n-1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329-342.  doi: 10.1007/s00200-010-0128-y.  Google Scholar
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