Designs, Codes and Cryptography

, Volume 72, Issue 2, pp 423–434 | Cite as

New sets of frequency-hopping sequences with optimal Hamming correlation

  • Wenli RenEmail author
  • Fang-Wei Fu
  • Zhengchun Zhou


Frequency-hopping spread spectrum and direct-sequence spread spectrum are two main spread coding technologies in communication systems. Frequency-hopping sequences are needed in FH-CDMA systems. In this paper, a construction of optimal sets of frequency-hopping sequences using cyclotomy and the Chinese remainder theorem is introduced. It generalizes some earlier constructions, and produces new optimal sets of frequency-hopping sequences.


Cyclotomy Hamming correlation frequency-hopping sequence  frequency-hopping spread spectrum 

Mathematics Subject Classification (2000)

94A55 11A41 11B99 



The authors would like to thank the Editor and anonymous reviewers for their valuable suggestions and comments that have much improved the quality of this paper.


  1. 1.
    Cao, Z., Ge, G., Miao, Y.: Combinatorial characterizations of one-coincidence frequency-hopping sequences. Des. Codes Cryptogr. 41, 177–184 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chu, W., Colbourn, C.J.: Optimal frequency-hopping sequences via cyclotomy. IEEE Trans. Inf. Theory 51, 1139–1141 (2005)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chung, J.H., Han, Y.K., Yang, K.: New classes of optimal frequency-hopping sequences by interleaving techniques. IEEE Trans. Inf. Theory 55, 5783–5791 (2009)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Ding, C.S., Moisio, M.J., Yuan, J.: Algebraic constrctions of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 53, 2606–2610 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Ding, C.S., Yin, J.: IEEE Trans. Inf. Theory. Sets of optimal frequency-hopping sequences 54, 3741–3745 (2008)MathSciNetGoogle Scholar
  6. 6.
    Ding, C.S., Fuji-Hara, Y., Fujiwara, Y., Jinbo, M., Mishima, M.: Sets of optimal frequency-hopping sequences: bounds and optimal constrctions. IEEE Trans. Inf. Theory 55, 3797–3804 (2009)Google Scholar
  7. 7.
    Fuji-Hara, R., Miao, Y., Mishima, M.: Optimal frequency-hopping sequences: a combinatorial approach. IEEE Trans. Inf. Theory 50, 2408–2420 (2004)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ge, G.N., Fuji-Hara, R.F., Miao, Y.: Further combinatorial constrcutions for optimal frequency-hopping sequences. J. Comb. Theory Ser. A 113, 1699–1718 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ge, G.N., Miao, Y., Yao, Z.: Optimal frequency-hopping sequences: auto and cross corelation properties. IEEE Trans. Inf. Theory 55, 867–879 (2009)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Han, Y.K., Yang, K.: On the Sidel’nikov sequences as frequency-hopping sequences. IEEE Trans. Inf. Theory 55, 4279–4285 (2009)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lempel, A., Greenberger, H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20, 90–94 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Liu, F., Peng, D.Y., Zhou, Z.C., Tang, X.H.: A new frequency-hopping sequence set based upon generalized cyclotomy. Des. Codes and Cryptogr. (2012, in press)Google Scholar
  13. 13.
    Peng, D.Y., Fan, P.Z.: Low bounds on the Hamming auto and cross-correlation of frequency-hopping sequences. IEEE Trans. Inf. Theory 50, 2149–2154 (2004)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Specification of the Bluetooth Systems-Core. The Bluetooth Special Interest Group (SIG).
  15. 15.
    Storer, T.: Cyclotomy and differrence sets. Markam, Chicago (1967)Google Scholar
  16. 16.
    Zhang Y., Ke P.H., Zhang S.Y.: Optimal frequency-hopping sequences based on cyclotomy. In: First International Workshop on Education Technology and Computer Science, vol. 1, pp. 1122–1126. (2009)Google Scholar
  17. 17.
    Zhou, Z.C., Tang, X.H., Peng, D.Y., Parampalli, U.: New constructions for optimal sets of frequency-hopping sequences. IEEE Trans. Inf. Theory 57, 3831–3840 (2011)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhou, Z.C., Tang, X.H., Niu, X.H., Parampalli, U.: New classes of frequency-hopping sequences with optimal partial correlation. IEEE Trans. Inf. Theory 58, 453–458 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Chern Institute of MathematicsNankai UniversityTianjinChina
  2. 2.Department of MathematicsDezhou UniversityDezhouChina
  3. 3.School of MathematicsSouthwest Jiaotong UniversityChengduChina

Personalised recommendations