# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021063
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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

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.

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.
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 $2^a10^b26^c$ $Q_8$
 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 $2^a10^b26^c$ $Q_8$
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 ${2^\alpha }{10^\beta }{26^\gamma }(q + 1)$ $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha ,\beta ,\gamma \ge 0,\;\\ \;q = 0\;{\rm{or}}\;q \equiv 1(\;\bmod \,4)\;{\rm{is}}\;{\rm{prime power}} \end{array}$ $Q_8$
 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 ${2^\alpha }{10^\beta }{26^\gamma }(q + 1)$ $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\alpha ,\beta ,\gamma \ge 0,\;\\ \;q = 0\;{\rm{or}}\;q \equiv 1(\;\bmod \,4)\;{\rm{is}}\;{\rm{prime power}} \end{array}$ $Q_8$
 [1] Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9 [2] Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021, 15 (1) : 23-33. doi: 10.3934/amc.2020040 [3] Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim, Dae-Woon Lim. New construction methods of quaternary periodic complementary sequence sets. Advances in Mathematics of Communications, 2010, 4 (1) : 61-68. doi: 10.3934/amc.2010.4.61 [4] Fanxin Zeng, Xiaoping Zeng, Zhenyu Zhang, Guixin Xuan. Quaternary periodic complementary/Z-complementary sequence sets based on interleaving technique and Gray mapping. Advances in Mathematics of Communications, 2012, 6 (2) : 237-247. doi: 10.3934/amc.2012.6.237 [5] Yang Yang, Xiaohu Tang, Guang Gong. Even periodic and odd periodic complementary sequence pairs from generalized Boolean functions. Advances in Mathematics of Communications, 2013, 7 (2) : 113-125. doi: 10.3934/amc.2013.7.113 [6] Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004 [7] Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475 [8] Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55 [9] Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049 [10] Ferruh Özbudak, Eda Tekin. Correlation distribution of a sequence family generalizing some sequences of Trachtenberg. Advances in Mathematics of Communications, 2021, 15 (4) : 647-662. doi: 10.3934/amc.2020087 [11] Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375 [12] Liqun Yao, Wenli Ren, Yong Wang, Chunming Tang. Z-complementary pairs with flexible lengths and large zero odd-periodic correlation zones. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021037 [13] Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117 [14] Huaning Liu, Xi Liu. On the correlation measures of orders $3$ and $4$ of binary sequence of period $p^2$ derived from Fermat quotients. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021008 [15] Yuhua Sun, Zilong Wang, Hui Li, Tongjiang Yan. The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$. Advances in Mathematics of Communications, 2013, 7 (4) : 409-424. doi: 10.3934/amc.2013.7.409 [16] Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115 [17] Bingsheng Shen, Yang Yang, Ruibin Ren. Three constructions of Golay complementary array sets. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022019 [18] Hua Liang, Jinquan Luo, Yuansheng Tang. On cross-correlation of a binary $m$-sequence of period $2^{2k}-1$ and its decimated sequences by $(2^{lk}+1)/(2^l+1)$. Advances in Mathematics of Communications, 2017, 11 (4) : 693-703. doi: 10.3934/amc.2017050 [19] Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015 [20] Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018

2020 Impact Factor: 0.935

## Metrics

• HTML views (173)
• Cited by (0)

• on AIMS