American Institute of Mathematical Sciences

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November  2013, 7(4): 475-484. doi: 10.3934/amc.2013.7.475

Correlation of binary sequence families derived from the multiplicative characters of finite fields

Zilong Wang 1, and  Guang Gong 2,

1. 

State Key Laboratory of Integrated Service Networks, Xidian University, Xi'an, Shanxi 710071, China
2. 

Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Received  November 2012 Published  October 2013

In this paper, new constructions of the binary sequence families of period $q-1$ with large family size and low correlation, derived from multiplicative characters of finite fields for odd prime powers, are proposed. For $m ≥ 2$, the maximum correlation magnitudes of new sequence families $\mathcal{S}_m$ are bounded by $(2m-2)\sqrt{q}+2m+2$, and the family sizes of $\mathcal{S}_m$ are given by $q-1$ for $m=2$, $2(q-1)-1$ for $m=3$, $(q^2-1)q^{\frac{m-4}{2}}$ for $m$ even, $m>2$, and $2(q-1)q^{\frac{m-3}{2}}$ for $m$ odd, $m>3$. It is shown that the known binary Sidel'nikov-based sequence families are equivalent to the new constructions for the case $m=2$.
Keywords: Binary sequences, multiplicative characters, correlation, Weil bound..
Mathematics Subject Classification: Primary: 94B25, 11T2.
Citation: 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
References:
[1]

P. Deligne, La conjecture de Weil I,, Publ. Math. IHES, 43 (1974), 273. Google Scholar

[2]

S. W. Golomb and G. Gong, Signal Design with Good Correlation: for Wireless Communications, Cryptography and Radar Applications,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511546907. Google Scholar

[3]

L. Goubin, C. Mauduit and A. Sárközy, Construction of large families of pseudorandom binary sequences,, J. Number Theory, 106 (2004), 56. doi: 10.1016/j.jnt.2003.12.002. Google Scholar

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Y. K. Han and K. Yang, New $M$-ary sequence families with low correlation and large size,, IEEE Trans. Inf. Theory, 55 (2009), 1815. doi: 10.1109/TIT.2009.2013040. 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

[6]

Y. Kim, J. Chung, J. S. No and H. Chung, New families of $M$-ary sequences with low correlation constructed from Sidel'nikov sequences,, IEEE Trans. Inf. Theory, 54 (2008), 3768. doi: 10.1109/TIT.2008.926428. Google Scholar

[7]

Y. J. Kim and H. Y. Song, Cross correlation of Sidel'nikov sequences and their constant multiples,, IEEE Trans. Inf. Theory, 53 (2007), 1220. doi: 10.1109/TIT.2006.890723. Google Scholar

[8]

Y. J. Kim, H. Y. Song, G. Gong and H. Chung, Crosscorrelation of $q$-ary power residue sequences of period $p$,, in Proc. IEEE ISIT, (2006), 311. doi: 10.1109/ISIT.2006.261604. Google Scholar

[9]

P. V. Kumar, T. Helleseth, A. R. Calderbank and A. R. Hammons, Large families of quaternary sequences with low correlation,, IEEE Trans. Inf. Theory, 42 (1996), 579. doi: 10.1109/18.485726. Google Scholar

[10]

V. M. Sidel'nikov, Some $k$-valued pseudo-random sequences and nearly equidistant codes,, Probl. Inf. Transm., 5 (1969), 12. Google Scholar

[11]

V. M. Sidel'nikov, On mutual correlation of sequences,, Soviet Math. Dokl, 12 (1971), 197. Google Scholar

[12]

D. Wan, Generators and irreducible polynomials over finite fields,, Math. Comput., 66 (1997), 1195. doi: 10.1090/S0025-5718-97-00835-1. Google Scholar

[13]

Z. Wang and G. Gong, New polyphase sequence families with low correlation derived from the Weil bound of exponential sums,, IEEE Trans. Inf. Theory, 59 (2013), 3990. doi: 10.1109/TIT.2013.2243496. Google Scholar

[14]

A. Weil, On some exponential sums,, Proc. Natl. Acad. Sci. USA, 34 (1948), 204. doi: 10.1073/pnas.34.5.204. Google Scholar

