Skip to main content
SpringerLink
Log in

The Merit Factor Problem for Binary Sequences

  • Conference paper
Book cover Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3857))

Abstract

Binary sequences with small aperiodic correlations play an important role in many applications ranging from radar to modulation and testing of systems. In 1977, M. Golay introduced the merit factor as a measure of the goodness of the sequence and conjectured an upper bound for this. His conjecture is still open. In this paper we survey the known results on the Merit Factor problem and comment on the recent experimental results by R.A.Kristiansen and M. Parker and by P. Borwein,K.-K.S.Choi and J. Jedwab.

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. Golay, M.J.E.: Sieves for low autocorrelation binary sequences. IEEE Trans. Inform. Theory IT-23(1), 43–51 (1977)

    Article  Google Scholar 

  2. Newman, D.J., Byrnes, J.S.: The L4 Norm of a Polynomial with Coeffcients ±1. Amer. Math. Monthly 97(1), 42–45 (1990)

    Article  MathSciNet  Google Scholar 

  3. Lindner, J.: Binary sequences up to length 40 with best possible autocorrelation function. Electron. Lett. 11, 507 (1975)

    Article  Google Scholar 

  4. Jungnickel, D., Pott, A.: Perfect and almost perfect sequences. Discrete Applied Math. 95(1), 331–359 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. de Groot, C., Würtz, D., Hoffmann, K.H.: Low autocorrelation binary sequences: Exact enumeration and optimization by evolutionary strategies. Optimization 23, 369–384 (1991)

    Article  Google Scholar 

  6. Mertens, S.: Exhaustive search for low-autocorrelation binary sequences. J. Phys. A 29, 473–481 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bernasconi, J.: Low autocorrelation binary sequences: Statistical mechanics and configuration space analysis. J. Phys. 48, 559–567 (1987)

    Google Scholar 

  8. Bencker, G.F.M., Claasen, T.A.C.M., Heimes, P.W.C.: Binary sequences with a maximally at amplitude spectrum. Phillips J. Res. 40(5), 289–304 (1985)

    Google Scholar 

  9. Høholdt, T., Elbrønd Jensen, H., Justesen, J.: Aperiodic Correlations and the Merit Factor of a class of Binary Sequences. IEEE Trans. Inform. Theory IT-31(4), 549–552 (1985)

    Article  Google Scholar 

  10. Rudin, W.: Some theorems on Fourier coefficients. Proc. Amer. Math. Soc. 10, 855–859 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  11. Baumert, L.D.: Cyclic Difference Sets. Lecture Notes in Mathematics, vol. 189. Springer, Berlin (1971)

    MATH  Google Scholar 

  12. Jensen, J.M., Elbrønd Jensen, H., Høholdt, T.: The Merit Factor of Binary Sequences Related to Difference Sets. IEEE Trans. Inform. Theory 37(3), 617–626 (1991)

    Article  MathSciNet  Google Scholar 

  13. Høholdtand, T., Elbrønd Jensen, H.: Determination of the Merit Factor of Legendre Sequences. IEEE Trans. Inform. Theory 34(1), 161–164 (1988)

    Article  Google Scholar 

  14. Golay, M.J.E.: The merit factor of Legendre sequences. IEEE Trans. Inform. Theory IT-29(6), 934–936 (1982)

    Google Scholar 

  15. Polya, G., Szegö, G.: Aufgeben und Lehrsätze aus der Analyse II. Springer, Berlin (1925)

    Google Scholar 

  16. McEliece, R.J.: Finite Fields for Computer Scientists and Engineers. Kluwer Academics, Boston (1987)

    MATH  Google Scholar 

  17. Lüke, H.D.: Sequences and arrays with perfect periodic correlation. IEEE Trans. Aerospace Electron. Systems 24(3), 287–294 (1988)

    Article  Google Scholar 

  18. Borwein, P., Choi, K.-K.S., Jedwab, J.: Binary sequences with merit factor greater than 6.34. IEEE-Trans.Inform. Theory 50, 3224–3249 (2004)

    MathSciNet  Google Scholar 

  19. Borwein, P., Ferguson, R., Knauer, J.: The merit factor of binary sequences (In preparation)

    Google Scholar 

  20. Knauer, J.: Merit Factor Records. Online Available: http://www.cecm.sfu.ca/~jknauer/labs/records.html

  21. Kristiansen, R., Parker, M.: Binary sequences with merit factor >6:3. IEEE Trans. Inform. Theory 50, 3385–3389 (2004)

    Article  MathSciNet  Google Scholar 

  22. Jedwab, J.: A Survey of the Merit Factor Problem for Binary Sequences (December 2004) (Preprint)

    Google Scholar 

  23. Høholdt, T.: The Merit Factor of Binary Sequences. In: Pott, A., et al. (eds.) Difference Sets, Sequences and Their Correlation Properties. NATO Science Series C, vol. 542, pp. 227–237. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  24. Littlewood, J.E.: On polynomials ∑ n ±zm, ∑n eα m izm, z = eθ i. J. London Math. Soc. 41, 367–376 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kirilusha, A., Narayanaswamy, G.: Construction of new asymptotic classes of binary sequences based on existing asymptotic classes. Summer Science Program Technical Report, Dept. Math. Comput. Science, University of Richmond (July 1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Technical University of Denmark, Bldg. 303, DK-2800, Lyngby, Denmark

    Tom Høholdt

Authors
  1. Tom Høholdt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, Honolulu, USA

    Marc P. C. Fossorier

  2. National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Information Security (RCIS),  

    Hideki Imai

  3. Department of Electrical and Computer Engineering, University of California, CA 95616, Davis,  

    Shu Lin

  4. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Høholdt, T. (2006). The Merit Factor Problem for Binary Sequences. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_4

Download citation

  • DOI: https://doi.org/10.1007/11617983_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31423-3

  • Online ISBN: 978-3-540-31424-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation