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)}

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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

Abstract
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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), 285-305. doi: 10.1016/j.ffa.2003.08.004.
[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.
[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.
[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.
[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.
[6]	R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Boston, 1983.
[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.
[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.
[9]	E. N. Müller, On the cross-correlation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289-295.
[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.
[11]	T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.

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.
[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.
[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.
[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.
[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.
[6]	R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Boston, 1983.
[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.
[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.
[9]	E. N. Müller, On the cross-correlation of sequences over $GF(p)$ with short periods, IEEE Trans. Inf. Theory, 45 (1999), 289-295.
[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.
[11]	T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967.

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