February  2021, 15(1): 23-33. doi: 10.3934/amc.2020040

Golay complementary sets with large zero odd-periodic correlation zones

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

2. 

State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China

Corresponding author: Yang Yang

Received  February 2019 Revised  June 2019 Published  February 2021 Early access  November 2019

Fund Project: This work was supported in part by the National Science Foundation of China under Grants 61771016, 61661146003 and 11571285

Golay complementary sets (GCSs) are widely used in different communication systems, i.e., GCSs could be used in OFDM systems to control peak-to-mean envelope power ratio (PMEPR). In this paper, inspired by the work on GCSs with large zero correlation zone given by Chen et al in 2018, we investigate the relationship between GCSs and zero odd-periodic correlation zone (ZOCZ) sequence sets, and present GCSs with flexible sequence set sizes, sequence lengths, large ZOCZ and low PMEPR. Those proposed sequences could be applied in OFDM system for synchronization.

Citation: 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
References:
[1]

C. Y. ChenC. H. Wang and C. C. Chao, Complete complementary codes and generalized Reed-Muller codes, IEEE Communi. Lett., 12 (2008), 849-851.  doi: 10.1109/LCOMM.2008.081189.

[2]

C. Y. Chen and S. W. Wu, Golay complementary sequence sets with large zero correlation zones, IEEE Trans. Communi., 66 (2018), 5197-5204.  doi: 10.1109/TCOMM.2018.2857485.

[3]

J. D. Coker and A. H. Tewfik, Simplified ranging systems using discrete wavelet decomposition, IEEE Trans. Signal Process., 58 (2010), 575-582.  doi: 10.1109/TSP.2009.2032949.

[4]

J. A. 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.

[5]

X. M. Deng and P. Z. Fan, Spreading sequence sets with zero correlation zone, Electron. Lett., 36 (2000), 993-994.  doi: 10.1049/el:20000720.

[6]

Evolved Universal Terrestrial Radio Access (E-UTRA) - Physical channels and modulation, 3GPP TS 36.211 V14.6, 2018.

[7]

P. Z. FanN. SuehiroN. Kuroyanagi and X. M. Deng, Class of binary sequences with zero correlation zone, Electron. Lett., 35 (1999), 777-779.  doi: 10.1049/el:19990567.

[8]

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

[9] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.
[10]

G. GongF. Huo and Y. Yang, Large zero autocorrelation zones of Golay sequences and their applications, IEEE Trans. Communi., 61 (2013), 3967-3979.  doi: 10.1109/TCOMM.2013.072813.120928.

[11]

G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay pairs and QAM Golay pairs, IEEE International Symposium on Information Theory, 2013, 3135–3139. doi: 10.1109/ISIT.2013.6620803.

[12]

T. Hayashi, A generalization of binary zero-correlation zone sequence sets constructed from Hadamard matrices, IEICE Trans. Fundam., E87-A (2004), 559-565. 

[13]

H. G. Hu and G. Gong, New sets of zero or low correlation zone sequences via interleaving techniques, IEEE Trans. Inform. Theory, 56 (2010), 1702-1713.  doi: 10.1109/TIT.2010.2040887.

[14]

K. M. Z. IslamT. Y. Al-Naffouri and N. Al-Dhahir, On optimum pilot design for comb-type OFDM transmission over doubly-selective channels, IEEE Trans. Communi., 59 (2011), 930-935.  doi: 10.1109/TCOMM.2011.020411.100151.

[15]

IEEE Standard for Local and Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11ad, 2014.

[16]

IEEE Standard for Local and Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11ax, 2018.

[17]

IEEE Standard for Local and Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11ay, 2018.

[18]

B. LongP. Zhang and J. Hu, A generalized QS-CDMA system and the design of new spreading codes, IEEE Trans. Veh. Tech., 47 (1998), 1268-1275.  doi: 10.1109/25.728516.

[19]

J. L. Massey and J. J. Uhran, Sub-baud coding, Proceedings of the Thirteenth Annual Allerton Conference on Circuit and System Theory, 1975,539–547.

