# American Institute of Mathematical Sciences

May  2015, 9(2): 199-210. doi: 10.3934/amc.2015.9.199

## A class of quaternary sequences with low correlation

 1 Information Security and National Computing Grid Laboratory, Southwest Jiaotong University, Chengdu, Sichuan 610031, China, China 2 Department of Informatics, University of Bergen, N-5020 Bergen

Received  March 2014 Revised  November 2014 Published  May 2015

A class of quaternary sequences $\mathbb{S}_{\lambda}$ had been proven to be optimal for some special values of $\lambda$. In this note, $\mathbb{S}_{\lambda}$ is investigated for all $\lambda$ by virtue of the $\mathbb{Z}_4$-valued quadratic forms over Galois rings. As a consequence, a new class of quaternary sequences with low correlation is obtained and the correlation distribution is also completely determined. It also turns out that the known optimal quaternary sequences $\mathbb{S}_{\lambda}$ for particular $\lambda$ can be easily obtained from our approach.
Citation: Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199
##### References:
 [1] S. Boztas, R. Hammons and P. V. Kumar, $4$-phase sequences with near-optimum correlation properties, IEEE Trans. Inf. Theory, 14 (1992), 1101-1113. doi: 10.1109/18.135649.  Google Scholar [2] S. Boztas and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span, IEEE Trans. Inf. Theory, 40 (1994), 532-537. doi: 10.1109/18.312181.  Google Scholar [3] E. H. Brown, Generalizations of the Kervaire invariant, Annals Math., 95 (1972), 368-383. doi: 10.2307/1970804.  Google Scholar [4] P. Fan and M. Darnell, Sequence Design for Communications Applications, John Wiley, 1996. Google Scholar [5] R. Gold, Maximal recursive sequences with $3$-valued recursive crosscorrelation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156. doi: 10.1109/TIT.1968.1054106.  Google Scholar [6] T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdman, 1998.  Google Scholar [7] W. Jiang, L. Hu, X. Tang and X. Zeng, New optimal quadriphase sequences with larger linear span, IEEE Trans. Inf. Theory, 55 (2009), 458-470. doi: 10.1109/TIT.2008.2008122.  Google Scholar [8] A. Johansen, T. Helleseth and X. Tang, The correlation distribution of quaternary sequences of period $2(2^n-1)$, IEEE Trans. Inf. Theory, 54 (2008), 3130-3139. doi: 10.1109/TIT.2008.924727.  Google Scholar [9] T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes, Coordinated Sci. Lab., Univ. Illinois Urbana-Champaign, Tech. Rep. R-285, 1966. Google Scholar [10] S. H. Kim and J. S. No, New families of binary sequences with low crosscorrelation property, IEEE Trans. Inf. Theory, 49 (2003), 3059-3065. doi: 10.1109/TIT.2003.818399.  Google Scholar [11] 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.  Google Scholar [12] N. Li, X. Tang, X. Zeng and L. Hu, On the correlation distributions of optimal quaternary sequence family $\mathcal U$ and optimal binary sequence family $\mathcal V$, IEEE Trans. Inf. Theory, 57 (2011), 3815-3824. doi: 10.1109/TIT.2011.2132670.  Google Scholar [13] K.-U. Schmidt, $\mathbb Z_4$-valued quadratic forms and quaternary sequence families, IEEE Trans. Inf. Theory, 55 (2009), 5803-5810. doi: 10.1109/TIT.2009.2032818.  Google Scholar [14] V. Sidelnikov, On mutual correlation of sequences, Soviet Math. Dokl., 12 (1971), 197-201.  Google Scholar [15] X. Tang and T. Helleseth, Generic construction of quaternary sequences of period $2N$ with low correlation from quaternary sequences of odd period $N$, IEEE Trans. Inf. Theory, 57 (2011), 2295-2300. doi: 10.1109/TIT.2011.2110290.  Google Scholar [16] X. Tang, T. Helleseth and P. Fan, A new optimal quaternary sequence family of length $2(2^n-1)$ obtained from the orthogonal transformation of families $\mathcal B$ and $\mathcal C$, Des. Codes Crypt., 53 (2009), 137-148. doi: 10.1007/s10623-009-9294-y.  Google Scholar [17] X. Tang, T. Helleseth, L. Hu and W. Jiang, Two new families of optimal binary sequences obtained from quaternary sequences, IEEE Trans. Inf. Theory, 55 (2009), 433-436. doi: 10.1109/TIT.2009.2013023.  Google Scholar [18] X. Tang and P. Udaya, A note on the optimal quadriphase sequences families, IEEE Trans. Inf. Theory, 53 (2007), 433-436. doi: 10.1109/TIT.2006.887502.  Google Scholar [19] X. Tang, P. Udaya and P. Fan, Quadriphase sequences obtained from binary quadratic form sequences, in Sequences and Their Applications - SETA 2004, 2005, 243-254. doi: 10.1007/11423461_17.  Google Scholar [20] P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings, Ph.D thesis, Dept. Elec. Eng., Indian Inst. Technol., Kanpur, 1992. Google Scholar [21] P. Udaya and M. U. Siddiqi, Optimal and suboptimal quadriphase sequences derived from maximal length sequences over $\mathbb Z_4$, Appl. Algebra Eng. Commun. Comput., 9 (1998), 161-191. doi: 10.1007/s002000050101.  Google Scholar [22] L. R. Welch, Lower bounds on the maximum crosscorrelation on the signals, IEEE Trans. Inf. Theory, 20 (1974), 397-399. doi: 10.1109/TIT.1974.1055219.  Google Scholar

