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Sequences pp 208-227 | Cite as

Combinatorial Designs Derived from Costas Arrays

  • Tuvi Etzion

Abstract

A Costas array is an n×n 0-1 permutation matrix such that all the \(\left[ {\begin{array}{*{20}{c}}n \\2\end{array}} \right]\) vectors connecting two ones in the matrix are distinct. Symmetry and periodicity have an important role in the known constructions for Costas arrays. We prove that some structures of symmetric (or periodic) Costas arrays are not possible, or exist for a limited number of cases. Using Costas arrays we can obtain other arrays which are symmetric and have 4-valued autocorrelation function. Finally, we give some constructions for plane-filling with Costas arrays, i.e., an n×n array with n symbols such that each symbol defines a Costas array.

Keywords

Permutation Matrix Primitive Element Primitive Root Combinatorial Design Bent Function 
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.

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References

  1. [1]
    G. S. Bloom and S. W. Golomb, Applications of numbered undirected graphs, IEEE Proceedings 65 (1977) 562–570.CrossRefGoogle Scholar
  2. [2]
    H. Chung and P. V. Kumar, Generalized Bent functions - some new general constructions and nonexistence tests, Submitted for publication.Google Scholar
  3. [3]
    J. P. Costas, A study of a class of detection waveforms having nearly ideal range- Doppler ambiguity properties, IEEE Proceedings 72 (1984) 996–1009.CrossRefGoogle Scholar
  4. [4]
    J. Denes and A. D. Keedwell, Latin Squares and their Applications ( Academic Press, New York, 1974 ).MATHGoogle Scholar
  5. [5]
    J. Denes and A. D. Keedwell, On the existence of singly-periodic Costas arrays and related questions, preprint.Google Scholar
  6. [6]
    T. Etzion, Combinatorial designs with Costas arrays’ properties, Annals of Discrete Mathematics, to appear.Google Scholar
  7. [7]
    T. Etzion, S. W. Golomb, and H. Taylor, Tuscan-k squares, Advances in Applied Mathematics, to appear.Google Scholar
  8. [8]
    R. Gagliardi, J. Robbins, and H. Taylor, Acquisition sequences in PPM communications, IEEE Transaction on Information Theory IT-33 (1987) 738–744.Google Scholar
  9. [9]
    R. Games, Crosscorrelation of M-sequences and GMW-sequences with the same primitive polynomial, Discrete Applied Math 12 (1985) 139–146.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    R. Games, An algebraic construction of sonar sequences using M-sequences, SIAM J. on Algebraic and Discrete Methods 8 (1987) 753–761.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    S. W. Golomb, Algebraic constructions for Costas arrays, Journal of Combinatorial Theory Series A 37 (1984) 13–21.Google Scholar
  12. [12]
    S. W. Golomb, T. Etzion, and H. Taylor, Polygonal path constructions for Tuscan-k squares, Ars Combinatoria, to appear.Google Scholar
  13. [13]
    S. W. Golomb and H. Taylor, Two-dimensional synchronization patterns for minimum ambiguity, IEEE Transactions on Information Theory IT-28 (1982) 600– 604.Google Scholar
  14. [14]
    S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays, IEEE Proceedings 72 (1984) 1143–1163.CrossRefMATHGoogle Scholar
  15. [15]
    S. W. Golomb and H. Taylor, Tuscan squares - a new family of combinatorial designs, Ars Combinatoria 20-B (1985) 115–132.Google Scholar
  16. [16]
    P. V. Kumar, On the existence of square, dot-matrix pattern having a specific 3- valued periodic-correlation function, IEEE Transactions on Information Theory, to appear.Google Scholar
  17. [17]
    O. Moreno, Costas arrays and a shifting property of its differences, Submmited for publication.Google Scholar
  18. [18]
    H. Taylor, Non-attacking rooks with distinct differences, Communication Science Institute, University of Southern California, Tech Report CSI-84-03-02.Google Scholar

Copyright information

© Springer-Verlag new York Inc. 1990

Authors and Affiliations

  • Tuvi Etzion
    • 1
  1. 1.Computer Science Department, TechnionHaifaIsrael

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