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New Families of Frequency-Hopping Sequences of Length mN Derived from the k-Fold Cyclotomy

  • Jin-Ho Chung
  • Kyeongcheol Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)

Abstract

Let N = p 1 ⋯ p k where p i , 1 ≤ i ≤ k, are odd primes such that p 1 < ⋯ < p k and p i  = M i f + 1 for some positive integers M i and f. In this paper, we construct frequency-hopping sequence (FHS) sets by using the properties of the k-fold cycltomy. We give FHS sets with length 2N and frequency set size (N − 1)/f, which are optimal with respect to the Peng-Fan bound if k = 1, and near-optimal if k ≥ 2. We also present near-optimal FHS sets with length mN and frequency set size (N − 1)/f + 1 for any integer m with 2 ≤ m ≤ M 1. The FHS sets constructed in this paper have new parameters not covered in the literature.

Keywords

Cyclotomic numbers frequency-hopping sequences  generalized cyclotomy Hamming correlation interleaved sequences 

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References

  1. 1.
    Specification of the Bluetooth Systems-Core. The Bluetooth Special Interest Group (SIG), http://www.bluetooth.com
  2. 2.
    Simon, M.K., Omura, J.K., Scholtz, R.A., Levitt, B.K.: Spread Spectrum Communications Handbook. McGraw-Hill Inc., New York (1994) (Revised Ed.)Google Scholar
  3. 3.
    Sarwate, D.V.: Reed-Solomon codes and the design of sequences for spread-spectrum multiple-access communications. In: Wicker, S.B., Bharagava, V.K. (eds.) Reed-Solomon Codes and Their Applications. IEEE Press, Piscataway (1994)Google Scholar
  4. 4.
    Gauss, C.F.: Disquisitiones Arithmeticae (1801); English translation, Yale, New Haven (1966)Google Scholar
  5. 5.
    Chu, W., Colbourn, C.J.: Optimal frequency-hopping sequences via cyclotomy. IEEE Trans. Inf. Theory 51(3), 1139–1141 (2005)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ding, C., Yin, J.: Sets of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 54(8), 3741–3745 (2008)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Han, Y.K., Yang, K.: On the Sidel’nikov sequences as frequency-hopping sequences. IEEE Trans. Inf. Theory 55(9), 4279–4285 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Chung, J.-H., Han, Y.K., Yang, K.: New classes of optimal frequency-hopping sequences by interleaving techniques. IEEE Trans. Inf. Theory 55(12), 5783–5791 (2009)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Chung, J.-H., Yang, K.: Optimal frequency-hopping sequences with new parameters. IEEE Trans. Inf. Theory 56(4), 1685–1693 (2010)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Storer, T.: Cylotomy and Difference Sets. Lectures in Advanced Mathematics. Markham, Chicago (1967)Google Scholar
  11. 11.
    Whiteman, A.L.: A family of difference sets. Illinois J. Math. 6, 107–121 (1962)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Ding, C., Helleseth, T.: New generalized cyclotomy and its applications. Finite Fields Their Appl. 4, 140–166 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chung, J.-H., Yang, K.: k-fold cyclotomy and its application to frequency-hopping sequences (submitted for publication)Google Scholar
  14. 14.
    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)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Fan, P.Z., Darnell, M.: Sequence Design for Communications Applications. Research Studies Press (RSP). John Wiley & Sons, London (1996)Google Scholar
  16. 16.
    Lempel, A., Greenberger, H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20(1), 90–94 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gong, G.: Theory and applications of q-ary interleaved sequences. IEEE Trans. Inf. Theory 41(2), 400–411 (1995)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jin-Ho Chung
    • 1
  • Kyeongcheol Yang
    • 1
  1. 1.Dept. of Electronics and Electrical EngineeringPohang University of Science and Technology (POSTECH)PohangKorea

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