American Institute of Mathematical Sciences

Advanced
February  2020, 14(1): 1-9. doi: 10.3934/amc.2020001

Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems

Chunlei Xie 1,2, and  Yujuan Sun 1,2,,

1. 

State Key Laboratory of Integrated Services Networks, Xidian University, Xi'an 710071 China
2. 

State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China

* Corresponding author: Yujuan Sun

Received  November 2017 Revised  November 2018 Published  August 2019

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant 61672414 and the National Cryptography Development Fund under Grant MMJJ20170113.

Orthogonal sequences can be assigned to a regular tessellation of hexagonal cells, typical for synchronised code-division multiple-access (S-CDMA) systems. In this paper, we first construct a new class of orthogonal sequences with increasing the number of users per cell to be $ 2^{m-2} $ for even number $ m\geq 4 $ (where $ 2^m $ is the length of the sequences). In addition, based on the above construction we construct a family of orthogonal sequences with zero correlation zone property which can be applied to the quasi-synchronous CDMA (QS-CMDA) spread spectrum systems.

Keywords: Boolean functions, CDMA, orthogonal sequences, semi-bent functions, zero correlation zone.
Mathematics Subject Classification: Primary: 11B50, 94A55; Secondary: 11B75.
Citation: Chunlei Xie, Yujuan Sun. Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems. Advances in Mathematics of Communications, 2020, 14 (1) : 1-9. doi: 10.3934/amc.2020001
References:
[1]

A. N. Akansu and R. Poluri, Walsh-like nonlinear phase orthogonal codes for direct sequence CDMA communications, IEEE Trans. on Signal Processing, 55 (2007), 3800-3806.  doi: 10.1109/TSP.2007.894229.

[2]

S. Chee, S. Lee and K. Kim, Semi-bent functions, Advances in Cryptology-ASIACRYPT'94, 917 (1994), 107-118. 

[3]

E. H. Dinan and B. Jabbari, Spreading codes for direct sequence CDMA and wideband CDMA cellular networks, IEEE Communications Magazine, 36 (1998), 48-54.  doi: 10.1109/35.714616.

[4]

P. Z. Fan, N. Suehiro, N. Kuroyanagi and X. Deng, Class of binary sequences with zero correlation zone, Electronics Letters, 35 (1999), 777-779.  doi: 10.1049/el:19990567.

[5] A. Goldsmith, Wireless Communications, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511841224.
[6]

O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.

[7]

D. H. Smith, R. P. War and S. Perkins, Gold codes, Hadamard partitions and the security of CDMA systems, Designs, Codes and Cryptography, 51 (2009), 231-243.  doi: 10.1007/s10623-008-9257-8.

[8]

D. H. Smith, F. H. Hunt and S. Perkins, Exploiting spatial separations in CDMA systems with correlation constrained sets of Hadamard matrices, IEEE Transactions on Information Theory, 56 (2010), 5757-5761.  doi: 10.1109/TIT.2010.2070310.

[9]

X. H. Tang and P. Fan, Bounds on aperiodic and odd correlations of spreading sequences with low or zero correlation zone, Electronics Letters, 37 (2001), 1201-1202.  doi: 10.1049/el:20010801.

[10]

X. Tang and W. H. Mow, Design of spreading codes for quasi-synchronous CDMA with intercell interference, IEEE Journal on Selected Areas in Communications, 24 (2006), 84-93. 

[11]

X. H. Tang, P. Z. Fan and J. Lindner, Multiple binary ZCZ sequence sets with good cross-correlation property based on complementary sequence sets, IEEE Transactions on Information Theory, 56 (2010), 4038-4045.  doi: 10.1109/TIT.2010.2050796.

[12]

K. Yang, Y.-K. Kim and P. V. Kumar, Quasi-orthogonal sequences for code-division multiple-access systems, IEEE Transactions on Information Theory, 46 (2000), 982-993.  doi: 10.1109/18.841175.

[13]

W.-G. Zhang, C.-L. Xie and E. Pasalic, Large sets of orthogonal sequences suitable for applications in CDMA systems, IEEE Transactions on Information Theory, 62 (2016), 3757-3767.  doi: 10.1109/TIT.2016.2550478.

[14]

Z. C. Zhou, D. Zhang, T. Helleseth and J. Wen, A construction of multiple optimal zcz sequence sets with good cross-correlation, IEEE Transactions on Information Theory, 64 (2018), 1340-1346.  doi: 10.1109/TIT.2017.2756845.

show all references

References:
[1]

A. N. Akansu and R. Poluri, Walsh-like nonlinear phase orthogonal codes for direct sequence CDMA communications, IEEE Trans. on Signal Processing, 55 (2007), 3800-3806.  doi: 10.1109/TSP.2007.894229.

[2]

S. Chee, S. Lee and K. Kim, Semi-bent functions, Advances in Cryptology-ASIACRYPT'94, 917 (1994), 107-118. 

[3]

E. H. Dinan and B. Jabbari, Spreading codes for direct sequence CDMA and wideband CDMA cellular networks, IEEE Communications Magazine, 36 (1998), 48-54.  doi: 10.1109/35.714616.

[4]

P. Z. Fan, N. Suehiro, N. Kuroyanagi and X. Deng, Class of binary sequences with zero correlation zone, Electronics Letters, 35 (1999), 777-779.  doi: 10.1049/el:19990567.

[5] A. Goldsmith, Wireless Communications, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511841224.
[6]

O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.

[7]

D. H. Smith, R. P. War and S. Perkins, Gold codes, Hadamard partitions and the security of CDMA systems, Designs, Codes and Cryptography, 51 (2009), 231-243.  doi: 10.1007/s10623-008-9257-8.

[8]

D. H. Smith, F. H. Hunt and S. Perkins, Exploiting spatial separations in CDMA systems with correlation constrained sets of Hadamard matrices, IEEE Transactions on Information Theory, 56 (2010), 5757-5761.  doi: 10.1109/TIT.2010.2070310.

[9]

X. H. Tang and P. Fan, Bounds on aperiodic and odd correlations of spreading sequences with low or zero correlation zone, Electronics Letters, 37 (2001), 1201-1202.  doi: 10.1049/el:20010801.

[10]

X. Tang and W. H. Mow, Design of spreading codes for quasi-synchronous CDMA with intercell interference, IEEE Journal on Selected Areas in Communications, 24 (2006), 84-93. 

[11]

X. H. Tang, P. Z. Fan and J. Lindner, Multiple binary ZCZ sequence sets with good cross-correlation property based on complementary sequence sets, IEEE Transactions on Information Theory, 56 (2010), 4038-4045.  doi: 10.1109/TIT.2010.2050796.

[12]

K. Yang, Y.-K. Kim and P. V. Kumar, Quasi-orthogonal sequences for code-division multiple-access systems, IEEE Transactions on Information Theory, 46 (2000), 982-993.  doi: 10.1109/18.841175.

[13]

W.-G. Zhang, C.-L. Xie and E. Pasalic, Large sets of orthogonal sequences suitable for applications in CDMA systems, IEEE Transactions on Information Theory, 62 (2016), 3757-3767.  doi: 10.1109/TIT.2016.2550478.

[14]

Z. C. Zhou, D. Zhang, T. Helleseth and J. Wen, A construction of multiple optimal zcz sequence sets with good cross-correlation, IEEE Transactions on Information Theory, 64 (2018), 1340-1346.  doi: 10.1109/TIT.2017.2756845.

Figure 1.  Assignment of orthogonal sets to a lattice of regular hexagonal cells
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Orthogonality between $ \overline{f_c} $ and $ \mathcal {H}_{a} $
$ \mathcal {H}_{00} $ $ \mathcal {H}_{10} $ $ \mathcal {H}_{01} $ $ \mathcal {H}_{11} $
$ \overline{f_{00}} $ $ \bot $ $ \bot $ $ \bot $
$ \overline{f_{10}} $ $ \bot $ $ \bot $
$ \overline{f_{01}} $ $ \bot $ $ \bot $
$ \overline{f_{11}} $ $ \bot $ $ \bot $ $ \bot $
Table Options

Download as excel

$ \mathcal {H}_{00} $ $ \mathcal {H}_{10} $ $ \mathcal {H}_{01} $ $ \mathcal {H}_{11} $
$ \overline{f_{00}} $ $ \bot $ $ \bot $ $ \bot $
$ \overline{f_{10}} $ $ \bot $ $ \bot $
$ \overline{f_{01}} $ $ \bot $ $ \bot $
$ \overline{f_{11}} $ $ \bot $ $ \bot $ $ \bot $
[1]

Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026
[2]

Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037
[3]

Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069
[4]

Wei-Wen Hu. Integer-valued Alexis sequences with large zero correlation zone. Advances in Mathematics of Communications, 2017, 11 (3) : 445-452. doi: 10.3934/amc.2017037
[5]

Jian Liu, Sihem Mesnager, Lusheng Chen. Variation on correlation immune Boolean and vectorial functions. Advances in Mathematics of Communications, 2016, 10 (4) : 895-919. doi: 10.3934/amc.2016048
[6]

Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260
[7]

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
[8]

Yu Zhou. On the distribution of auto-correlation value of balanced Boolean functions. Advances in Mathematics of Communications, 2013, 7 (3) : 335-347. doi: 10.3934/amc.2013.7.335
[9]

Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $. Advances in Mathematics of Communications, 2022, 16 (3) : 485-501. doi: 10.3934/amc.2020121
[10]

Constanza Riera, Pantelimon Stănică. Landscape Boolean functions. Advances in Mathematics of Communications, 2019, 13 (4) : 613-627. doi: 10.3934/amc.2019038
[11]

Claude Carlet, Khoongming Khoo, Chu-Wee Lim, Chuan-Wen Loe. On an improved correlation analysis of stream ciphers using multi-output Boolean functions and the related generalized notion of nonlinearity. Advances in Mathematics of Communications, 2008, 2 (2) : 201-221. doi: 10.3934/amc.2008.2.201
[12]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21
[13]

Sihem Mesnager, Fengrong Zhang, Yong Zhou. On construction of bent functions involving symmetric functions and their duals. Advances in Mathematics of Communications, 2017, 11 (2) : 347-352. doi: 10.3934/amc.2017027
[14]

Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305
[15]

Claude Carlet. Parameterization of Boolean functions by vectorial functions and associated constructions. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022013
[16]

Sihem Mesnager, Gérard Cohen. Fast algebraic immunity of Boolean functions. Advances in Mathematics of Communications, 2017, 11 (2) : 373-377. doi: 10.3934/amc.2017031
[17]

Claude Carlet, Serge Feukoua. Three basic questions on Boolean functions. Advances in Mathematics of Communications, 2017, 11 (4) : 837-855. doi: 10.3934/amc.2017061
[18]

Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041
[19]

Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043
[20]

Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249

2021 Impact Factor: 1.015

Tools

Metrics

  • PDF downloads (657)
  • HTML views (487)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy