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On the equivalence of several classes of quaternary sequences with optimal autocorrelation and length $ 2p$

  • * Corresponding author: Pinhui Ke

    * Corresponding author: Pinhui Ke 

The authors are supported by National Natural Science Foundation of China (No. 61772292, 61772476, 61771016), Natural Science Foundation of Fujian Province (No. 2019J01273) and Fujian Normal University Innovative Research Team (IRTL1207)

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  • Quaternary sequences with optimal autocorrelation property are preferred in applications. Cyclotomic classes of order 4 are widely used in the constructions of quaternary sequences due to the convenience of defining a quaternary sequence with the cyclotomic classes of order 4 as its support set. Recently, several classes of optimal quaternary sequences of period $ 2p $, which are all closely related to the cyclotomic classes of order 4 with respect to $ \mathbb{Z}_p $ were introduced in the literature. However, less attention has been paid to the equivalence between these known results. In this paper, we introduce the unified form of this kind of quaternary sequences to classify these known results and then conclude the unified forms of these optimal quaternary sequences. By doing this, we disclose the relationship between the optimal quaternary sequences derived from different methods in the literature on one hand. And on the other hand, when the new obtained optimal quaternary sequence period is $ 2p $ and the cyclotomic classes of order 4 are involved, the methods and the results given in this paper can be used to identify if the sequence is new or not.

    Mathematics Subject Classification: Primary: 94A55, 11T71; Secondary: 05B10.

    Citation:

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  • Table 1.  Known Optimal Quaternary Sequences of Length $ 2p $

    Sequence Construction (Sketch) Constrains
    Chung et al.
    [12] [2]
    $ \mathbf{s}=\phi^{-1}(\mathbf{a}, L^{p}(\mathbf{b})) $
    $ \mathbf{b}=\mathbf{a} $ or $ \mathbf{a}+1 $
    $ p\equiv 5\pmod 8 $
    $ \mathbf{a} $ is DHM sequence of length $ p $
    Su et al.
    [17]
    $ \mathbf{s}=\phi^{-1}(\mathbf{c}, \mathbf{d})) $
    $ \mathbf{c}=I(\mathbf{a}_{0}, e(0)+L^{\lambda}(\mathbf{a}_{1})) $
    $ \mathbf{d}=I(e(1)+\mathbf{a}_{2}, e(2)+L^{\lambda}(\mathbf{a}_{3})) $
    $ p\equiv 1\pmod 4 $, $ \lambda=\frac{p+1}{2} $
    $ (e(0), e(1), e(2))\in \mathbb{Z}_2^3 $
    $ \mathbf{a}_i\in\{\mathbf{s_1}, \mathbf{s_2}, \cdots, \mathbf{s_6}\}, 0\le i\le 3 $
    Shen et al.
    [14]
    $ \mbox{Supp}_{\mathbf{s}}(0)=\psi(\{0\}\times D_{i_0}\cup \{1\}\times D_{j_0})\cup \{0\} $
    $ \mbox{Supp}_{\mathbf{s}}(2)=\psi(\{0\}\times D_{i_2}\cup \{1\}\times D_{j_2})\cup \{p\} $
    $ \mbox{Supp}_{\mathbf{s}}(t)=\psi(\{0\}\times D_{i_t}\cup \{1\}\times D_{j_t}), \ \mbox{for} \ t=1, 3 $
    $ p\equiv 1\pmod 4 $
    $ i_l\neq i_m $ and $ j_l\neq j_m $, if $ l\neq m $
    Luo et al.
    [11]
    $ \mathbf{s}=I(\mathbf{u}_c, L^{\lambda}(\mathbf{v}_c)+2) $
    $ u_c(0) $ and $ u_c(i)=j, \mbox{if}\ i\in D_{(m+j)\pmod 4} $
    $ v_c(0)=c $ and $ v_c(i)=j, \mbox{if}\ i\in D_{(n+3j)\pmod 4} $
    $ p\equiv 1\pmod 4 $, $ \lambda=\frac{p+1}{2} $
    $ m, n, c\in \mathbb{Z}_4 $
    $ f $ is odd and $ m-n\equiv 2c\pmod 4 $ or
    $ f $ is even and $ m-n\equiv 2c+2\pmod 4 $
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