Aperiodic/periodic complementary sequence pairs over quaternions

Zhen Li; Cuiling Fan; Wei Su; Yanfeng Qi

doi:10.3934/amc.2021063

Article Contents

2023, Volume 17, Issue 6: 1493-1506. Doi: 10.3934/amc.2021063

This issue Previous Article Niederreiter cryptosystems using quasi-cyclic codes that resist quantum Fourier sampling Next Article On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

Aperiodic/periodic complementary sequence pairs over quaternions

1.
School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China
2.
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China
3.
School of Economics and Information Engineering, Southwestern University of Finance and Economics, Chengdu, 610074, China
4.
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China

^*Corresponding author: Cuiling Fan
^*Corresponding author: Cuiling Fan

Received: June 2021

Revised: August 2021

Early access: December 2021

Published: December 2023

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Abstract

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.

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

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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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