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Open AccessOpen Access JUSTC Mathematics 11 May 2022

Almost balanced and uncorrelated quaternary sequence pairs of even length

  • 1.

    Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

  • 2.

    Key Laboratory of Electromagnetic Space Information, University of Science and Technology of China, Hefei 230027, China

Funds:  The work is supported by Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200), USTC Research Funds of the Double First-Class Initiative (Grant No. YD0010002004), and the Fundamental Research Funds for the Central Universities (Grant No. WK0010000068).
Cite this:
https://doi.org/10.52396/JUSTC-2021-0234
More Information
  • Author Bio:

    Yi Ouyang is a professor at the University of Science and Technology of China (USTC). He received the PhD degree in Mathematics from the University of Minnesota in 2000. His research interests include Fontaine theory of p-adic representations and (φ, Γ)-modules and its connection to Iwasawa theory, theory of elliptic curves and its application in cryptography, class groups and class numbers of number fields and function fields

    Sen Wang is working toward the PhD degree at the School of Mathematical Sciences, University of Science and Technology of China. His area of interests include sequence design and its applications

  • Corresponding author: E-mail: senwang11@163.com
  • Received Date: 11 November 2021
  • Accepted Date: 09 March 2022
    • Available Online: 11 May 2022
    Abstract

  • Abstract

    Given a partition of $ \mathbb{Z}_N^* $ into four subsets, we present a generic construction of uncorrelated quaternary sequence pairs of length $ 2N $ using the interleaved technique based on this partition. By choosing partitions arising from cyclotomic classes of order 4 and 8 over $ \mathbb{Z}_p $, we construct uncorrelated quaternary sequence pairs of length $ 2p $, which are almost balanced and have low autocorrelation, except at a few positions.

    Graphical abstract

    The autocorrelation $ R_u(\tau), R_v(\tau) $ and cross-correlation $ R_{u,v}(\tau), R_{v,u}(\tau) $ of sequence pair $ (u,v) $.

    Abstract

    Given a partition of $ \mathbb{Z}_N^* $ into four subsets, we present a generic construction of uncorrelated quaternary sequence pairs of length $ 2N $ using the interleaved technique based on this partition. By choosing partitions arising from cyclotomic classes of order 4 and 8 over $ \mathbb{Z}_p $, we construct uncorrelated quaternary sequence pairs of length $ 2p $, which are almost balanced and have low autocorrelation, except at a few positions.

    • Public Summary

    • Given a partition of $\mathbb{Z}_N^* $ into four subsets, a generic construction of uncorrelated quaternary sequence pairs of length $ 2N $ was proposed.
    • Choose the partition of $\mathbb{Z}_N^* $ from cyclotomic classes of order 4 and 8. The sequence pairs obtained are uncorrelated, almost balanced and with low autocorrelation, except at a few positions.