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##### References:
 [1] S. Boztas, R. Hammons and P. V. Kumar, $4$-phase sequences with near-optimum correlation properties, IEEE Trans. Inf. Theory, 14 (1992), 1101-1113. doi: 10.1109/18.135649.  Google Scholar [2] S. Boztas and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span, IEEE Trans. Inf. Theory, 40 (1994), 532-537. doi: 10.1109/18.312181.  Google Scholar [3] E. H. Brown, Generalizations of the Kervaire invariant, Annals Math., 95 (1972), 368-383. doi: 10.2307/1970804.  Google Scholar [4] P. Fan and M. Darnell, Sequence Design for Communications Applications, John Wiley, 1996. Google Scholar [5] R. Gold, Maximal recursive sequences with $3$-valued recursive crosscorrelation functions, IEEE Trans. Inf. Theory, 14 (1968), 154-156. doi: 10.1109/TIT.1968.1054106.  Google Scholar [6] T. Helleseth and P. V. Kumar, Sequences with low correlation, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdman, 1998.  Google Scholar [7] W. Jiang, L. Hu, X. Tang and X. Zeng, New optimal quadriphase sequences with larger linear span, IEEE Trans. Inf. Theory, 55 (2009), 458-470. doi: 10.1109/TIT.2008.2008122.  Google Scholar [8] A. Johansen, T. Helleseth and X. Tang, The correlation distribution of quaternary sequences of period $2(2^n-1)$, IEEE Trans. Inf. Theory, 54 (2008), 3130-3139. doi: 10.1109/TIT.2008.924727.  Google Scholar [9] T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes, Coordinated Sci. Lab., Univ. Illinois Urbana-Champaign, Tech. Rep. R-285, 1966. Google Scholar [10] S. H. Kim and J. S. No, New families of binary sequences with low crosscorrelation property, IEEE Trans. Inf. Theory, 49 (2003), 3059-3065. doi: 10.1109/TIT.2003.818399.  Google Scholar [11] 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.  Google Scholar [12] N. Li, X. Tang, X. Zeng and L. Hu, On the correlation distributions of optimal quaternary sequence family $\mathcal U$ and optimal binary sequence family $\mathcal V$, IEEE Trans. Inf. Theory, 57 (2011), 3815-3824. doi: 10.1109/TIT.2011.2132670.  Google Scholar [13] K.-U. Schmidt, $\mathbb Z_4$-valued quadratic forms and quaternary sequence families, IEEE Trans. Inf. Theory, 55 (2009), 5803-5810. doi: 10.1109/TIT.2009.2032818.  Google Scholar [14] V. Sidelnikov, On mutual correlation of sequences, Soviet Math. Dokl., 12 (1971), 197-201.  Google Scholar [15] X. Tang and T. Helleseth, Generic construction of quaternary sequences of period $2N$ with low correlation from quaternary sequences of odd period $N$, IEEE Trans. Inf. Theory, 57 (2011), 2295-2300. doi: 10.1109/TIT.2011.2110290.  Google Scholar [16] X. Tang, T. Helleseth and P. Fan, A new optimal quaternary sequence family of length $2(2^n-1)$ obtained from the orthogonal transformation of families $\mathcal B$ and $\mathcal C$, Des. Codes Crypt., 53 (2009), 137-148. doi: 10.1007/s10623-009-9294-y.  Google Scholar [17] X. Tang, T. Helleseth, L. Hu and W. Jiang, Two new families of optimal binary sequences obtained from quaternary sequences, IEEE Trans. Inf. Theory, 55 (2009), 433-436. doi: 10.1109/TIT.2009.2013023.  Google Scholar [18] X. Tang and P. Udaya, A note on the optimal quadriphase sequences families, IEEE Trans. Inf. Theory, 53 (2007), 433-436. doi: 10.1109/TIT.2006.887502.  Google Scholar [19] X. Tang, P. Udaya and P. Fan, Quadriphase sequences obtained from binary quadratic form sequences, in Sequences and Their Applications - SETA 2004, 2005, 243-254. doi: 10.1007/11423461_17.  Google Scholar [20] P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings, Ph.D thesis, Dept. Elec. Eng., Indian Inst. Technol., Kanpur, 1992. Google Scholar [21] P. Udaya and M. U. Siddiqi, Optimal and suboptimal quadriphase sequences derived from maximal length sequences over $\mathbb Z_4$, Appl. Algebra Eng. Commun. Comput., 9 (1998), 161-191. doi: 10.1007/s002000050101.  Google Scholar [22] L. R. Welch, Lower bounds on the maximum crosscorrelation on the signals, IEEE Trans. Inf. Theory, 20 (1974), 397-399. doi: 10.1109/TIT.1974.1055219.  Google Scholar
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