Skip to main content
SpringerLink
Log in

Improved Singleton bound on frequency hopping sequences and optimal constructions

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

A Correction to this article was published on 16 February 2019

This article has been updated

Abstract

Frequency hopping (FH) sequences play an important role in FH spread spectrum communication systems. In this paper, a new theoretical bound on the FH sequences with respect to the size of the frequency slot set, the sequence length, the family size, and the maximum periodic Hamming correlation is established. The new bound is tighter than the Singleton bounds on the FH sequences derived by Ding et al. (IEEE Trans Inf Theory 55:3297–3304, 2009) and Yang et al. (IEEE Trans Inf Theory 57:7605–7613, 2011) (first by Sarwate in: Glisic and Leppanen (eds) Code division multiple access communications, Springer, Boston, 1995). In addition, the new bound employes the M\(\ddot{\text {o}}\)bius function. Then, more optimal FH sequence sets are obtained from Reed–Solomon codes. By utilizing the properties of cyclic codes, a new class of optimal FH sequence sets is obtained whose parameters meet the new bound. Further, two new constructions of FH sequence sets are presented. More new FH sequence sets are obtained by choosing proper base sequence sets. Meanwhile, the FH sequence sets constructed are optimal with respect to the new bound.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Change history

  • 16 February 2019

    The original version of this article unfortunately contained a mistake in an equation.

References

  1. Bao J.J., Ji L.J.: New families of optimal frequency hopping sequence sets. IEEE Trans. Inf. Theory 62, 5209–5224 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  2. Chu W., Colbourn C.J.: Optimal frequency-hopping sequences via cyclotomy. IEEE Trans. Inf. Theory 51, 1139–1141 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  3. Chung J.H., Gong G., Yang K.: New families of optimal frequency-hopping sequences of composite lengths. IEEE Trans. Inf. Theory 60, 3688–3697 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  4. Chung J.H., Han Y.K., Yang K.: New classes of optimal frequency-hopping sequences by interleaving techniques. IEEE Trans. Inf. Theory 45, 5783–5791 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  5. Chung J.H., Yang K.: Optimal frequency-hopping sequences with new parameters. IEEE Trans. Inf. Theory 56, 1685–1693 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding C., Fuji-Hara R., Fujiwara Y., Jimbo M., Mishima M.: Sets of frequency hopping sequences: bounds and optimal constructions. IEEE Trans. Inf. Theory 55, 3297–3304 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding C., Moisio M.J., Yuan J.: Algebraic constructions of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 53, 2606–2610 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding C., Yang Y., Tang X.H.: Optimal sets of frequency hopping sequences from linear cyclic codes. IEEE Trans. Inf. Theory 55, 3605–3612 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  9. Fan P.Z., Darnell M.: Sequence Design for Communications Applications. Research Studies Press (RSP), John Wiley and Sons, London (1996).

    Google Scholar 

  10. Fan P.Z., Lee M.H., Peng D.Y.: New family of hopping sequences for time/frequency-hopping CDMA systems. IEEE Trans. Wirel. Commun. 4, 2836–2842 (2005).

    Article  Google Scholar 

  11. Fuji-Hara R., Miao Y., Mishima M.: Optimal frequency hopping sequences: a combinatorial approach. IEEE Trans. Inf. Theory 50, 2408–2420 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  12. Ge G., Miao Y., Yao Z.: Optimal frequency hopping sequences: auto- and cross-correlation properties. IEEE Trans. Inf. Theory 55, 867–879 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  13. Golomb S.W.: A mathematical theory of discrete classification. In: Proc. 4th London Symp. Information Theory. London, UK, pp. 404–425 (1961).

  14. Golomb S.W., Gong G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography and Radar. Cambridge University Press, Cambridge (2005).

    Book  MATH  Google Scholar 

  15. Golomb S.W., Gordon B., Welch L.R.: Comma-free codes. Can. J. Math. 10, 202–209 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  16. Kumar P.V.: Frequency-hopping code sequence designs having large linear span. IEEE Trans. Inf. Theory 34, 146–151 (1988).

