# American Institute of Mathematical Sciences

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

 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.

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.  Google Scholar [2] S. Chee, S. Lee and K. Kim, Semi-bent functions, Advances in Cryptology-ASIACRYPT'94, 917 (1994), 107-118.   Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. Goldsmith, Wireless Communications, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511841224.  Google Scholar [6] O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] S. Chee, S. Lee and K. Kim, Semi-bent functions, Advances in Cryptology-ASIACRYPT'94, 917 (1994), 107-118.   Google Scholar [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.  Google Scholar [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.  Google Scholar [5] A. Goldsmith, Wireless Communications, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511841224.  Google Scholar [6] O. S. Rothaus, On "bent" functions, Journal of Combinatorial Theory, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
Assignment of orthogonal sets to a lattice of regular hexagonal cells
Orthogonality between $\overline{f_c}$ and $\mathcal {H}_{a}$
 $\mathcal {H}_{00}$ $\mathcal {H}_{10}$ $\mathcal {H}_{01}$ $\mathcal {H}_{11}$ $\overline{f_{00}}$ $\bot$ $\bot$ $\bot$ $\overline{f_{10}}$ $\bot$ $\bot$ $\overline{f_{01}}$ $\bot$ $\bot$ $\overline{f_{11}}$ $\bot$ $\bot$ $\bot$
 $\mathcal {H}_{00}$ $\mathcal {H}_{10}$ $\mathcal {H}_{01}$ $\mathcal {H}_{11}$ $\overline{f_{00}}$ $\bot$ $\bot$ $\bot$ $\overline{f_{10}}$ $\bot$ $\bot$ $\overline{f_{01}}$ $\bot$ $\bot$ $\overline{f_{11}}$ $\bot$ $\bot$ $\bot$
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