The Density of Ternary Barker Sequences

  • Tomas Boothby
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7280)


Ternary Barker sequences are sequences whose elements are in -1,0,1 for which every aperiodic offset autocorrelation has magnitude at most 1. Despite promising properties, they have received little attention from both the signals and mathematics communities. In this paper, we demonstrate the existence of ternary Barker sequences to answer a question of Millar. We enumerate ternary Barker sequences of length up to 44 and summarize some features of these sequences. Of primary interest is the density, or proportion of nonzero entries in a sequence. We also briefly examine the relation between density and merit factor.


Nonzero Entry Binary Sequence FPGA Implementation Merit Factor Ternary Sequence 
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.
    Aigner, M.: Proofs from the book. Springer, Berlin (2010)CrossRefGoogle Scholar
  2. 2.
    Balaji, N., Subba Rao, K., Srinivasa Rao, M.: FPGA implementation of ternary pulse compression sequences with superior merit factors. International Journal of Circuits, Systems and Signal Processing 3, 47–54 (2009)Google Scholar
  3. 3.
    Balaji, N., Subba Rao, K., Srinivasa Rao, M.: FPGA implementation of the ternary pulse compression sequences with good discrimination factor values. In: International Conference on Computer Engineering and Technology, vol. 2, pp. 353–357 (2009)Google Scholar
  4. 4.
    Barker, R.H.: Group synchronizing of binary digital systems. Communication Theory, 273–287 (1953)Google Scholar
  5. 5.
    Bose, R., Chowla, S.: Theorems in the additive theory of numbers. Commentarii Mathematici Helvetici 37, 141–147 (1962), doi:10.1007/BF02566968MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Drakakis, K.: A review of the available construction methods for Golomb rulers. Advances in Mathematics of Communications 3, 235–250 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Golay, M.J.E.: The merit factor of long low autocorrelation binary sequences. IEEE Trans. Inform. Theory IT-28, 543–549 (1982)CrossRefGoogle Scholar
  8. 8.
    Jedwab, J.: A Survey of the Merit Factor Problem for Binary Sequences. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 30–55. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Jedwab, J.: What can be used instead of a Barker sequence? Contemporary Mathematics 461, 153–178 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jedwab, J., Katz, D., Schmidt, K.U.: Littlewood polynomials with small L4 norm (submitted)Google Scholar
  11. 11.
    Leung, K.H., Schmidt, B.: New restrictions on possible orders of circulant Hadamard matrices. In: Designs, Codes and Cryptography, pp. 1–9 (2011), doi:10.1007/s10623-011-9493-1Google Scholar
  12. 12.
    Millar, G.: Circulant weighing matrices. Master’s thesis (2009)Google Scholar
  13. 13.
    Moharir, P.S.: Signal design. International Journal of Electronics 41, 381–398 (1976)CrossRefGoogle Scholar
  14. 14.
    Moharir, P.S.: Ternary Barker codes. Electronics Letters 10, 460–461 (1974)CrossRefGoogle Scholar
  15. 15.
    Moharir, P.S., Verma, S.K., Venkata Rao, K.: Ternary pulse compression sequences. J. Inst. Electron. Telecomn. Engrs., 1–8 (1985)Google Scholar
  16. 16.
    Turyn, R.J.: Optimum codes study. Final report. Contract AF19(604)-5473. Sylvania Electronic Systems (1960)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tomas Boothby
    • 1
  1. 1.Simon Fraser UniversityBurnabyCanada

Personalised recommendations