Two classes of optimal frequency-hopping sequences with new parameters

  • Shanding XuEmail author
  • Xiwang Cao
  • Guangkui Xu
  • Chunming Tang
Original Paper


Direct-sequence spread spectrum and frequency-hopping (FH) spread spectrum are two main spread-coding technologies. Frequency-hopping sequences (FHSs) achieving the well-known Lempel–Greenberger bound play an important part in FH code-division multiple-access systems. Our objective is to construct more FHSs with new parameters attaining the above bound. In this paper, two classes of FHSs are proposed by means of two partitions of \({{\mathbb {Z}}_{v}}\), where v is an odd positive integer. It is shown that all the constructed FHSs are optimal with respect to the Lempel–Greenberger bound. By choosing appropriate injective functions, infinitely many optimal FHSs can be recursively obtained. Above all, these FHSs have new parameters which are not covered in the former literature.


Frequency-hopping sequence Maximal periodic Hamming out-of-phase autocorrelation Optimal 

Mathematics Subject Classification

94A05 94A55 



This work was partially supported by the National Natural Science Foundation of China (Grant No. 11771007, 11601177 and 61572027). The first author was also supported by the Funding of Jiangsu Innovation Program for Graduate Education (Grant No. KYZZ15_0090), the Funding for Outstanding Doctoral Dissertation in NUAA (Grant No. BCXJ16-08), the Open Project Program of Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University (Grant No. GDSXJCKX2016-07), the Funding of Nanjing Institute of Technology (Grant No. CKJB201606), the Nature Science Foundation of Jiangsu Province (Grant No. BK20160771) and the Fundamental Research Funds for the Central Universities.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Shanding Xu
    • 1
    • 2
    • 5
    Email author
  • Xiwang Cao
    • 1
    • 4
  • Guangkui Xu
    • 3
  • Chunming Tang
    • 5
  1. 1.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.Department of Mathematics and physicsNanjing Institute of TechnologyNanjingPeople’s Republic of China
  3. 3.School of Mathematical ScienceHuainan Normal UniversityHuainanPeople’s Republic of China
  4. 4.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingPeople’s Republic of China
  5. 5.Key Laboratory of Mathematics and Interdisciplinary Sciences, Guangdong Higher Education InstitutesGuangzhou UniversityGuangzhouPeople’s Republic of China

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