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On crosscorrelation of a binary $m$sequence of period $2^{2k}1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$
A new nonbinary sequence family with low correlation and large size
1.  School of Mathematical Sciences, Huaiyin Normal University, Huaian 223300, China 
2.  School of Mathematics & Computation Science, Anqing Normal University, Anqing 246133, China 
3.  School of Mathematics and Statistics, & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China 
4.  School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, China 
Let $p$ be an odd prime, $n≥q3$ and $k$ positive integers with $e=\gcd(n,k)$. In this paper, a new family $\mathcal{S}$ of $p$ary sequences with period $N=p^n1$ is proposed. The sequences in $\mathcal{S}$ are constructed by adding a $p$ary sequence to its two decimated sequences with different phase shifts. The correlation distribution among sequences in $\mathcal{S}$ is completely determined. It is shown that the maximum magnitude of nontrivial correlations of $\mathcal{S}$ is upper bounded by $p^e\sqrt{N+1}+1$, and the family size of $\mathcal{S}$ is $N^2$. Our sequence family has a large family size and low correlation.
References:
[1] 
S. T. Choi, T. Lim, J. S. No, H. Chung, On the crosscorrelation of a $p$ary msequence of period $p^{2m}1$ and its decimated sequence by $\frac{(p^{m}+1)^{2}}{2(p+1)}$, IEEE Trans. Inf. Theory, 58 (2012), 18731879. doi: 10.1109/TIT.2011.2177573. 
[2] 
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), 28472867. doi: 10.1109/TIT.2002.804044. 
[3] 
T. Helleseth, Some results about the crosscorrelation function between two maximallinear sequence, Discrete Math., 16 (1976), 209232. doi: 10.1016/0012365X(76)90100X. 
[4] 
T. Kasami, Weight distribution of BoseChaudhuriHocquenghem codes, in Combinatorial Mathematics and Its Applications, Chapel Hill, NC: Univ. North Carolina Press, 1969,335357. 
[5] 
T. Kasami, Weight Distribution Formular for Some Class of Cyclic Codes, Coordinated Science Lab., Univ. Illinois at UrbanaChampaign, Urbana, IL, Tech. Rep. R285(AD 637524), 1966. 
[6] 
J. Y. Kim, S. T. Choi, J. S. No, H. Chung, A new family of $p$ary sequences of period $(p^n1)/2$ with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 38253830. doi: 10.1109/TIT.2011.2133730. 
[7] 
D. S. Kim, H. J. Chae, H. Y. Song, A generalizaton of the family of $p$ary decimated sequences with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 76147617. doi: 10.1109/TIT.2011.2159576. 
[8] 
P. V. Kumar, O. Moreno, Primephase sequences with periodic correlation properites better than binary sequences, IEEE Trans. Inf. Theory, 37 (1991), 603616. 
[9] 
H. Liang, Y. Tang, The cross correlation distribution of a $p$ary $m$sequence of period $p^m1$ and its decimated sequences by $(p^k+1)(p^m+1)/4$, Finite Fields Appl., 31 (2015), 137161. doi: 10.1016/j.ffa.2014.10.005. 
[10] 
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, AddisonWesley, Reading, MA, 1983. 
[11] 
S. C. Liu, J. J. Komo, Nonbinary Kasami sequences over $GF(p)$, IEEE Trans. Inf. Theory, 38 (1992), 14091412. doi: 10.1109/18.144728. 
[12] 
J. Luo, K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 53325344. doi: 10.1109/TIT.2008.2006424. 
[13] 
J. Luo, K. Feng, Cyclic codes and sequences from generalized CoulterMatthews function, IEEE Trans. Inf. Theory, 54 (2008), 53455353. doi: 10.1109/TIT.2008.2006394. 
[14] 
J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with sixvalued cross correlation, in Proceeding of IWSDA'11, 2011, 4447. doi: 10.1109/IWSDA.2011.6159435. 
[15] 
E. N. Muller, On the crosscorrelation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289295. doi: 10.1109/18.746820. 
[16] 
G. J. Ness, T. Helleseth, A. Kholosha, On the correlation distribution of the CoulterMatthews decimation, IEEE Trans. Inf. Theory, 52 (2006), 22412247. doi: 10.1109/TIT.2006.872857. 
[17] 
E. Y. Seo, Y. S. Kim, J. S. No, D. J. Shin, Crosscorrelation distribution of pary msequence of period $p^{4k}1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$, IEEE Trans. Inf. Theory, 54 (2008), 31403149. doi: 10.1109/TIT.2008.924694. 
[18] 
Y. Sun, Z. Wang, H. Li, T. Yan, The crosscorrelation 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), 409424. doi: 10.3934/amc.2013.7.409. 
[19] 
Y. Xia, X. Zeng, L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d=\frac{p^n+1}{p+1}\frac{p^n1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329342. doi: 10.1007/s002000100128y. 
