# American Institute of Mathematical Sciences

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 $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$

 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.
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), 285-305. 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), 1873-1879. 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), 1473-1481. 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), 209-232. 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), Elsevier Science Publishers, 1998, 1765-1853.  Google Scholar [6] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Boston, 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), 5332-5344. 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), 44-47. 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), 289-295. 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), 3140-3149. doi: 10.1109/TIT.2008.924694.  Google Scholar [11] T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.  Google Scholar

show all references

##### References:
 [1] A. W. Bluher, On $x^{q+1}+ax+b$, Finite Fields Appl., 10 (2004), 285-305. 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), 1873-1879. 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), 1473-1481. 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), 209-232. 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), Elsevier Science Publishers, 1998, 1765-1853.  Google Scholar [6] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Boston, 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), 5332-5344. 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), 44-47. 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), 289-295. 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), 3140-3149. doi: 10.1109/TIT.2008.924694.  Google Scholar [11] T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 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] 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 [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] 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 [10] 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 [11] 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 [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] Qing Liu, Bingo Wing-Kuen Ling, Qingyun Dai, Qing Miao, Caixia Liu. Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1993-2011. doi: 10.3934/jimo.2020055 [14] 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 [15] 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 [16] Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & 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 & 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 & 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

2020 Impact Factor: 0.935