On the Aperiodic Hamming Correlation of Frequency-Hopping Sequences from Norm Functions

  • Zhengchun Zhou
  • Xiaohu Tang
  • Yang Yang
  • Udaya Parampalli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)

Abstract

Frequency-hopping sequences (FHSs) are needed in frequency hopping code-division multiple-access (FH-CDMA) systems. Aperiodic Hamming correlation of FHSs matters in real applications, while it received little attraction in the literature compared with periodic Hamming correlation. In this paper, we study the aperiodic Hamming correlation of a family of FHSs via norm functions by Ding, Moisio and Yuan (IEEE Trans Inform Theory 53: 2606-2610, 2007). Bounds on their aperiodic Hamming correlation are established based on the calculation and estimation of some exponential sums over finite fields.

Keywords

Aperiodic correlation Hamming correlation frequency- hopping sequences exponential sum hybrid sum 

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References

  1. 1.
    Beaulieu, N.C., Young, D.J.: Designing time-hopping ultrawidebandwidth receivers for multiuser interference environments. Proc. IEEE 97, 255–284 (2009)CrossRefGoogle Scholar
  2. 2.
    Carlet, C., Feng, K.: An Infinite Class of Balanced Functions with Optimal Algebraic Immunity, Good Immunity to Fast Algebraic Attacks and Good Nonlinearity. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 425–440. Springer, Heidelberg (2008)CrossRefGoogle 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 45, 5783–5791 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ding, C., Moisio, M.J., Yuan, J.: Algebraic constructions of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 53, 2606–2610 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ding, C., Yin, J.: Sets of optimal frequency-hopping sequences. IEEE Trans. Inf. Theory 54, 3741–3745 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ding, C., Fuji-Hara, R., Fujiwara, F., Jimbo, M., Mishima, M.: Sets of frequency hopping sequences: bounds and optimal constructions. IEEE Trans. Inf. Theory 55, 3297–3304 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Eun, Y.C., Jin, S.Y., Hong, Y.P., Song, H.Y.: Frequency hopping sequences with optimal partial autocorrelation properties. IEEE Trans. Inf. Theory 50, 2438–2442 (2004)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ge, G., Miao, Y., Yao, Z.H.: Optimal frequency hopping sequences: auto- and cross-correlation properties. IEEE Trans. Inf. Theory 55, 867–879 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Helleseth, T., Kumar, P.V.: Sequences with low correlation. In: Pless, V., Huffman, C. (eds.) Handbook of Coding Theory, pp. 1767–1853. Elservier, Amsterdam (1998)Google Scholar
  10. 10.
    Lempel, A., Greenberger, A.: Families of sequence with optimal Hamming correlation properties. IEEE Trans. Inf. Theory 20, 90–94 (1974)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press (1997)Google Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sarwate, D.V.: An upper bound on the aperiodic autocorrelation function for a maximal-length sequence. IEEE Trans. Inf. Theory IT-30, 685–687 (1984)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Simon, M.K., Omura, J.K., Scholtz, R.A.: Spread Spectrum Communications Handbook. McGraw-Hill, Inc. (2002)Google Scholar
  15. 15.
    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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhou, Z.C., Tang, X.H., Niu, X.H., Udaya, P.: New classes of frequency-hopping sequences with optimal partial correlation. IEEE Trans. on Inf. Theory 58, 453–458 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zhou, Z.C., Tang, X.H., Yang, Y., Udaya, P.: A hybrid incomplete exponential sum with application to aperiodic Hamming correlation of some frequency-hopping sequences. Submitted to IEEE Trans. Inf. TheoryGoogle Scholar
  19. 19.
    Wang, Q.: Optimal sets of frequency hopping sequences with large linear spans. IEEE Trans. Inf. Theory 56, 1729–1736 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Zhengchun Zhou
    • 1
    • 2
  • Xiaohu Tang
    • 2
  • Yang Yang
    • 2
  • Udaya Parampalli
    • 3
  1. 1.School of MathematicsSouthwest Jiaotong UniversityChengduPRC
  2. 2.Institute of Mobile CommunicationsSouthwest Jiaotong UniversityChengduPRC
  3. 3.Department of Computer Science and Software EngineeringUniversity of MelbourneAustralia

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