[20] 
Y. Xia, S. Chen, A new family of $p$ary sequences with low correlation constructed from decimated sequences, IEEE Trans. Inf. Theory, 58 (2012), 60376046. doi: 10.1109/TIT.2012.2201132. 
[21] 
N. Y. Yu, G. Gong, A new binary sequence family with low correlation and large size, IEEE Trans. Inf. Theory, 52 (2006), 16241636. doi: 10.1109/TIT.2006.871062. 
show all references
References:
[1] 
S. T. Choi, T. Lim, J. S. No, H. Chung, On the crosscorrelation of a $p$ary msequence of period $p^{2m}1$ and its decimated sequence by $\frac{(p^{m}+1)^{2}}{2(p+1)}$, IEEE Trans. Inf. Theory, 58 (2012), 18731879. doi: 10.1109/TIT.2011.2177573. 
[2] 
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), 28472867. doi: 10.1109/TIT.2002.804044. 
[3] 
T. Helleseth, Some results about the crosscorrelation function between two maximallinear sequence, Discrete Math., 16 (1976), 209232. doi: 10.1016/0012365X(76)90100X. 
[4] 
T. Kasami, Weight distribution of BoseChaudhuriHocquenghem codes, in Combinatorial Mathematics and Its Applications, Chapel Hill, NC: Univ. North Carolina Press, 1969,335357. 
[5] 
T. Kasami, Weight Distribution Formular for Some Class of Cyclic Codes, Coordinated Science Lab., Univ. Illinois at UrbanaChampaign, Urbana, IL, Tech. Rep. R285(AD 637524), 1966. 
[6] 
J. Y. Kim, S. T. Choi, J. S. No, H. Chung, A new family of $p$ary sequences of period $(p^n1)/2$ with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 38253830. doi: 10.1109/TIT.2011.2133730. 
[7] 
D. S. Kim, H. J. Chae, H. Y. Song, A generalizaton of the family of $p$ary decimated sequences with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 76147617. doi: 10.1109/TIT.2011.2159576. 
[8] 
P. V. Kumar, O. Moreno, Primephase sequences with periodic correlation properites better than binary sequences, IEEE Trans. Inf. Theory, 37 (1991), 603616. 
[9] 
H. Liang, Y. Tang, The cross correlation distribution of a $p$ary $m$sequence of period $p^m1$ and its decimated sequences by $(p^k+1)(p^m+1)/4$, Finite Fields Appl., 31 (2015), 137161. doi: 10.1016/j.ffa.2014.10.005. 
[10] 
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, AddisonWesley, Reading, MA, 1983. 
[11] 
S. C. Liu, J. J. Komo, Nonbinary Kasami sequences over $GF(p)$, IEEE Trans. Inf. Theory, 38 (1992), 14091412. doi: 10.1109/18.144728. 
[12] 
J. Luo, K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 53325344. doi: 10.1109/TIT.2008.2006424. 
[13] 
J. Luo, K. Feng, Cyclic codes and sequences from generalized CoulterMatthews function, IEEE Trans. Inf. Theory, 54 (2008), 53455353. doi: 10.1109/TIT.2008.2006394. 
[14] 
J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with sixvalued cross correlation, in Proceeding of IWSDA'11, 2011, 4447. doi: 10.1109/IWSDA.2011.6159435. 
[15] 
E. N. Muller, On the crosscorrelation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289295. doi: 10.1109/18.746820. 
[16] 
G. J. Ness, T. Helleseth, A. Kholosha, On the correlation distribution of the CoulterMatthews decimation, IEEE Trans. Inf. Theory, 52 (2006), 22412247. doi: 10.1109/TIT.2006.872857. 
[17] 
E. Y. Seo, Y. S. Kim, J. S. No, D. J. Shin, Crosscorrelation distribution of pary msequence of period $p^{4k}1$ and its decimated sequences by $(\frac{p^{2k}+1}{2})^{2}$, IEEE Trans. Inf. Theory, 54 (2008), 31403149. doi: 10.1109/TIT.2008.924694. 
[18] 
Y. Sun, Z. Wang, H. Li, T. Yan, The crosscorrelation 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), 409424. doi: 10.3934/amc.2013.7.409. 
[19] 
Y. Xia, X. Zeng, L. Hu, Further crosscorrelation properties of sequences with the decimation factor $d=\frac{p^n+1}{p+1}\frac{p^n1}{2}$, Appl. Algebra Eng. Commun. Comput., 21 (2010), 329342. doi: 10.1007/s002000100128y. 
[20] 
Y. Xia, S. Chen, A new family of $p$ary sequences with low correlation constructed from decimated sequences, IEEE Trans. Inf. Theory, 58 (2012), 60376046. doi: 10.1109/TIT.2012.2201132. 
[21] 
N. Y. Yu, G. Gong, A new binary sequence family with low correlation and large size, IEEE Trans. Inf. Theory, 52 (2006), 16241636. doi: 10.1109/TIT.2006.871062. 
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