Crosscorrelation Properties of Binary Sequences with Ideal Two-Level Autocorrelation

  • Nam Yul Yu
  • Guang Gong
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4086)


For odd n, binary sequences of period 2 n –1 with ideal two-level autocorrelation are investigated with respect to 3- or 5-valued crosscorrelation property between them. At most 5-valued crosscorrelation of m-sequences is first discussed, which is linked to crosscorrelation of some other binary two-level autocorrelation sequences. Then, several theorems and conjectures are established for describing 3- or 5-valued crosscorrelation of a pair of binary two-level autocorrelation sequences.


Binary Sequence Cyclic Code Quadratic Residue Weight Enumerator Trace Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Antweiler, M.: Cross-correlation of p-ary GMW sequences. IEEE Trans. Inform. Theory 40, 1253–1261 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chang, A., Gaal, P., Golomb, S.W., Gong, G., Helleseth, T., Kumar, P.V.: On a conjectured ideal autocorrelation sequence and a related triple-error correcting cyclic code. IEEE Trans. Inform. Theory 46(2), 680–687 (2000)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dillon, J.F., Dobbertin, H.: New cyclic difference sets with Singer parameters. Finite Fields and Their Applications 10, 342–389 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dillon, J.F.: Multiplicative difference sets via additive characters. Designs, Codes and Cryptography 17, 225–235 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Games, R.A.: Crosscorrelation of m-sequences and GMW-sequences with the same primitive polynomial. Discrete Applied Mathematics 12, 139–146 (1985)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gold, R.: Maximal recursive sequences with 3-valued recursive cross-correlation functions. IEEE Trans. Inform. Theory 14, 154–156 (1968)MATHCrossRefGoogle Scholar
  7. 7.
    Gong, G., Golomb, S.W.: The decimation-Hadamard transform of two-level autocorrelation sequences. IEEE Trans. Inform. Theory 48(4), 853–865 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gordon, B., Mills, W.H., Welch, L.R.: Some new difference sets. Canadian Journal of Mathematics 14(4), 614–625 (1962)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Helleseth, T., Kumar, P.V.: Sequences with Low Correlation. In: Pless, V., Huffmann, C. (eds.) A chapter in Handbook of Coding Theory. Elsevier Science Publishers, Amsterdam (1998)Google Scholar
  10. 10.
    Hertel, D.: Cross-correlation properties of perfect binary sequences. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 208–219. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Kasami, T.: Weight enumerators for several classes of subcodes of the 2nd-order Reed-Muller codes. Information and Control 18, 369–394 (1971)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Maschietti, A.: Difference sets and hyperovals. Designs, Codes and Cryptography 14, 89–98 (1998)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Niho, Y.: Multi-valued cross-correlation functions between two maximal linear recursive sequences. Ph.D. Dissertation. University of Southern California (1972)Google Scholar
  14. 14.
    No, J.S., Chung, H.C., Yun, M.S.: Binary pseudorandom sequences of period 2m − 1 with ideal autocorrelation generated by the polynomial z d + (z + 1)d. IEEE Trans. Inform. Theory 44(3), 1278–1282 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    No, J.S., Golomb, S.W., Gong, G., Lee, H.K., Gaal, P.: Binary pseudorandom sequences of period 2m − 1 with ideal autocorrelation. IEEE Trans. Inform. Theory 44(2), 814–817 (1998)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sidelnikov, V.M.: On mutual correlation of sequences. Soviet Math. Dokl 12, 197–201 (1971)Google Scholar
  17. 17.
    Welch, L.R.: Lower bounds on the maximum cross correlation of signals. IEEE Trans. Inform. Theory IT-20, 397–399 (1974)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nam Yul Yu
    • 1
  • Guang Gong
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterlooCanada

Personalised recommendations