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L. R. Welch, Lower bounds on the minimum correlation of signal,, IEEE Trans. Inf. Theory, 20 (1974), 397. Google Scholar

[16]

N. Y. Yu and G. Gong, Multiplicative characters, the Weil Bound, and polyphase sequence families with low correlation,, IEEE Trans. Inf. Theory, 56 (2010), 6376. doi: 10.1109/TIT.2010.2079590. Google Scholar

[17]

N. Y. Yu and G. Gong, New construction of $M$-ary sequence families with low correlation from the structure of Sidelnikov sequences,, IEEE Trans. Inf. Theory, 56 (2010), 4061. doi: 10.1109/TIT.2010.2050793. Google Scholar

show all references

References:
[1]

P. Deligne, La conjecture de Weil I,, Publ. Math. IHES, 43 (1974), 273. Google Scholar

[2]

S. W. Golomb and G. Gong, Signal Design with Good Correlation: for Wireless Communications, Cryptography and Radar Applications,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511546907. Google Scholar

[3]

L. Goubin, C. Mauduit and A. Sárközy, Construction of large families of pseudorandom binary sequences,, J. Number Theory, 106 (2004), 56. doi: 10.1016/j.jnt.2003.12.002. Google Scholar

[4]

Y. K. Han and K. Yang, New $M$-ary sequence families with low correlation and large size,, IEEE Trans. Inf. Theory, 55 (2009), 1815. doi: 10.1109/TIT.2009.2013040. 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]

Y. Kim, J. Chung, J. S. No and H. Chung, New families of $M$-ary sequences with low correlation constructed from Sidel'nikov sequences,, IEEE Trans. Inf. Theory, 54 (2008), 3768. doi: 10.1109/TIT.2008.926428. Google Scholar

[7]

Y. J. Kim and H. Y. Song, Cross correlation of Sidel'nikov sequences and their constant multiples,, IEEE Trans. Inf. Theory, 53 (2007), 1220. doi: 10.1109/TIT.2006.890723. Google Scholar

[8]

Y. J. Kim, H. Y. Song, G. Gong and H. Chung, Crosscorrelation of $q$-ary power residue sequences of period $p$,, in Proc. IEEE ISIT, (2006), 311. doi: 10.1109/ISIT.2006.261604. Google Scholar

[9]

P. V. Kumar, T. Helleseth, A. R. Calderbank and A. R. Hammons, Large families of quaternary sequences with low correlation,, IEEE Trans. Inf. Theory, 42 (1996), 579. doi: 10.1109/18.485726. Google Scholar

[10]

V. M. Sidel'nikov, Some $k$-valued pseudo-random sequences and nearly equidistant codes,, Probl. Inf. Transm., 5 (1969), 12. Google Scholar

[11]

V. M. Sidel'nikov, On mutual correlation of sequences,, Soviet Math. Dokl, 12 (1971), 197. Google Scholar

[12]

D. Wan, Generators and irreducible polynomials over finite fields,, Math. Comput., 66 (1997), 1195. doi: 10.1090/S0025-5718-97-00835-1. Google Scholar

[13]

Z. Wang and G. Gong, New polyphase sequence families with low correlation derived from the Weil bound of exponential sums,, IEEE Trans. Inf. Theory, 59 (2013), 3990. doi: 10.1109/TIT.2013.2243496. Google Scholar

[14]

A. Weil, On some exponential sums,, Proc. Natl. Acad. Sci. USA, 34 (1948), 204. doi: 10.1073/pnas.34.5.204. Google Scholar

[15]

L. R. Welch, Lower bounds on the minimum correlation of signal,, IEEE Trans. Inf. Theory, 20 (1974), 397. Google Scholar

[16]

N. Y. Yu and G. Gong, Multiplicative characters, the Weil Bound, and polyphase sequence families with low correlation,, IEEE Trans. Inf. Theory, 56 (2010), 6376. doi: 10.1109/TIT.2010.2079590. Google Scholar

[17]

N. Y. Yu and G. Gong, New construction of $M$-ary sequence families with low correlation from the structure of Sidelnikov sequences,, IEEE Trans. Inf. Theory, 56 (2010), 4061. doi: 10.1109/TIT.2010.2050793. Google Scholar

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