Skip to main content
Springer Nature Link
Log in

New families of asymptotically optimal doubly periodic arrays with ideal correlation constraints

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

We present q new asymptotically optimal families of doubly periodic arrays with ideal auto and cross correlation constraints, derived from the Moreno-Maric construction for frequency hopping applications. These new families possess the same properties that make the Moreno-Maric construction suitable for communications systems and digital watermarking, size (q+1)×(q+1), weight ω=q+1, family size q−2, and correlation 2, where q is a power of a prime. These new families are asymptotically optimal.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bömer, L., Antweiler, M., Schotten, H.: Quadratic residue arrays. Frequenz 47(7–8), 190–196 (1993)

    Google Scholar 

  2. Chung, F.R.K., Salehi, J.A., Wei, V.K.-W.: Correction to optical orthogonal codes: Design, analysis, and applications (may 89 595–604). IEEE Trans. Inf. Theory 38(4), 1429 (1992)

    MathSciNet  Google Scholar 

  3. Chung, H., Vijay Kumar, P.: Optical orthogonal codes-new bounds and an optimal construction. IEEE Trans. Inf. Theory 36(4), 866 (1990)

    Article  MATH  Google Scholar 

  4. Costas, J.P.: Medium Constraints on Sonar Design and Performance, pp. 68A–68L. FASCON Convention Record (1975)

  5. Golomb, S.W., Taylor, H.: Two-dimensional syncronization patterns for minimum ambiguity. IEEE Trans. Inf. Theory IT–28, 600–604 (1982)

    Article  MathSciNet  Google Scholar 

  6. Gong, G.: Constructions of multiple shift-distinct signal sets with low correlation. In: IEEE International Symposium on Information Theory, ISIT, pp 2306–2310 (2007)

  7. Johnson, S.M.: A new upper bound for error-correcting codes. IEEE Trans. Inf. Theory IT(8), 203–207 (1962)

    Article  Google Scholar 

  8. Kim, S., Yu, K., Park, N.: A new family of space/wavelength/time spread three dimesnional optical code for OCDMA networks. J. Lightwave Technol. 18(4), 502–511 (2000)

    Article  Google Scholar 

  9. Mendez, A.J., Gagliardi, R.M.: Code division multiple access (CDMA) enhancement of wavelength division multiple access (wdm) systems. In: IEEE International Conference Communication, pp 271–276, Seatle, WA (1995)

  10. Milewski, A.: Periodic sequences with optimal properties for channel estimation and fast start-up equalization. IBM J. Res. Dev. 27(5), 426–431 (1983)

    Article  Google Scholar 

  11. Moreno, O., Games, R.A., Taylor, H.: Sonar sequences from Costas arrays and the best known sonar sequences with up to 100 symbols. IEEE Trans. Inf. Theory 39, 1985–1987 (November 1993)

  12. Moreno, O., Maric, S.V.: A new family of frequency-hop codes. IEEE Trans. Commun. 48(8), 1241–1244 (2000)

    Article  MATH  Google Scholar 

  13. Moreno, O., Ortiz Ubarri, J.: Group permutable constant weight codes. In: IEEE Proceedings Information Theory Workshop, Dublin, Ireland (2010)

  14. Nguyen, Q.A., Gyorfi, L., Massey, J.L.: Construction of binary constant-weight cyclic codes and cyclically permutable codes. IEEE Trans. Inf. Theory 38(3), 940–949 (1992)

    Article  MATH  Google Scholar 

  15. Ortiz-Ubarri, J., Moreno, O., Tirkel, A.: Three-dimensional periodic optical orthogonal code for OCDMA systems. In: IEEE Information Theory Workshop (ITW), pp 170–174 (2011)

  16. Ortiz-Ubarri, J., Moreno, O., Tirkel, A.Z., Arce-Nazario, R., Golomb, S.W.: Algebraic symmetries of generic (m+1)-dimensional periodic Costas arrays, Vol. 59 (2013)

  17. Park, E., Mendez, A.J, Garmire, E.M.: Temporal/spatial optical CDMA networks: Design, demonstration and comparison with temporal network. IEEE Photon. Technol. Letter 4, 1160–1162 (1992)

    Article  Google Scholar 

  18. Schotten, H.D., Lüke, H.D.: New perfect and w-cyclic-perfect sequences. In: Proceedings 1996 IEEE International Symposium on Information Theory, vol. 12 (1996)

  19. Tirkel, A.Z., Rankin, G.A., van Schyndel, R.M., Ho, W.J., Mee, N.R.A., Osborne, C.F.: Electronic water mark. In: Digital Image Computing, Technology and Applications (DICTA’93), pp 666–673, Macquarie University, Sydney (1993)

  20. Andrew Tirkel. Personal communication

  21. Tirkel, A., Hall, T.: New matrices with good auto and cross-correlation. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E89–A(9), 2315–2321 (2006)

    Article  Google Scholar 

Download references

Acknowledgments

The author would like to thank Dr. Oscar Moreno, Dr. Andrew Tirkel, Dr. Guang Gong, Dr. Ivelisse Rubio, and the reviewers for their useful comments.

Author information

Authors and Affiliations

  1. Computer Science Department, University of Puerto Rico, San Juan, PR, Puerto Rico

    José Ortiz-Ubarri

Authors
  1. José Ortiz-Ubarri
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to José Ortiz-Ubarri.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ortiz-Ubarri, J. New families of asymptotically optimal doubly periodic arrays with ideal correlation constraints. Cryptogr. Commun. 7, 403–414 (2015). https://doi.org/10.1007/s12095-015-0122-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-015-0122-0

Keywords

Mathematics Subject Classification (2010)