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November  2017, 11(4): 671-691. doi: 10.3934/amc.2017049

A new nonbinary sequence family with low correlation and large size

Hua Liang 1, , Wenbing Chen 2,, , Jinquan Luo 3, and  Yuansheng Tang 4,

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

* Corresponding author

Received  May 2015 Revised  February 2016 Published  November 2017

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 $N=p^n-1$ 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.

Keywords: Sequence family, m-sequence, cross-correlation, exponential sum.
Mathematics Subject Classification: Primary: 94A55; Secondary: 11T71.
Citation: Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049
References:
[1]

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.

[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), 2847-2867.  doi: 10.1109/TIT.2002.804044.

[3]

T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequence, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.

[4]

T. Kasami, Weight distribution of Bose-Chaudhuri-Hocquenghem codes, in Combinatorial Mathematics and Its Applications, Chapel Hill, NC: Univ. North Carolina Press, 1969,335-357.

[5]

T. Kasami, Weight Distribution Formular for Some Class of Cyclic Codes, Coordinated Science Lab., Univ. Illinois at Urbana-Champaign, Urbana, IL, Tech. Rep. R-285(AD 637524), 1966.

[6]

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), 3825-3830.  doi: 10.1109/TIT.2011.2133730.

[7]

D. S. Kim, H. J. Chae and H. Y. Song, A generalizaton of the family of $p$-ary decimated sequences with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 7614-7617.  doi: 10.1109/TIT.2011.2159576.

[8]

P. V. Kumar and O. Moreno, Prime-phase sequences with periodic correlation properites better than binary sequences, IEEE Trans. Inf. Theory, 37 (1991), 603-616. 

[9]

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.

[10]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, MA, 1983.

[11]

S. C. Liu and J. J. Komo, Nonbinary Kasami sequences over $GF(p)$, IEEE Trans. Inf. Theory, 38 (1992), 1409-1412.  doi: 10.1109/18.144728.

[12]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.  doi: 10.1109/TIT.2008.2006424.

[13]

J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353.  doi: 10.1109/TIT.2008.2006394.

[14]

J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation, in Proceeding of IWSDA'11, 2011, 44-47. 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), 289-295.  doi: 10.1109/18.746820.

[16]

G. J. Ness, T. Helleseth and A. Kholosha, On the correlation distribution of the Coulter-Matthews decimation, IEEE Trans. Inf. Theory, 52 (2006), 2241-2247.  doi: 10.1109/TIT.2006.872857.

[17]

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.

[18]

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.

[19]

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.

[20]

Y. Xia and S. Chen, A new family of $p$-ary sequences with low correlation constructed from decimated sequences, IEEE Trans. Inf. Theory, 58 (2012), 6037-6046.  doi: 10.1109/TIT.2012.2201132.

[21]

N. Y. Yu and G. Gong, A new binary sequence family with low correlation and large size, IEEE Trans. Inf. Theory, 52 (2006), 1624-1636.  doi: 10.1109/TIT.2006.871062.

show all references

References:
[1]

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.

[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), 2847-2867.  doi: 10.1109/TIT.2002.804044.

[3]

T. Helleseth, Some results about the cross-correlation function between two maximal-linear sequence, Discrete Math., 16 (1976), 209-232.  doi: 10.1016/0012-365X(76)90100-X.

[4]

T. Kasami, Weight distribution of Bose-Chaudhuri-Hocquenghem codes, in Combinatorial Mathematics and Its Applications, Chapel Hill, NC: Univ. North Carolina Press, 1969,335-357.

[5]

T. Kasami, Weight Distribution Formular for Some Class of Cyclic Codes, Coordinated Science Lab., Univ. Illinois at Urbana-Champaign, Urbana, IL, Tech. Rep. R-285(AD 637524), 1966.

[6]

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), 3825-3830.  doi: 10.1109/TIT.2011.2133730.

[7]

D. S. Kim, H. J. Chae and H. Y. Song, A generalizaton of the family of $p$-ary decimated sequences with low correlation, IEEE Trans. Inf. Theory, 57 (2011), 7614-7617.  doi: 10.1109/TIT.2011.2159576.

[8]

P. V. Kumar and O. Moreno, Prime-phase sequences with periodic correlation properites better than binary sequences, IEEE Trans. Inf. Theory, 37 (1991), 603-616. 

[9]

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.

[10]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, MA, 1983.

[11]

S. C. Liu and J. J. Komo, Nonbinary Kasami sequences over $GF(p)$, IEEE Trans. Inf. Theory, 38 (1992), 1409-1412.  doi: 10.1109/18.144728.

[12]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.  doi: 10.1109/TIT.2008.2006424.

[13]

J. Luo and K. Feng, Cyclic codes and sequences from generalized Coulter-Matthews function, IEEE Trans. Inf. Theory, 54 (2008), 5345-5353.  doi: 10.1109/TIT.2008.2006394.

[14]

J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross correlation, in Proceeding of IWSDA'11, 2011, 44-47. 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), 289-295.  doi: 10.1109/18.746820.

[16]

G. J. Ness, T. Helleseth and A. Kholosha, On the correlation distribution of the Coulter-Matthews decimation, IEEE Trans. Inf. Theory, 52 (2006), 2241-2247.  doi: 10.1109/TIT.2006.872857.

[17]

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.

[18]

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.

[19]

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.

[20]

Y. Xia and S. Chen, A new family of $p$-ary sequences with low correlation constructed from decimated sequences, IEEE Trans. Inf. Theory, 58 (2012), 6037-6046.  doi: 10.1109/TIT.2012.2201132.

[21]

N. Y. Yu and G. Gong, A new binary sequence family with low correlation and large size, IEEE Trans. Inf. Theory, 52 (2006), 1624-1636.  doi: 10.1109/TIT.2006.871062.

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