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:
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References:
  S. Boztas, R. Hammons and P. V. Kumar, $4$-phase sequences with near-optimum correlation properties,, IEEE Trans. Inf. Theory, 14 (1992), 1101. doi: 10.1109/18.135649. Google Scholar  S. Boztas and P. V. Kumar, Binary sequences with Gold-like correlation but larger linear span,, IEEE Trans. Inf. Theory, 40 (1994), 532. doi: 10.1109/18.312181. Google Scholar  E. H. Brown, Generalizations of the Kervaire invariant,, Annals Math., 95 (1972), 368. doi: 10.2307/1970804. Google Scholar  P. Fan and M. Darnell, Sequence Design for Communications Applications,, John Wiley, (1996). Google Scholar  R. Gold, Maximal recursive sequences with $3$-valued recursive crosscorrelation functions,, IEEE Trans. Inf. Theory, 14 (1968), 154. doi: 10.1109/TIT.1968.1054106. Google Scholar  T. Helleseth and P. V. Kumar, Sequences with low correlation,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998). Google Scholar  W. Jiang, L. Hu, X. Tang and X. Zeng, New optimal quadriphase sequences with larger linear span,, IEEE Trans. Inf. Theory, 55 (2009), 458. doi: 10.1109/TIT.2008.2008122. Google Scholar  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. doi: 10.1109/TIT.2008.924727. Google Scholar  T. Kasami, Weight Distribution Formula for Some Class of Cyclic Codes,, Coordinated Sci. Lab., (1966). Google Scholar  S. H. Kim and J. S. No, New families of binary sequences with low crosscorrelation property,, IEEE Trans. Inf. Theory, 49 (2003), 3059. doi: 10.1109/TIT.2003.818399. Google Scholar  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  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. doi: 10.1109/TIT.2011.2132670. Google Scholar  K.-U. Schmidt, $\mathbb Z_4$-valued quadratic forms and quaternary sequence families,, IEEE Trans. Inf. Theory, 55 (2009), 5803. doi: 10.1109/TIT.2009.2032818. Google Scholar  V. Sidelnikov, On mutual correlation of sequences,, Soviet Math. Dokl., 12 (1971), 197. Google Scholar  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. doi: 10.1109/TIT.2011.2110290. Google Scholar  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. doi: 10.1007/s10623-009-9294-y. Google Scholar  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. doi: 10.1109/TIT.2009.2013023. Google Scholar  X. Tang and P. Udaya, A note on the optimal quadriphase sequences families,, IEEE Trans. Inf. Theory, 53 (2007), 433. doi: 10.1109/TIT.2006.887502. Google Scholar  X. Tang, P. Udaya and P. Fan, Quadriphase sequences obtained from binary quadratic form sequences,, in Sequences and Their Applications - SETA 2004, (2004), 243. doi: 10.1007/11423461_17. Google Scholar  P. Udaya, Polyphase and Frequency Hopping Sequences Obtained from Finite Rings,, Ph.D thesis, (1992). Google Scholar  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. doi: 10.1007/s002000050101. Google Scholar  L. R. Welch, Lower bounds on the maximum crosscorrelation on the signals,, IEEE Trans. Inf. Theory, 20 (1974), 397. doi: 10.1109/TIT.1974.1055219. Google Scholar
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