Skip to main content
SpringerLink
Log in

On the Aperiodic Correlation Function of Galois Ring m-Sequences

  • Conference paper
  • First Online:
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2227))

Abstract

We define Gauss-like sums over the Galois Ring GR(4, r) and bound them using the Cauchy-Schwarz inequality. These sums are then used to obtain an upper bound on the aperiodic correlation function of quadriphase m-sequences constructed from GR(4, r). Our first bound δ1 has a simple derivation and is better than the previous upper bound of Shanbag et. al. for small values of N. We then make use of a result of Shanbag et. al. to improve our bound which gives rise to a bound δimproved which is better than the bound of Shanbag et. al. These results can be used as a benchmark while searching for the best phases 3—termed auto3—optimal phases 3—of such quadriphase sequences for use in spread spectrum communication systems. The bounds can also be applied to many other classes of non binary sequences.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Bozta¢, R. Hammons, and P. V. Kumar. 4-phase sequences with near-optimum correlation properties. IEEE, Trans. Inform. Theory, 38:1101–1113, 1992.

    Article  Google Scholar 

  2. S. W. Golomb. Shift Register Sequences. Aegean Park Press: California, U.S.A, 1982.

    Google Scholar 

  3. P. V. Kumar, T. Helleseth, and A. R. Calderbank. An Upper bound for some Exponential Sums Over Galois Rings and Applications. IEEE Trans. Inform. Theory, 41:456–468, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. V. Kumar, T. Helleseth, A. R. Calderbank, and A. R. Hammons, Jr. Large families of quaternary sequences with low correlation. IEEE Trans. Inform. Theory, 42:579–592, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  5. V.S. Pless and W.C. Huffman, editors. Handbookof Coding Theory, Volume 2. New York: Elsevier, 1998.

    Google Scholar 

  6. D. Sarwate. An Upper Bound on the Aperiodic Autocorrelation Function for a Maximal-Length Sequence. IEEE Trans. Inform. Theory, 30:685–687, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. G. Shanbhag, P. V. Kumar, and T. Helleseth. Upper bound for a hybrid sum over galois rings with application to aperiodic correlation of some q-ary sequences. IEEE Trans. Inform. Theory, 42:250–254, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Solé. A Quaternary Cyclic Code, and a Familly of Quadriphase Sequences with low Correlation Properties. Lect. Notes Computer Science, 388:193–201, 1989.

    Google Scholar 

  9. P. Udaya and M. U. Siddiqi. Large Linear Complexity Sequences over Z4 for Quadriphase Modulated Communication Systems having Good Correlation Properties. IEEE International Symposium on Information Theory, Budapest, Hungary, June 23–29, 1991., 1991.

    Google Scholar 

  10. P. Udaya and M. U. Siddiqi. Optimal and Suboptimal Quadriphase Sequences Derived from Maximal Length Sequences over Z4. Appl. Algebra Engrg. Comm. Comput., 9:161–191, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  11. I.M. Vinogradov. Elements of Number Theory. Dover, New York, 1954.

    MATH  Google Scholar 

  12. P. V. Kumar, and O. Moreno. Prime-phase Sequences with Periodic Correlation Properties Better Than Binary Sequences IEEE Trans. Inform. Theory, 37:603–616, 1991.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science and Software Engineering, University of Melbourne, Melbourne, Australia

    P. Udaya1

  2. Dept. of Mathematics, RMIT University, Melbourne, Australia

    S. Bozta¢

Authors
  1. P. Udaya1
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Bozta¢
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, RMIT University, GPO Box 2476V, 3001, Melbourne, Australia

    Serdar Boztaş

  2. Department of Computing, Macquarie University, 2109, NSW, Australia

    Igor E. Shparlinski

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Udaya1, P., Bozta¢, S. (2001). On the Aperiodic Correlation Function of Galois Ring m-Sequences. In: Boztaş, S., Shparlinski, I.E. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2001. Lecture Notes in Computer Science, vol 2227. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45624-4_24

Download citation

  • DOI: https://doi.org/10.1007/3-540-45624-4_24

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42911-1

  • Online ISBN: 978-3-540-45624-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation