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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
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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 $
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$ \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 $
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