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Aperiodic/periodic complementary sequence pairs over quaternions

  • *Corresponding author: Cuiling Fan

    *Corresponding author: Cuiling Fan 
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  • Aperodic (or called Golay)/Periodic complementary pairs (GCPs/ PCPs) are pairs of sequences whose aperiodic/periodic autocorrelation sums are zero everywhere, except at the zero shift. In this paper, we introduce GCPs/PCPs over the quaternion group $ Q_8 $, which is a generalization of quaternary GCPs/PCPs. Some basic properties of autocorrelations of $ Q_8 $-sequences are also obtained. We present three types of constructions for GCPs and PCPs over $ Q_8 $. The main ideas of these constructions are to consider pairs of a $ Q_8 $-sequence and its reverse, pairs of interleaving of sequence, or pairs of Kronecker product of sequences. By choosing suitable sequences in these constructions, we obtain new parameters for GCPs and PCPs, which have not been reported before.

    Mathematics Subject Classification: Primary: 94A05; Secondary: 60G35.

    Citation:

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  • Table 1.  Comparing with Some Known GCPs

    Reference Length Parameters Alphabet
    [28] $ 2^a10^b26^c $ $ a,b,c $ are nonnegative integers $ \{\pm1\} $
    [8] $ 2^m $ $ m\geq0 $ $ \{\pm1\} $
    [13,18] $ 2^{\alpha+\mu}3^\beta5^\gamma11^\eta13^\zeta $ $\begin{array}{*{20}{c}} {\alpha ,\beta ,\gamma ,\eta ,\zeta \ge 0,}\\ {\beta + \gamma + \eta + \zeta \le \alpha + 2\mu + 1}\\ {\mu \le \gamma + \zeta } \end{array}$ $ \{\pm1,\pm i\} $
    Corollary 4 $ 2^t(N_1+N_2) $ $t,a,b,c\geq0, N_1,N_2$ are of the form $2^a10^b26^c$ $ Q_8 $
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    Table 2.  Comparing with Some Known PCPs

    Ref. Length Parameters Alphabet
    [9] $ r $ $ r \in \{ 34,50,58,68,72,74,82,122,164,202,226\} $ $ \{\pm1,\pm i\} $
    [31] $ L $ $ L $ is the length of Binary PCPs $ \{\pm1,\pm i\} $
    [32] $ sN $ $ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;s \in \{ 1,2,4,8,16\} ,\;\\ {\rm{2}}N\;{\rm{is}}\;{\rm{the}}\;{\rm{length}}\;{\rm{of}}\;{\rm{Odd}}\;{\rm{Binary}}\;{\rm{PCPs}} \end{array} $ $ \{\pm1,\pm i\} $
    Corollary 5 ${2^\alpha }{10^\beta }{26^\gamma }(q + 1)$ $ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha ,\beta ,\gamma \ge 0,\;\\ \;q = 0\;{\rm{or}}\;q \equiv 1(\;\bmod \,4)\;{\rm{is}}\;{\rm{prime power}} \end{array} $ $ Q_8 $
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