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

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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

Abstract
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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), 273-307.
[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.
[3]	L. Goubin, C. Mauduit and A. Sárközy, Construction of large families of pseudorandom binary sequences, J. Number Theory, 106 (2004), 56-69. doi: 10.1016/j.jnt.2003.12.002.
[4]	Y. K. Han and K. Yang, New $M$-ary sequence families with low correlation and large size, IEEE Trans. Inf. Theory, 55 (2009), 1815-1823. doi: 10.1109/TIT.2009.2013040.
[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]	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), 3768-3774. doi: 10.1109/TIT.2008.926428.
[7]	Y. J. Kim and H. Y. Song, Cross correlation of Sidel'nikov sequences and their constant multiples, IEEE Trans. Inf. Theory, 53 (2007), 1220-1224. doi: 10.1109/TIT.2006.890723.
[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, 311-315. doi: 10.1109/ISIT.2006.261604.
[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), 579-592. doi: 10.1109/18.485726.
[10]	V. M. Sidel'nikov, Some $k$-valued pseudo-random sequences and nearly equidistant codes, Probl. Inf. Transm., 5 (1969), 12-16.
[11]	V. M. Sidel'nikov, On mutual correlation of sequences, Soviet Math. Dokl, 12 (1971), 197-201.
[12]	D. Wan, Generators and irreducible polynomials over finite fields, Math. Comput., 66 (1997), 1195-1212. doi: 10.1090/S0025-5718-97-00835-1.
[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), 3990-3998. doi: 10.1109/TIT.2013.2243496.
[14]	A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. USA, 34 (1948), 204-207. doi: 10.1073/pnas.34.5.204.
[15]	L. R. Welch, Lower bounds on the minimum correlation of signal, IEEE Trans. Inf. Theory, 20 (1974), 397-399.
[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), 6376-6387. doi: 10.1109/TIT.2010.2079590.
[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), 4061-4070. doi: 10.1109/TIT.2010.2050793.

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References:

[1]	P. Deligne, La conjecture de Weil I, Publ. Math. IHES, 43 (1974), 273-307.
[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.
[3]	L. Goubin, C. Mauduit and A. Sárközy, Construction of large families of pseudorandom binary sequences, J. Number Theory, 106 (2004), 56-69. doi: 10.1016/j.jnt.2003.12.002.
[4]	Y. K. Han and K. Yang, New $M$-ary sequence families with low correlation and large size, IEEE Trans. Inf. Theory, 55 (2009), 1815-1823. doi: 10.1109/TIT.2009.2013040.
[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]	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), 3768-3774. doi: 10.1109/TIT.2008.926428.
[7]	Y. J. Kim and H. Y. Song, Cross correlation of Sidel'nikov sequences and their constant multiples, IEEE Trans. Inf. Theory, 53 (2007), 1220-1224. doi: 10.1109/TIT.2006.890723.
[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, 311-315. doi: 10.1109/ISIT.2006.261604.
[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), 579-592. doi: 10.1109/18.485726.
[10]	V. M. Sidel'nikov, Some $k$-valued pseudo-random sequences and nearly equidistant codes, Probl. Inf. Transm., 5 (1969), 12-16.
[11]	V. M. Sidel'nikov, On mutual correlation of sequences, Soviet Math. Dokl, 12 (1971), 197-201.
[12]	D. Wan, Generators and irreducible polynomials over finite fields, Math. Comput., 66 (1997), 1195-1212. doi: 10.1090/S0025-5718-97-00835-1.
[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), 3990-3998. doi: 10.1109/TIT.2013.2243496.
[14]	A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. USA, 34 (1948), 204-207. doi: 10.1073/pnas.34.5.204.
[15]	L. R. Welch, Lower bounds on the minimum correlation of signal, IEEE Trans. Inf. Theory, 20 (1974), 397-399.
[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), 6376-6387. doi: 10.1109/TIT.2010.2079590.
[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), 4061-4070. doi: 10.1109/TIT.2010.2050793.

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