Sequences pp 195-207 | Cite as

On Pseudo-Random Arrays Constructed from Patterns with distinct Differences

  • Tuvi Etzion
Conference paper


A few constructions of infinite arrays such that, in each (2 n +n-1)×(2 n +1) subarray, each n×2 binary matrix appears exactly once, are given. In other constructions each n×2 binary nonzero matrix appears exactly once. The constructions are using patterns with distinct differences, and although the arrays are not linear they have some similar properties to m-sequences.


Primitive Element Random Array Binary Matrice Polygonal Path Perfect 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.


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  1. [1]
    S. W. Golomb, Shift Register Sequences. Laguna Hills, CA: Aegean Park Press, 1982.Google Scholar
  2. [2]
    I. S. Reed and R. M. Stewart, “Note on the existence of perfect maps” IRE Trans, on Inform. Theory, vol. 8, pp. 10–12, January 1962.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    B. Gordon, “On the existence of perfect maps” IEEE Trans, on Inform. Theory, vol. 12, pp. 486–487, October 1966.CrossRefGoogle Scholar
  4. [4]
    T. Nomura, H. Miyakawa, H. Imai, and A. Fukuda, “A theory of two-dimensional linear recurring arrays”, IEEE Trans, on Inform. Theory, vol. 18, pp. 775–785, November 1972.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    F. J. Macwilliams, and N. J. A. Sloane, “Pseudo-random sequences and arrays”, Proceedings of the IEEE, vol. 64, pp. 1715–1729, December 1976.MathSciNetCrossRefGoogle Scholar
  6. [6]
    S. L. Ma, “A note on binary arrays with a certain window property”, IEEE Trans, on Inform. Theory, vol. 30, pp. 774–775, September 1984.CrossRefMATHGoogle Scholar
  7. [7]
    C. T. Fan, S. M. Fan, S. L. Ma, and M. K. Siu, “On de Bruijn arrays”, Ars Combinatorial vol. 19A, pp. 205–213, May 1985.MathSciNetGoogle Scholar
  8. [8]
    T. Etzion, “Constructions for Perfect Maps and Pseudo-Random Arrays”, IEEE Trans, on Inform. Theory, to appear.Google Scholar
  9. [9]
    J. H. Van Lint, F. J. Macwilliams, and N. J. A. Sloane, “On pseudo-random arrays”, SIAM J. Appl. Math. vol. 36, pp. 62–72, February 1979.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    R. B. Baneiji, “The construction of binary matrices with distinct submatrices”, IEEE Trans, on Computers, vol. C-27, pp. 162–164, February 1978.CrossRefGoogle Scholar
  11. [11]
    J. Denes and A. D. Keedwell, “A new construction of two-dimensional array with the window property”, submitted for publication.Google Scholar
  12. [12]
    S. W. Golomb and H. Taylor, “Tuscan squares - a new family of combinatorial designs”, Ars Combinatorial vol. 20-B, pp. 115–132, December 1985.MathSciNetGoogle Scholar
  13. [13]
    G. S. Bloom and S. W. Golomb, “Applications of numbered undirected graph”, IEEE proceedings, vol. 65, pp. 562–570, April 1977.CrossRefGoogle Scholar
  14. [14]
    J. P. Costas, “A study of a class of detection waveforms having nearly ideal range- Doppler ambiguity properties”, IEEE proceedings, vol. 72, pp. 996–1009, August 1984.CrossRefGoogle Scholar
  15. [15]
    E. N. Gilbert, “Latin squares which contain no repeated diagrams”, SIAM Review, vol. 8, pp. 189–198, April 1965.CrossRefGoogle Scholar
  16. [16]
    S. W. Golomb, T. Etzion, and H. Taylor, “Polygonal path constructions for Tuscan- k squares”,Ars Combinatoria, to appear.Google Scholar
  17. [17]
    T. Etzion, “ On hamiltonian decomposition of K n*, patterns with distinct differences, and Tuscan squares”, submitted for publication.Google Scholar
  18. [18]
    S. W. Golomb and H. Taylor, “Two-dimensional synchronization patterns for minimum ambiguity”, IEEE Trans, on Information Theory, vol. 28, pp. 600–604, July 1982.CrossRefGoogle Scholar
  19. [19]
    S. W. Golomb, “Algebraic constructions for Costas arrays”, J. Combin. Theory, Ser. A, vol. 37, pp. 13–21, July 1984.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    S. W. Golomb and H. Taylor, “Constructions and properties of Costas arrays”, IEEE Proceedings, vol. 72, pp. 1143–1163, September 1984.CrossRefMATHGoogle Scholar
  21. [21]
    J. E. H. Elliott and A. T. Butson, “Relative difference sets”, Illinois J. Math., vol. 10, pp. 517–531, 1966.MathSciNetMATHGoogle Scholar
  22. [22]
    P. V. Kumar, “On the existence of square, dot-matrix patterns having a specific 3- valued periodic-correlation function”,IEEE Trans, on Information Theory, to appear.Google Scholar
  23. [23]
    N. G. de Bruijn, “A combinatorial problem, Nederl. Akad. Wetensch. Proc., vol. 49, pp. 758–764, 1946.MathSciNetGoogle Scholar
  24. [24]
    H. M. Fredricksen, “A class of non-linear de Bruijn cycles”, J. Combin. Theory, Ser. A, vol. 19, pp. 192–199, September 1975.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    H. M. Fredricksen, “A survey of full length nonlinear shift register cycle algorithms”, SIAM Review, vol. 24, pp. 195–221, April 1982.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    T. Etzion and A. Lempel, “Algorithms for the generation of full-length shift-register sequences”, IEEE Trans, on Inform. Theory, vol. 30, pp. 480–484, May 1984.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    R. K. Guy, Unsolved Problems in Number Theory, Berlin, Heidelberg, New York: Springer-Verlag 1981.MATHGoogle Scholar
  28. [28]
    M. R. Schroeder, Number Theory in Science and Communication, Berlin, Heidelberg, New York, Tokyo: Springer-Verlag 1986.MATHGoogle 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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