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A design and flexible assignment of orthogonal binary sequence sets for (QS)-CDMA systems

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Abstract

Boolean functions naturally induce binary sequences of length \(2^m\) and a large number of such orthogonal sequences is required in the design of code-division multiple-access (CDMA) systems. In this paper, Boolean functions are used to construct nonlinear phase orthogonal sequence sets for CDMA communications. For even m, employing carefully designed an m-variable Boolean function with five-valued Walsh spectra, one can get 16 different orthogonal sequence sets with sequence length \(2^m\). These sequence sets are assigned to a lattice of regular hexagonal cells, and we can ensure the orthogonality of adjacent cells. Moreover, the cross-correlation values between the sequences in a given cell and the sequences in non-neighbouring cells belong to \(\{0,\pm 2^{\frac{m}{2}}, \pm 2^{\frac{m}{2}+1}\}\). On the other hand, the cardinality of the sequences sets is \(2^{m-3}\) thus implying a trade-off between the quality of communication and the number of users assigned to each cell. This method can be improved so that the number of users is increased to \(2^{m-2}\) in one half of the network while preserving the orthogonality between adjacent cells and the same level of low cross-correlation values to the non-neighbouring cells.

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Acknowledgements

WeiGuo Zhang is partly supported by the National Natural Science Foundation of China (Grant 61972303, 62272360). Enes Pasalic is partly supported by the Slovenian Research Agency (research program P1-0404 and research projects J1-9108, J1-1694).

Author information

Authors and Affiliations

  1. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, 710071, China

    WeiGuo Zhang

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

    WeiGuo Zhang

  3. University of Primorska, FAMNIT & IAM, Koper, Slovenia

    Enes Pasalic

  4. Leonard N. Stern School of Business, New York University, New York, NY, 10012, USA

    Yiran Liu

  5. School of Cyberspace Security, Dongguan University of Technology, Dongguan, 523808, China

    Liupiao Zhang

  6. School of Information Engineering, Chang’an University, Xi’an, 710064, China

    Chunlei Xie

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  1. WeiGuo Zhang
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  2. Enes Pasalic
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  3. Yiran Liu
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Correspondence to WeiGuo Zhang.

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Communicated by K.-U. Schmidt.

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Appendix

Appendix

The sets of orthogonal sequences \(H_{000}\), \(H_{001}\), \(\ldots \), \(H_{111}\) in Example 1 is given as below:

\(H_{000}\):

(++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++)

(++++++++——–++++++++——–++++++++——–++++++++——–)

(++++++++++++++++—————-++++++++++++++++—————-)

(++++++++—————-++++++++++++++++—————-++++++++)

(++++++++++++++++++++++++++++++++——————————–)

(++++++++——–++++++++—————-++++++++——–++++++++)

(++++++++++++++++——————————–++++++++++++++++)

(++++++++—————-++++++++——–++++++++++++++++——–)

\(H_{001}\):

(+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-)

(+-+-+-+–+-+-+-++-+-+-+–+-+-+-++-+-+-+–+-+-+-++-+-+-+–+-+-+-+)

(+-+-+-+-+-+-+-+–+-+-+-+-+-+-+-++-+-+-+-+-+-+-+–+-+-+-+-+-+-+-+)

(+-+-+-+–+-+-+-+-+-+-+-++-+-+-+-+-+-+-+–+-+-+-+-+-+-+-++-+-+-+-)

(+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+–+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+)

(+-+-+-+–+-+-+-++-+-+-+–+-+-+-+-+-+-+-++-+-+-+–+-+-+-++-+-+-+-)

(+-+-+-+-+-+-+-+–+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-++-+-+-+-+-+-+-+-)

(+-+-+-+–+-+-+-+-+-+-+-++-+-+-+–+-+-+-++-+-+-+-+-+-+-+–+-+-+-+)

\(H_{010}\):

(++–++–++–++–++–++–++–++–++–++–++–++–++–++–++–++–)

(++–++—-++–++++–++—-++–++++–++—-++–++++–++—-++–++)

(++–++–++–++—-++–++–++–++++–++–++–++—-++–++–++–++)

(++–++—-++–++–++–++++–++–++–++—-++–++–++–++++–++–)

(++–++–++–++–++–++–++–++—-++–++–++–++–++–++–++–++)

(++–++—-++–++++–++—-++–++–++–++++–++—-++–++++–++–)

(++–++–++–++—-++–++–++–++–++–++–++–++++–++–++–++–)