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    • References(15)
  • [1]
    Fan P, Darnell M. Sequence Design for Communications Applications. London: Research Studies Press, 1996.
    [2]
    Golomb S, Gong G. Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar. Cambridge, UK: Cambridge University Press, 2005.
    [3]
    Cusick T, Ding C, Renvall A. Stream Ciphers and Number Theory. Amsterdam: North-Holland Publishing, 1998.
    [4]
    Luo G, Cao X, Shi M, et al. Three new constructions of asymptotically optimal periodic quasi-complementary sequence sets with small alphabet sizes. IEEE Trans. Inf. Theory, 2021, 67 (8): 5168–5177. doi: 10.1109/TIT.2021.3068474
    [5]
    Shi M, Qian L, Helleseth T, et al. Five-weight codes from three-valued correlation of M-sequences. Adv. Math. Commun., 2021: doi 10.3934/amc.2021022. doi: 10.3934/amc.2021022
    [6]
    Luo L, Ma W, Zhao F. Binary sequence pairs of period p m−1 with optimal three-level correlation. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2018, E101-A (8): 1263–1266. doi: 10.1587/transfun.E101.A.1263
    [7]
    Shen X, Jia Y, Song X. Constructions of binary sequence pairs of period 3p with optimal three-level correlation. IEEE Commun. Lett., 2017, 21 (10): 2150–2153. doi: 10.1109/LCOMM.2017.2700845
    [8]
    Peng X, Ren J, Xu C, et al. New families of binary sequence pairs with three-level correlation and odd composite length. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2016, E99-A (4): 874–879. doi: 10.1587/transfun.E99.A.874
    [9]
    Yang Y, Tang X, Zhou Z. The autocorrelation magnitude of balanced binary sequence pairs of prime period N≡1 (mod 4) with optimal cross-correlation. IEEE Commun. Lett., 2015, 19 (4): 585–588. doi: 10.1109/LCOMM.2015.2402278
    [10]
    Li M, Peng W, Bai P, et al. Construction of several classes of binary and quaternary complete sequences and sequence pairs. Wirel. Pers. Commun., 2015, 82 (1): 113–122. doi: 10.1007/s11277-014-2197-x
    [11]
    Peng X, Xu C. New construction of quaternary sequence pairs with even period and three-level correlation. In: Proceedings of the Fifth International Workshop on Signal Design and Its Applications in Communications. IEEE, 2011: 72–75.
    [12]
    Yang Y, Tang X. Balanced quaternary sequences pairs of odd period with (almost) optimal autocorrelation and cross-correlation. IEEE Commun. Lett., 2014, 18 (8): 1327–1330. doi: 10.1109/LCOMM.2014.2328603
    [13]
    Zhou Z, Li J, Yang Y, et al. Two constructions of quaternary periodic complementary pairs. IEEE Commun. Lett., 2018, 22 (12): 2507–2510. doi: 10.1109/LCOMM.2018.2876530
    [14]
    Bouniakowsky V. Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la de composition des entiers en facteurs. Sc. Math. Phys., 1857, 6: 305–329.
    [15]
    Shen X, Jia Y, Song X, et al. New construction methods for binary sequence pairs of period pq with ideal two-level correlation. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2018, E101-A (4): 704–712. doi: 10.1587/transfun.E101.A.704
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    [1]
    Fan P, Darnell M. Sequence Design for Communications Applications. London: Research Studies Press, 1996.
    [2]
    Golomb S, Gong G. Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar. Cambridge, UK: Cambridge University Press, 2005.
    [3]
    Cusick T, Ding C, Renvall A. Stream Ciphers and Number Theory. Amsterdam: North-Holland Publishing, 1998.
    [4]
    Luo G, Cao X, Shi M, et al. Three new constructions of asymptotically optimal periodic quasi-complementary sequence sets with small alphabet sizes. IEEE Trans. Inf. Theory, 2021, 67 (8): 5168–5177. doi: 10.1109/TIT.2021.3068474
    [5]
    Shi M, Qian L, Helleseth T, et al. Five-weight codes from three-valued correlation of M-sequences. Adv. Math. Commun., 2021: doi 10.3934/amc.2021022. doi: 10.3934/amc.2021022
    [6]
    Luo L, Ma W, Zhao F. Binary sequence pairs of period p m−1 with optimal three-level correlation. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2018, E101-A (8): 1263–1266. doi: 10.1587/transfun.E101.A.1263
    [7]
    Shen X, Jia Y, Song X. Constructions of binary sequence pairs of period 3p with optimal three-level correlation. IEEE Commun. Lett., 2017, 21 (10): 2150–2153. doi: 10.1109/LCOMM.2017.2700845
    [8]
    Peng X, Ren J, Xu C, et al. New families of binary sequence pairs with three-level correlation and odd composite length. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2016, E99-A (4): 874–879. doi: 10.1587/transfun.E99.A.874
    [9]
    Yang Y, Tang X, Zhou Z. The autocorrelation magnitude of balanced binary sequence pairs of prime period N≡1 (mod 4) with optimal cross-correlation. IEEE Commun. Lett., 2015, 19 (4): 585–588. doi: 10.1109/LCOMM.2015.2402278
    [10]
    Li M, Peng W, Bai P, et al. Construction of several classes of binary and quaternary complete sequences and sequence pairs. Wirel. Pers. Commun., 2015, 82 (1): 113–122. doi: 10.1007/s11277-014-2197-x
    [11]
    Peng X, Xu C. New construction of quaternary sequence pairs with even period and three-level correlation. In: Proceedings of the Fifth International Workshop on Signal Design and Its Applications in Communications. IEEE, 2011: 72–75.
    [12]
    Yang Y, Tang X. Balanced quaternary sequences pairs of odd period with (almost) optimal autocorrelation and cross-correlation. IEEE Commun. Lett., 2014, 18 (8): 1327–1330. doi: 10.1109/LCOMM.2014.2328603
    [13]
    Zhou Z, Li J, Yang Y, et al. Two constructions of quaternary periodic complementary pairs. IEEE Commun. Lett., 2018, 22 (12): 2507–2510. doi: 10.1109/LCOMM.2018.2876530
    [14]
    Bouniakowsky V. Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la de composition des entiers en facteurs. Sc. Math. Phys., 1857, 6: 305–329.
    [15]
    Shen X, Jia Y, Song X, et al. New construction methods for binary sequence pairs of period pq with ideal two-level correlation. IEICE Trans. Fundam. Electron. Commun. Comput. Sci., 2018, E101-A (4): 704–712. doi: 10.1587/transfun.E101.A.704
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