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Two classes of optimal frequency-hopping sequences with new parameters

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

Abstract

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.

Keywords

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

Mathematics Subject Classification

94A05 94A55 

Notes

Acknowledgements

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.

References

  1. 1.
    Lempel, A., Greenberger, H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20(1), 90–94 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kumar, P.: Frequency-hopping code sequence designs having large linear span. IEEE Trans. Inf. Theory 34(1), 146–151 (1988)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Udaya, P., Siddiqi, M.: Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings. IEEE Trans. Inf. Theory 44(4), 1492–1503 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Peng, D., Fan, P.: Lower bounds on the Hamming auto- and cross- correlations of frequency-hopping sequences. IEEE Trans. Inf. Theory 50(9), 2149–2154 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fuji-Hara, R., Miao, Y., Mishima, M.: Optimal frequency hopping sequences: a combinatorial approach. IEEE Trans. Inf. Theory 50(10), 2408–2420 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fan, P., Lee, M., Peng, D.: New family of hopping sequences for time/frequency-hopping CDMA systems. IEEE Trans. Wirel. Commun. 4(6), 2836–2842 (2005)CrossRefGoogle Scholar
  7. 7.
    Chu, W., Colbourn, C.: Optimal frequency-hopping sequences via cyclotomy. IEEE Trans. Inf. Theory 51(3), 1139–1141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ge, G., Fuji-Hara, R., Miao, Y.: Further combinatorial constructions for optimal frequency-hopping sequences. J. Combinat. Theory A 113(8), 1699–1718 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ding, C., Moisio, M., Yuan, J.: Algebraic constructions of optimal frequency hopping sequences. IEEE Trans. Inf. Theory 53(7), 2606–2610 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, C., Yin, J.: Sets of optimal frequency hopping sequences. IEEE Trans. Inf. Theory 54(8), 3741–3745 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ge, G., Miao, Y., Yao, Z.: Optimal frequency hopping sequences: auto- and cross-correlation properties. IEEE Trans. Inf. Theory 55(2), 867–879 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Han, Y., Yang, K.: On the Sidel’nikov sequences as frequency-hopping sequences. IEEE Trans. Inf. Theory 55(9), 4279–4285 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chung, J., Yang, K.: Optimal frequency-hopping sequences with new parameters. IEEE Trans. Inf. Theory 56(4), 1685–1693 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chung, J., Yang, K.: \(k\)-fold cyclotomy and its application to frequency-hopping sequences. IEEE Trans. Inf. Theory 57(4), 2306–2317 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zeng, X., Cai, H., Tang, X., Yang, Y.: A class of optimal frequency hopping sequences with new parameters. IEEE Trans. Inf. Theory 58(7), 4899–4907 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zeng, X., Cai, H., Tang, X., Yang, Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory 59(5), 3237–3248 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cai, H., Zeng, X., Helleseth, T., Tang, X., Yang, Y.: A new construction of zero-difference balanced functions and its applications. IEEE Trans. Inf. Theory 59(8), 5008–5015 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, F., Peng, D., Zhou, Z., Tang, X.: New constructions of optimal frequency hopping sequences with new parameters. Adv. Math. Commun. 7(1), 91–101 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Su, M.: New optimum frequency hopping sequences derived from fermat quotients. In: The Sixth International Workshop on Signal Design and Its Applications, Communications, pp. 166–169 (2013)Google Scholar
  20. 20.
    Cai, H., Zhou, Z., Yang, Y., Tang, X.: A new construction of frequency-hopping sequences with optimal partial Hamming correlation. IEEE Trans. Inf. Theory 60(9), 5782–5790 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Xu, S., Cao, X., Xu, G.: Recursive construction of optimal frequency-hopping sequence sets. IET Commun. 10(9), 1080–1086 (2016)CrossRefGoogle Scholar
  22. 22.
    Cai, H., Zhou, Z., Tang, X., Miao, Y.: Zero-difference balanced functions with new parameters and their applications. IEEE Trans. Inf. Theory 63(7), 4379–4387 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  24. 24.
    Ding, C., Pei, D., Salomaa, A.: Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography. World Scientific, Singapore (1996)CrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Shanding Xu
    • 1
    • 2
    • 5
  • 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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