(++–++—-++–++–++–++++–++—-++–++++–++–++–++—-++–++)

\(H_{011}\):

(+–++–++–++–++–++–++–++–++–++–++–++–++–++–++–++–+)

(+–++–+-++–++-+–++–+-++–++-+–++–+-++–++-+–++–+-++–++-)

(+–++–++–++–+-++–++–++–++-+–++–++–++–+-++–++–++–++-)

(+–++–+-++–++–++–++-+–++–++–++–+-++–++–++–++-+–++–+)

(+–++–++–++–++–++–++–++–+-++–++–++–++–++–++–++–++-)

(+–++–+-++–++-+–++–+-++–++–++–++-+–++–+-++–++-+–++–+)

(+–++–++–++–+-++–++–++–++–++–++–++–++-+–++–++–++–+)

(+–++–+-++–++–++–++-+–++–+-++–++-+–++–++–++–+-++–++-)

\(H_{100}\):

(++++—-++++—-++++—-++++—-++++—-++++—-++++—-++++—-)

(++++——–++++++++——–++++++++——–++++++++——–++++)

(++++—-++++——–++++—-++++++++—-++++——–++++—-++++)

(++++——–++++—-++++++++—-++++——–++++—-++++++++—-)

(++++—-++++—-++++—-++++——–++++—-++++—-++++—-++++)

(++++——–++++++++——–++++—-++++++++——–++++++++—-)

(++++—-++++——–++++—-++++—-++++—-++++++++—-++++—-)

(++++——–++++—-++++++++——–++++++++—-++++——–++++)

\(H_{101}\):

(+-+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-+–+-+)

(+-+–+-+-+-++-+-+-+–+-+-+-++-+-+-+–+-+-+-++-+-+-+–+-+-+-++-+-)

(+-+–+-++-+–+-+-+-++-+–+-++-+-+-+–+-++-+–+-+-+-++-+–+-++-+-)

(+-+–+-+-+-++-+–+-++-+-+-+–+-++-+–+-+-+-++-+–+-++-+-+-+–+-+)

(+-+–+-++-+–+-++-+–+-++-+–+-+-+-++-+–+-++-+–+-++-+–+-++-+-)

(+-+–+-+-+-++-+-+-+–+-+-+-++-+–+-++-+-+-+–+-+-+-++-+-+-+–+-+)

(+-+–+-++-+–+-+-+-++-+–+-++-+–+-++-+–+-++-+-+-+–+-++-+–+-+)

(+-+–+-+-+-++-+–+-++-+-+-+–+-+-+-++-+-+-+–+-++-+–+-+-+-++-+-)

\(H_{110}\):

(++—-++++—-++++—-++++—-++++—-++++—-++++—-++++—-++)

(++—-++–++++–++—-++–++++–++—-++–++++–++—-++–++++–)

(++—-++++—-++–++++—-++++–++—-++++—-++–++++—-++++–)

(++—-++–++++—-++++–++—-++++—-++–++++—-++++–++—-++)

(++—-++++—-++++—-++++—-++–++++—-++++—-++++—-++++–)

(++—-++–++++–++—-++–++++—-++++–++—-++–++++–++—-++)

(++—-++++—-++–++++—-++++—-++++—-++++–++—-++++—-++)

(++—-++–++++—-++++–++—-++–++++–++—-++++—-++–++++–)

\(H_{111}\):

(+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-+–+-++-)

(+–+-++–++-+–++–+-++–++-+–++–+-++–++-+–++–+-++–++-+–+)

(+–+-++-+–+-++–++-+–+-++-+–++–+-++-+–+-++–++-+–+-++-+–+)

(+–+-++–++-+–+-++-+–++–+-++-+–+-++–++-+–+-++-+–++–+-++-)

(+–+-++-+–+-++-+–+-++-+–+-++–++-+–+-++-+–+-++-+–+-++-+–+)

(+–+-++–++-+–++–+-++–++-+–+-++-+–++–+-++–++-+–++–+-++-)

(+–+-++-+–+-++–++-+–+-++-+–+-++-+–+-++-+–++–+-++-+–+-++-)

(+–+-++–++-+–+-++-+–++–+-++–++-+–++–+-++-+–+-++–++-+–+)

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Zhang, W., Pasalic, E., Liu, Y. et al. A design and flexible assignment of orthogonal binary sequence sets for (QS)-CDMA systems. Des. Codes Cryptogr. 91, 373–389 (2023). https://doi.org/10.1007/s10623-022-01113-4

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