[20]

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

[21]

M. B. Pursley, Introduction to Digital Communications., Pearson Prentice Hall, 2005.

[22]

M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication - Part I: System analysis, IEEE Trans. Inform. Theory, 25 (1977), 795-799.  doi: 10.1109/TCOM.1977.1093915.

[23]

M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication - Part II: Code sequence analysis, IEEE Trans. Inform. Theory, 25 (1977), 800-803.  doi: 10.1109/TCOM.1977.1093916.

[24]

A. Rathinakumar and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE. Trans. Inform. Theory, 52 (2006), 3817-3826.  doi: 10.1109/TIT.2006.878171.

[25]

J. R. Seberry, B. J. Wysocki and T. A. Wysocki, On a use of Golay sequences for asynchronous DS CDMA applications, in Advanced Signal Processing for Communication Systems, The International Series in Engineering and Computer Science, 703, Springer, Boston, 2002,183–196. doi: 10.1007/0-306-47791-2_14.

[26]

X. H. TangP. Z. Fan and S. Matsufuji, Lower bounds on the maximum correlation of sequence set with low or zero correlation zone, Electron. Lett., 36 (2000), 551-552. 

[27]

J. D. Yang, X. Jin, K. Y. Song, J. S. No and D. J. Shin, Multicode MIMO systems with quaternary LCZ and ZCZ sequences, IEEE Trans. Veh. Technol., 2008, 2334–2341. doi: 10.1109/PIMRC.2007.4394294.

[28]

Y. Yang, F. Huo and G. Gong, Large zero odd periodic autocorrelation zone of Golay sequences and QAM Golay sequences, ISIT, 2012, 1029–1033. doi: 10.1109/ISIT.2012.6283006.

[29]

W. C. Zhang, F. Zeng, X. H. Long and M. X. Xie, Improved mutually orthogonal ZCZ polyphase sequence sets and their applications in OFDM frequency synchronization, Proc. Int. Conf. on Wireless Commun. Netw. and Mobile Comput., 2010, 1–5. doi: 10.1109/WICOM.2010.5600748.

[30]

R. Q. Zhang, X. Cheng, M. Ma and B. L. Jiao, Interference-avoidance pilot design using ZCZ sequences for multi-cell MIMO-OFDM systems, Proc. IEEE Global Commun. Conf., 2012, 5056–5061. doi: 10.1109/GLOCOM.2012.6503922.

[31]

Z. C. ZhouX. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique, IEEE Trans. Inform. Theory, 54 (2008), 4267-4273.  doi: 10.1109/TIT.2008.928256.

[32]

Z. C. ZhouD. ZhangT. Helleseth and J. Wen, A construction of multiple optimal ZCZ sequence sets with good cross-correlation, IEEE Trans. Inform. Theory, 64 (2018), 1340-1346.  doi: 10.1109/TIT.2017.2756845.

show all references

References:
[1]

C. Y. ChenC. H. Wang and C. C. Chao, Complete complementary codes and generalized Reed-Muller codes, IEEE Communi. Lett., 12 (2008), 849-851.  doi: 10.1109/LCOMM.2008.081189.

[2]

C. Y. Chen and S. W. Wu, Golay complementary sequence sets with large zero correlation zones, IEEE Trans. Communi., 66 (2018), 5197-5204.  doi: 10.1109/TCOMM.2018.2857485.

[3]

J. D. Coker and A. H. Tewfik, Simplified ranging systems using discrete wavelet decomposition, IEEE Trans. Signal Process., 58 (2010), 575-582.  doi: 10.1109/TSP.2009.2032949.

[4]

J. A. 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.

[5]

X. M. Deng and P. Z. Fan, Spreading sequence sets with zero correlation zone, Electron. Lett., 36 (2000), 993-994.  doi: 10.1049/el:20000720.

[6]

Evolved Universal Terrestrial Radio Access (E-UTRA) - Physical channels and modulation, 3GPP TS 36.211 V14.6, 2018.

[7]

P. Z. FanN. SuehiroN. Kuroyanagi and X. M. Deng, Class of binary sequences with zero correlation zone, Electron. Lett., 35 (1999), 777-779.  doi: 10.1049/el:19990567.