    Article  MathSciNet  Google Scholar 

  17. Lam A.W., Sarwate D.V.: Time-hopping and frequency-hopping multiple-access packet communications. IEEE Trans. Commun. 38, 875–888 (1990).

    Article  Google Scholar 

  18. Lempel A., Greenberger H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20, 90–94 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  19. Ling S., Xing C.: Coding Theory: A First Course. Cambridge University Press, Cambridge (2004).

    Book  Google Scholar 

  20. Liu X., Peng D.Y.: Sets of frequency hopping sequences under aperiodic Hamming correlation: upper bound and optimal constructions. Adv. Math. Commun. 8, 359–373 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu X., Peng D.Y.: Frequency hopping sequences with optimal aperiodic Hamming correlation by interleaving techniques. Adv. Math. Commun. 11, 151–159 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu X., Peng D.Y., Han H.Y.: Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties. Des. Codes Cryptogr. 73, 167–176 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  23. Liu X., Zhou L.: New bound on partial Hamming correlation of low-hit-zone frequency hopping sequences and optimal constructions. IEEE Commun. Lett. 22, 878–881 (2018).

    Article  Google Scholar 

  24. Liu X., Zhou L., Zeng Q.: No-hit-zone frequency hopping sequence sets with respect to aperiodic Hamming correlation. Electron. Lett. 54, 212–213 (2018).

    Article  Google Scholar 

  25. Peng D.Y., Fan P.Z.: Lower bounds on the Hamming auto- and cross correlations of frequency-hopping sequences. IEEE Trans. Inf. Theory 50, 2149–2154 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  26. Sarwate D.V.: Optimum PN sequences for CDMA systems. In: Glisic S.G., Leppanen P.A. (eds.) Code Division Multiple Access Communications, pp. 53–78. Springer, Boston, MA (1995).

    Chapter  Google Scholar 

  27. Song H.Y., Reed I.S., Golomb S.W.: On the nonperiodic cyclic equivalent classes of Reed-Solomon codes. IEEE Trans. Inf. Theory 39, 1431–1434 (1993).

    Article  MATH  Google Scholar 

  28. Udaya P., Siddiqi M.U.: Optimal large linear span frequency hopping patterns derived from polynomial residue class rings. IEEE Trans. Inf. Theory 44, 1492–1503 (1998).

    Article  MATH  Google Scholar 

  29. Wang Q.: Optimal sets of frequency hopping sequences with large linear spans. IEEE Trans. Inf. Theory 56, 1729–1736 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang Y., Tang X.H., Udaya P., Peng D.Y.: New bound on frequency hopping sequence sets and its optimal constructions. IEEE Trans. Inf. Theory 57, 7605–7613 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  31. Zeng X.Y., Cai H., Tang X.H., Yang Y.: A class of optimal frequency hopping sequences with new parameters. IEEE Trans. Inf. Theory 58, 4899–4907 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  32. Zeng X.Y., Cai H., Tang X.H., Yang Y.: Optimal frequency hopping sequences of odd length. IEEE Trans. Inf. Theory 59, 3237–3248 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhou Z.C., Tang X.H., Peng D.Y., Udaya P.: New constructions for optimal sets of frequency-hopping sequences. IEEE Trans. Inf. Theory 57, 3831–3840 (2011).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the Postdoctoral Foundation of University of Electronic Science and Technology of China. The material in this paper was presented in part at the Eighth International Conference on Sequences and Their Applications, Melbourne, Australia, November 2014.

Author information

Authors and Affiliations

  1. National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Xing Liu & Liang Zhou

  2. College of Electrical Engineering and Information Technology, Sichuan University, Chengdu, 610065, China

    Xing Liu

Authors
  1. Xing Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liang Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing Liu.

Additional information

Communicated by C. J. Colbourn.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of this article was revised: the conversion error in Eq. 14 was corrected.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Zhou, L. Improved Singleton bound on frequency hopping sequences and optimal constructions. Des. Codes Cryptogr. 87, 1713–1733 (2019). https://doi.org/10.1007/s10623-018-0572-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-018-0572-4

Keywords

Mathematics Subject Classification

Search

Navigation