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

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

* 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.

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. ChoiT. LimJ. 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. KimS. T. ChoiJ. 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. KimH. 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. NessT. 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. SeoY. S. KimJ. 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. SunZ. WangH. 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. XiaX. 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. ChoiT. LimJ. 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. KimS. T. ChoiJ. 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. KimH. 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. NessT. 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. SeoY. S. KimJ. 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. SunZ. WangH. 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. XiaX. 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.

[1]

Yuhua Sun, Zilong Wang, Hui Li, Tongjiang Yan. The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$. Advances in Mathematics of Communications, 2013, 7 (4) : 409-424. doi: 10.3934/amc.2013.7.409

[2]

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

[3]

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

[4]

Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375

[5]

Ferruh Özbudak, Eda Tekin. Correlation distribution of a sequence family generalizing some sequences of Trachtenberg. Advances in Mathematics of Communications, 2021, 15 (4) : 647-662. doi: 10.3934/amc.2020087

[6]

Mehmet Duran Toksari, Emel Kizilkaya Aydogan, Berrin Atalay, Saziye Sari. Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1795-1807. doi: 10.3934/jimo.2021044

[7]

Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475

[8]

Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55

[9]

Samuel T. Blake, Thomas E. Hall, Andrew Z. Tirkel. Arrays over roots of unity with perfect autocorrelation and good ZCZ cross-correlation. Advances in Mathematics of Communications, 2013, 7 (3) : 231-242. doi: 10.3934/amc.2013.7.231

[10]

Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9

[11]

Huaning Liu, Xi Liu. On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021008

[12]

Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004

[13]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[14]

Xiujie Zhang, Xianhua Niu, Xin Tan. Constructions of optimal low hit zone frequency hopping sequence sets with large family size. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021071

[15]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015

[16]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018

[17]

Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial and Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451

[18]

Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135

[19]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533

[20]

Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023

2021 Impact Factor: 1.015

Metrics

  • PDF downloads (156)
  • HTML views (379)
  • Cited by (0)

[Back to Top]