[8]

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

[9] S. W. Golomb and G. Gong, Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511546907.
[10]

G. GongF. Huo and Y. Yang, Large zero autocorrelation zones of Golay sequences and their applications, IEEE Trans. Communi., 61 (2013), 3967-3979.  doi: 10.1109/TCOMM.2013.072813.120928.

[11]

G. Gong, F. Huo and Y. Yang, Large zero autocorrelation zone of Golay pairs and QAM Golay pairs, IEEE International Symposium on Information Theory, 2013, 3135–3139. doi: 10.1109/ISIT.2013.6620803.

[12]

T. Hayashi, A generalization of binary zero-correlation zone sequence sets constructed from Hadamard matrices, IEICE Trans. Fundam., E87-A (2004), 559-565. 

[13]

H. G. Hu and G. Gong, New sets of zero or low correlation zone sequences via interleaving techniques, IEEE Trans. Inform. Theory, 56 (2010), 1702-1713.  doi: 10.1109/TIT.2010.2040887.

[14]

K. M. Z. IslamT. Y. Al-Naffouri and N. Al-Dhahir, On optimum pilot design for comb-type OFDM transmission over doubly-selective channels, IEEE Trans. Communi., 59 (2011), 930-935.  doi: 10.1109/TCOMM.2011.020411.100151.

[15]

IEEE Standard for Local and Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11ad, 2014.

[16]

IEEE Standard for Local and Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11ax, 2018.

[17]

IEEE Standard for Local and Metropolitan Area Networks-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Standard 802.11ay, 2018.

[18]

B. LongP. Zhang and J. Hu, A generalized QS-CDMA system and the design of new spreading codes, IEEE Trans. Veh. Tech., 47 (1998), 1268-1275.  doi: 10.1109/25.728516.

[19]

J. L. Massey and J. J. Uhran, Sub-baud coding, Proceedings of the Thirteenth Annual Allerton Conference on Circuit and System Theory, 1975,539–547.

[20]

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

[21]

M. B. Pursley, Introduction to Digital Communications., Pearson Prentice Hall, 2005.

[22]

M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication - Part I: System analysis, IEEE Trans. Inform. Theory, 25 (1977), 795-799.  doi: 10.1109/TCOM.1977.1093915.

[23]

M. B. Pursley, Performance evaluation for phase-coded spread-spectrum multiple-access communication - Part II: Code sequence analysis, IEEE Trans. Inform. Theory, 25 (1977), 800-803.  doi: 10.1109/TCOM.1977.1093916.

[24]

A. Rathinakumar and A. K. Chaturvedi, A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences, IEEE. Trans. Inform. Theory, 52 (2006), 3817-3826.  doi: 10.1109/TIT.2006.878171.

[25]

J. R. Seberry, B. J. Wysocki and T. A. Wysocki, On a use of Golay sequences for asynchronous DS CDMA applications, in Advanced Signal Processing for Communication Systems, The International Series in Engineering and Computer Science, 703, Springer, Boston, 2002,183–196. doi: 10.1007/0-306-47791-2_14.

[26]

X. H. TangP. Z. Fan and S. Matsufuji, Lower bounds on the maximum correlation of sequence set with low or zero correlation zone, Electron. Lett., 36 (2000), 551-552. 

[27]

J. D. Yang, X. Jin, K. Y. Song, J. S. No and D. J. Shin, Multicode MIMO systems with quaternary LCZ and ZCZ sequences, IEEE Trans. Veh. Technol., 2008, 2334–2341. doi: 10.1109/PIMRC.2007.4394294.

[28]

Y. Yang, F. Huo and G. Gong, Large zero odd periodic autocorrelation zone of Golay sequences and QAM Golay sequences, ISIT, 2012, 1029–1033. doi: 10.1109/ISIT.2012.6283006.

[29]

W. C. Zhang, F. Zeng, X. H. Long and M. X. Xie, Improved mutually orthogonal ZCZ polyphase sequence sets and their applications in OFDM frequency synchronization, Proc. Int. Conf. on Wireless Commun. Netw. and Mobile Comput., 2010, 1–5. doi: 10.1109/WICOM.2010.5600748.

