American Institute of Mathematical Sciences

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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

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

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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

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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

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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

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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

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

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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

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

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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

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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

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