# 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. 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

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##### 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
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