American Institute of Mathematical Sciences

Advanced
November  2013, 7(4): 409-424. doi: 10.3934/amc.2013.7.409

The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$

Yuhua Sun 1, , Zilong Wang 2, , Hui Li 2, and  Tongjiang Yan 1,

1. 

College of Sciences, China University of Petroleum, 66 Changjiang Xilu, Qingdao, Shandong 266580, China, China
2. 

State Key Laboratory of Integrated Service Networks, Xidian University, 2 Taibai Nanlu, Xi'an, Shannxi 710071, China, China

Received  May 2012 Published  October 2013

Families of $m-$sequences with low correlation property have important applications in communication systems. In this paper, for a prime $p\equiv 1\ \mathrm{mod}\ 4$ and an odd integer $k$, we study the cross correlation between a $p$-ary $m$-sequence $\{s_t\}$ of period $p^n-1$ and its decimated sequence $\{s_{dt}\}$, where $d=\frac{(p^k+1)^2}{2(p^e+1)}$, $e|k$ and $n = 2k$. Using quadratic form polynomial theory, we obtain the distribution of the cross correlation which is six-valued. Specially, our results show that the magnitude of the cross correlation is upper bounded by $2\sqrt{p^n}+1$ for $p=5$ and $e=1$, which is meaningful in CDMA communication systems.
Keywords: Cross correlation, p-ary m-sequence, quadratic form., decimated sequence.
Mathematics Subject Classification: Primary: 94A55; Secondary: 11B5.
Citation: 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
References:
[1]

A. W. Bluher, On $x^{q+1}+ax+b$,, Finite Fields Appl., 10 (2004), 285. doi: 10.1016/j.ffa.2003.08.004. Google Scholar

[2]

S. T. Choi, 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. doi: 10.1109/TIT.2011.2177573. Google Scholar

[3]

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. doi: 10.1109/18.923728. Google Scholar

[4]

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

[5]

T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory (eds. V. Pless and C. Huffman), (1998), 1765. Google Scholar

[6]

R. Lidl and H. Niederreiter, Finite Fields,, Addison-Wesley, (1983). Google Scholar

[7]

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

[8]

J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross-correlation,, in Proceedings of IWSDA'11, (2011), 44. doi: 10.1109/IWSDA.2011.6159435. Google Scholar

[9]

E. N. Müller, On the cross-correlation of sequences over $GF(p)$ with short periods,, IEEE Trans. Inf. Theory, 45 (1999), 289. Google Scholar

[10]

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. doi: 10.1109/TIT.2008.924694. Google Scholar

[11]

T. Storer, Cyclotomy and Difference Sets,, Markham, (1967). Google Scholar

show all references

References:
[1]

A. W. Bluher, On $x^{q+1}+ax+b$,, Finite Fields Appl., 10 (2004), 285. doi: 10.1016/j.ffa.2003.08.004. Google Scholar

[2]

S. T. Choi, 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. doi: 10.1109/TIT.2011.2177573. Google Scholar

[3]

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. doi: 10.1109/18.923728. Google Scholar

[4]

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

[5]

T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory (eds. V. Pless and C. Huffman), (1998), 1765. Google Scholar

[6]

R. Lidl and H. Niederreiter, Finite Fields,, Addison-Wesley, (1983). Google Scholar

[7]

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

[8]

J. Luo, T. Helleseth and A. Kholosha, Two nonbinary sequences with six-valued cross-correlation,, in Proceedings of IWSDA'11, (2011), 44. doi: 10.1109/IWSDA.2011.6159435. Google Scholar

[9]

E. N. Müller, On the cross-correlation of sequences over $GF(p)$ with short periods,, IEEE Trans. Inf. Theory, 45 (1999), 289. Google Scholar

[10]

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. doi: 10.1109/TIT.2008.924694. Google Scholar

[11]

T. Storer, Cyclotomy and Difference Sets,, Markham, (1967). Google Scholar

[1]

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
[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]

Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019
[5]

Lanqiang Li, Shixin Zhu, Li Liu. The weight distribution of a class of p-ary cyclic codes and their applications. Advances in Mathematics of Communications, 2019, 13 (1) : 137-156. doi: 10.3934/amc.2019008
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[13]

Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018
[14]

Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial & Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451
[15]

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
[16]

Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533
[17]

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
[18]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
[19]

Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115
[20]

Alexander Zeh, Antonia Wachter. Fast multi-sequence shift-register synthesis with the Euclidean algorithm. Advances in Mathematics of Communications, 2011, 5 (4) : 667-680. doi: 10.3934/amc.2011.5.667

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences