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doi: 10.3934/amc.2021063
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Aperiodic/periodic complementary sequence pairs over quaternions

Zhen Li 1,2, , Cuiling Fan 1,, , Wei Su 3, and  Yanfeng Qi 4,

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

Received  June 2021 Revised  August 2021 Early access December 2021

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.

Keywords: Aperiod correlation, Periodic correlation, Golay sequence, Complementary sequence pair, Complementary sequence set, Quaternions.
Mathematics Subject Classification: Primary: 94A05; Secondary: 60G35.
Citation: Zhen Li, Cuiling Fan, Wei Su, Yanfeng Qi. Aperiodic/periodic complementary sequence pairs over quaternions. Advances in Mathematics of Communications, doi: 10.3934/amc.2021063
References:
[1]

S. Acevedo and H. Dietrich, Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in ${C_n \times Q_8}$, Cryptogr. Commun, 10 (2018), 357-368.  doi: 10.1007/s12095-017-0224-y.

[2]

S. Acevedo and T. Hall, Perfect sequences of unbounded lengths over the basic quaternions, Sequences and Their Applications – SETA 2012, 7280 (2012), 159-167.  doi: 10.1007/978-3-642-30615-0_15.

[3]

L. Bomer and M. Antweiler, Periodic complementary binary sequences, IEEE Trans. Inf. Theory, 36 (1990), 1487-1494.  doi: 10.1109/18.59954.

[4]

P. Borwein and R. Ferguson, A complete description of Golay pairs for lengths up to 100, Math. Comput, 73 (2004), 967-985.  doi: 10.1090/S0025-5718-03-01576-X.

[5]

C. Bright, I. Kotsireas and V. Ganesh, New infinite families of perfect quaternion sequences and Williamson sequences, IEEE Trans. Inform. Theory, 66 (2020), 7739-7751.  doi: 10.1109/TIT.2020.3016510.

[6]

C. Chen, Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM, IEEE Trans. Inform. Theory, 62 (2016), 7538-7545.  doi: 10.1109/TIT.2016.2613994.

[7]

R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: Structure and applications, Discret. Math, 252 (2002), 73-89.  doi: 10.1016/S0012-365X(01)00162-5.

[8]

J. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417.  doi: 10.1109/18.796380.

[9]

D. Doković and I. Kotsireas, Some new periodic Golay pairs, Numer. Algorithms, 69 (2015), 523-530.  doi: 10.1007/s11075-014-9910-4.

[10]

D. Doković, I. Kotsireas, R. Daniel and J. Sawada, Charm bracelets and their application to the construction of periodic Golay pairs, Discret. Appl. Math., 188 (2015), 32-40.  doi: 10.1016/j.dam.2015.03.001.

[11]

K. Feng, P. Shiue and Q. Xiang, On aperiodic and periodic complementary binary sequences, IEEE Trans. Inf. Theory, 45 (1999), 296-303.  doi: 10.1109/18.746823.

[12]

F. Fiedler, Small Golay sequences, Adv. Math. Commun, 7 (2013), 379-407.  doi: 10.3934/amc.2013.7.379.

[13]

R. Frank, Polyphase complementary codes, IEEE Trans. Inf. Theory, 26 (1980), 641-647.  doi: 10.1109/TIT.1980.1056272.

[14]

H. Ganapathy, D. Pados and G. Karystinos, New bounds and optimal binary signature sets-part I: Periodic total squared correlation, IEEE Trans. Commun, 59 (2011), 1123-1132. 

[15]

S. Georgiou, S. Stylianou, K. Drosou and C. Koukouvinos, Construction of orthogonal and nearly orthogonal designs for computer experiments, Biometrika, 101 (2014), 741-747.  doi: 10.1093/biomet/asu021.

[16]

M. Golay, Static multislit spectrometry and its application to the panoramic display of infrared spectra$^*$, J. Opt. Soc. Am, 41 (1951), 468-472. 

[17]

M. Golay, Complementary series, IRE Trans., 7 (1961), 82-87.  doi: 10.1109/tit.1961.1057620.

[18]

W. Holzmann and H. Kharaghani, A computer search for complex Golay sequences, Australas. J. Comb, 10 (1994), 251-258. 

[19]

P. Kumari, J. Choi, N. Prelcic and R. Heath, IEEE802.11 ad-based radar: An approach to joint vehicular communication-radar system, IEEE Trans. Veh. Technol, 67 (2018), 3012-3027. 

[20]

O. Kuznetsov, Perfect sequences over the real quaternions, 2009 Fourth International Workshop on Signal Design and its Applications in Communications, (2009), 8–11. doi: 10.1109/IWSDA.2009.5346443.

[21]

M. Nazarathy, S. Newton, R. Giffard, D. Moberly, F. Sischka, W. Trutna and S. Foster, Real-time long range complementary correlation optical time domain reflectometer, J. Lightw. Technol, 7 (1989), 24-38. 

[22]

K. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Trans. Inf. Theory, 46 (2000), 104-120.  doi: 10.1109/18.817512.

[23]

A. Pezeshki, A. Calderbank, W. Moran and S. Howard, Doppler resilient Golay complementary waveforms, IEEE Trans. Inf. Theory, 54 (2008), 4254-4266.  doi: 10.1109/TIT.2008.928292.

[24]

K. Schmidt, Complementary sets, generalized Reed-Muller codes, and power control for OFDM, IEEE Trans. Inf. Theory, 53 (2007), 808-814.  doi: 10.1109/TIT.2006.889723.

[25]

K. Schmidt, Sequences with small correlation, Des. Codes Cryptogr., 78 (2016), 237-267.  doi: 10.1007/s10623-015-0154-7.

[26]

P. Spasojevic and C. Georghiades, Complementary sequences for ISI channel estimation, IEEE Trans. Inf. Theory, 47 (2001), 1145-1152.  doi: 10.1109/18.915670.

[27]

C. Tseng and C. Liu, Complementary sets of sequences, IEEE Trans. Inf. Theory, 18 (1972), 644-652.  doi: 10.1109/tit.1972.1054860.

[28]

R. Turyn, Synthesis of power efficient multitone signals with flat amplitude spectrum, J. Comb. Theory, Ser. A, 16 (1991), 313-333. 

[29]

S. Wang and A. Abdi, MIMOISI channel estimation using uncorrelated Golay complementary sets of polyphase sequences, IEEE Trans. Veh. Technol, 56 (2007), 3024-3039. 

[30]

G. Wang, A. Adhikary, Z. Zhou and Y. Yang, Generalized constructions of complementary sets of sequences of lengths non-power-of-two, IEEE Signal Process. Lett, 27 (2020), 136-140. 

[31]

F. Zeng, X. Zeng, Z. Zhang and G. Xuan, Quaternary periodic complementary/z-complementary sequence sets based on interleaving technique and Gray mapping, Adv. Math. Commun, 6 (2012), 237-247.  doi: 10.3934/amc.2012.6.237.

[32]

Z. Zhou, J. Li, Y. Yang and S. Hu, Two constructions of quaternary periodic complementary pairs, IEEE Commun. Lett, 22 (2018), 2507-2510. 

show all references

References:
[1]

S. Acevedo and H. Dietrich, Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in ${C_n \times Q_8}$, Cryptogr. Commun, 10 (2018), 357-368.  doi: 10.1007/s12095-017-0224-y.

[2]

S. Acevedo and T. Hall, Perfect sequences of unbounded lengths over the basic quaternions, Sequences and Their Applications – SETA 2012, 7280 (2012), 159-167.  doi: 10.1007/978-3-642-30615-0_15.

[3]

L. Bomer and M. Antweiler, Periodic complementary binary sequences, IEEE Trans. Inf. Theory, 36 (1990), 1487-1494.  doi: 10.1109/18.59954.

[4]

P. Borwein and R. Ferguson, A complete description of Golay pairs for lengths up to 100, Math. Comput, 73 (2004), 967-985.  doi: 10.1090/S0025-5718-03-01576-X.

[5]

C. Bright, I. Kotsireas and V. Ganesh, New infinite families of perfect quaternion sequences and Williamson sequences, IEEE Trans. Inform. Theory, 66 (2020), 7739-7751.  doi: 10.1109/TIT.2020.3016510.

[6]

C. Chen, Complementary sets of non-power-of-two length for peak-to-average power ratio reduction in OFDM, IEEE Trans. Inform. Theory, 62 (2016), 7538-7545.  doi: 10.1109/TIT.2016.2613994.

[7]

R. Craigen, W. Holzmann and H. Kharaghani, Complex Golay sequences: Structure and applications, Discret. Math, 252 (2002), 73-89.  doi: 10.1016/S0012-365X(01)00162-5.

[8]

J. Davis and J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, IEEE Trans. Inform. Theory, 45 (1999), 2397-2417.  doi: 10.1109/18.796380.

[9]

D. Doković and I. Kotsireas, Some new periodic Golay pairs, Numer. Algorithms, 69 (2015), 523-530.  doi: 10.1007/s11075-014-9910-4.

[10]

D. Doković, I. Kotsireas, R. Daniel and J. Sawada, Charm bracelets and their application to the construction of periodic Golay pairs, Discret. Appl. Math., 188 (2015), 32-40.  doi: 10.1016/j.dam.2015.03.001.

[11]

K. Feng, P. Shiue and Q. Xiang, On aperiodic and periodic complementary binary sequences, IEEE Trans. Inf. Theory, 45 (1999), 296-303.  doi: 10.1109/18.746823.

[12]

F. Fiedler, Small Golay sequences, Adv. Math. Commun, 7 (2013), 379-407.  doi: 10.3934/amc.2013.7.379.

[13]

R. Frank, Polyphase complementary codes, IEEE Trans. Inf. Theory, 26 (1980), 641-647.  doi: 10.1109/TIT.1980.1056272.

[14]

H. Ganapathy, D. Pados and G. Karystinos, New bounds and optimal binary signature sets-part I: Periodic total squared correlation, IEEE Trans. Commun, 59 (2011), 1123-1132. 

[15]

S. Georgiou, S. Stylianou, K. Drosou and C. Koukouvinos, Construction of orthogonal and nearly orthogonal designs for computer experiments, Biometrika, 101 (2014), 741-747.  doi: 10.1093/biomet/asu021.

[16]

M. Golay, Static multislit spectrometry and its application to the panoramic display of infrared spectra$^*$, J. Opt. Soc. Am, 41 (1951), 468-472. 

[17]

M. Golay, Complementary series, IRE Trans., 7 (1961), 82-87.  doi: 10.1109/tit.1961.1057620.

[18]

W. Holzmann and H. Kharaghani, A computer search for complex Golay sequences, Australas. J. Comb, 10 (1994), 251-258. 

[19]

P. Kumari, J. Choi, N. Prelcic and R. Heath, IEEE802.11 ad-based radar: An approach to joint vehicular communication-radar system, IEEE Trans. Veh. Technol, 67 (2018), 3012-3027. 

[20]

O. Kuznetsov, Perfect sequences over the real quaternions, 2009 Fourth International Workshop on Signal Design and its Applications in Communications, (2009), 8–11. doi: 10.1109/IWSDA.2009.5346443.

[21]

M. Nazarathy, S. Newton, R. Giffard, D. Moberly, F. Sischka, W. Trutna and S. Foster, Real-time long range complementary correlation optical time domain reflectometer, J. Lightw. Technol, 7 (1989), 24-38. 

[22]

K. Paterson, Generalized Reed-Muller codes and power control in OFDM modulation, IEEE Trans. Inf. Theory, 46 (2000), 104-120.  doi: 10.1109/18.817512.

[23]

A. Pezeshki, A. Calderbank, W. Moran and S. Howard, Doppler resilient Golay complementary waveforms, IEEE Trans. Inf. Theory, 54 (2008), 4254-4266.  doi: 10.1109/TIT.2008.928292.

[24]

K. Schmidt, Complementary sets, generalized Reed-Muller codes, and power control for OFDM, IEEE Trans. Inf. Theory, 53 (2007), 808-814.  doi: 10.1109/TIT.2006.889723.

[25]

K. Schmidt, Sequences with small correlation, Des. Codes Cryptogr., 78 (2016), 237-267.  doi: 10.1007/s10623-015-0154-7.

[26]

P. Spasojevic and C. Georghiades, Complementary sequences for ISI channel estimation, IEEE Trans. Inf. Theory, 47 (2001), 1145-1152.  doi: 10.1109/18.915670.

[27]

C. Tseng and C. Liu, Complementary sets of sequences, IEEE Trans. Inf. Theory, 18 (1972), 644-652.  doi: 10.1109/tit.1972.1054860.

[28]

R. Turyn, Synthesis of power efficient multitone signals with flat amplitude spectrum, J. Comb. Theory, Ser. A, 16 (1991), 313-333. 

[29]

S. Wang and A. Abdi, MIMOISI channel estimation using uncorrelated Golay complementary sets of polyphase sequences, IEEE Trans. Veh. Technol, 56 (2007), 3024-3039. 

[30]

G. Wang, A. Adhikary, Z. Zhou and Y. Yang, Generalized constructions of complementary sets of sequences of lengths non-power-of-two, IEEE Signal Process. Lett, 27 (2020), 136-140. 

[31]

F. Zeng, X. Zeng, Z. Zhang and G. Xuan, Quaternary periodic complementary/z-complementary sequence sets based on interleaving technique and Gray mapping, Adv. Math. Commun, 6 (2012), 237-247.  doi: 10.3934/amc.2012.6.237.

[32]

Z. Zhou, J. Li, Y. Yang and S. Hu, Two constructions of quaternary periodic complementary pairs, IEEE Commun. Lett, 22 (2018), 2507-2510. 

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 $
Table Options

Download as excel

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 $
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 $
Table Options

Download as excel

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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