[30]

R. Q. Zhang, X. Cheng, M. Ma and B. L. Jiao, Interference-avoidance pilot design using ZCZ sequences for multi-cell MIMO-OFDM systems, Proc. IEEE Global Commun. Conf., 2012, 5056–5061. doi: 10.1109/GLOCOM.2012.6503922.

[31]

Z. C. ZhouX. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique, IEEE Trans. Inform. Theory, 54 (2008), 4267-4273.  doi: 10.1109/TIT.2008.928256.

[32]

Z. C. ZhouD. ZhangT. Helleseth and J. Wen, A construction of multiple optimal ZCZ sequence sets with good cross-correlation, IEEE Trans. Inform. Theory, 64 (2018), 1340-1346.  doi: 10.1109/TIT.2017.2756845.

Table 1.  Comparison of GCSs/GCPs with Large ZCZ/ZOCZ Property
Parameters Constraints Ref.
$ (2,2^m,2^{m-2}) $ Golay-ZCZ $ g_m\in \{\frac{H}{2},0\} $, $ H\equiv0\; \mathrm{mod}\; 2 $ [10,11]
$ (2,2^m,2^{\pi(2)-1}) $ Golay-ZCZ $ \begin{array}{l} g_m\in \{0,\frac{H}{2}\},H\equiv 0\; \mathrm{mod}\; 2\pi(1)=m \end{array} $ [2]
$ (2^k,2^m,2^{\pi_1(2)-1}) $ Golay-ZCZ $ \begin{array}{l}g_m\in \{0,\frac{H}{2}\}, H\equiv0\; \mathrm{mod}\; 2, \pi_\alpha(1)=m-\alpha+1, \;\mbox{ for }\; 1\le\alpha\le k\end{array} $ [2]
$ (2,2^m,2^{\pi(2)-1}) $ Golay-ZOCZ $ \begin{array}{l} c_m\in \{\frac{H}{4},\frac{3H}{4}\},H\equiv0\; \mathrm{mod}\; 4, \pi(1)=m \end{array} $ Thm. 1
$ (2^k,2^m,2^{\pi_1(2)-1}) $ Golay-ZOCZ $ \begin{array}{l}g_m\in \{\frac{H}{4},\frac{3H}{4}\}, H\equiv0\; \mathrm{mod}\; 4, \pi_\alpha(1)=m-\alpha+1, \;\mbox{ for }\; 1\le\alpha\le k\end{array} $ Thm. 2
Parameters Constraints Ref.
$ (2,2^m,2^{m-2}) $ Golay-ZCZ $ g_m\in \{\frac{H}{2},0\} $, $ H\equiv0\; \mathrm{mod}\; 2 $ [10,11]
$ (2,2^m,2^{\pi(2)-1}) $ Golay-ZCZ $ \begin{array}{l} g_m\in \{0,\frac{H}{2}\},H\equiv 0\; \mathrm{mod}\; 2\pi(1)=m \end{array} $ [2]
$ (2^k,2^m,2^{\pi_1(2)-1}) $ Golay-ZCZ $ \begin{array}{l}g_m\in \{0,\frac{H}{2}\}, H\equiv0\; \mathrm{mod}\; 2, \pi_\alpha(1)=m-\alpha+1, \;\mbox{ for }\; 1\le\alpha\le k\end{array} $ [2]
$ (2,2^m,2^{\pi(2)-1}) $ Golay-ZOCZ $ \begin{array}{l} c_m\in \{\frac{H}{4},\frac{3H}{4}\},H\equiv0\; \mathrm{mod}\; 4, \pi(1)=m \end{array} $ Thm. 1
$ (2^k,2^m,2^{\pi_1(2)-1}) $ Golay-ZOCZ $ \begin{array}{l}g_m\in \{\frac{H}{4},\frac{3H}{4}\}, H\equiv0\; \mathrm{mod}\; 4, \pi_\alpha(1)=m-\alpha+1, \;\mbox{ for }\; 1\le\alpha\le k\end{array} $ Thm. 2
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