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Journal of Applied and Industrial Mathematics

, Volume 13, Issue 2, pp 280–289 | Cite as

On a Construction of Easily Decodable Sub-de Bruijn Arrays

  • D. A. MakarovEmail author
  • A. D. YashunskyEmail author
Article

Abstract

We consider the two-dimensional generalizations of de Bruijn sequences; i.e., the integer-valued arrays whose all fragments of a fixed size (windows) are different. For these arrays, dubbed sub-de Bruijn, we consider the complexity of decoding; i.e., the determination of a position of a window with given content in an array. We propose a construction of arrays of arbitrary size with arbitrary windows where the number of different elements in the array is of an optimal order and the complexity of decoding a window is linear.

Keywords

de Bruijn sequence de Bruijn array decoding complexity 

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Notes

Acknowledgments

A. D. Yashunsky expresses his gratitude to V. D. Yashunsky for drawing the author’s attention to the problem of constructing de Bruijn arrays.

References

  1. 1.
    N. G de Bruijn, “A Combinatorial Problem,” Proc. Nederl. Akad. Wet. 49 (7), 758–764 (1946).MathSciNetzbMATHGoogle Scholar
  2. 2.
    J. Burns and C. J. Mitchell, “Coding Schemes for Two-Dimensional Position Sensing,” Res. Rep. HPL-92–19 (HP Lab., Bristol, 1992). Available at http://www.hpl.hp.com/techreports/92/HPL-92-19.pdf (accessed February 25, 2019).Google Scholar
  3. 3.
    J. Dénes and A. D. Keedwell, “A New Construction of Two-Dimensional Arrays with the Window Property,” IEEE Trans. Inform. Theory 36 (4), 873–876 (1990).CrossRefzbMATHGoogle Scholar
  4. 4.
    G. Hurlbert and G. Isaak, “On the de Bruijn Torus Problem,” J. Combin. Theory Ser. A 64 (1), 50–62 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    K. G. Paterson, “New Classes of Perfect Maps I,” J. Combin. Theory Ser. A 73 (2), 302–334 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    K. G. Paterson, “New Classes of Perfect Maps II,” J. Combin. Theory Ser. A 73 (2), 335–345 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    C. J. Mitchell, “Aperiodic and Semi-Periodic Perfect Maps,” IEEE Trans. Inform. Theory 41 (1), 88–95 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    T. Etzion, Sequence Folding, Lattice Tiling, and Multidimensional Coding (Cornell Univ. Libr. e-Print Archive, arXiv:0911.1745, 2009).zbMATHGoogle Scholar
  9. 9.
    R. Berkowitz and S. Kopparty, “Robust Positioning Patterns,” in Proceedings of 27th Annual ACM-SIAM Symposium on Discrete Algorithms SODA’16, Arlington, VA, USA, January 10–12, 2016 (SIAM, Philadelphia, PA, 2016), pp. 1937–1951.Google Scholar
  10. 10.
    A. M. Bruckstein, T. Etzion, R. Giryes, N. Gordon, R. J. Holt, and D. Shuldiner, “Simple and Robust Binary Self-Location Patterns,” IEEE Trans. Inform. Theory 58 (7), 4884–4889 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. A. Lloyd and J. Burns, “Finding the Position of a Subarray in a Pseudo-Random Array,” in Cryptography and Coding III (Clarendon Press, Oxford, 1993).Google Scholar
  12. 12.
    C. J. Mitchell and K. G. Paterson, “Decoding Perfect Maps,” Des. Codes Cryptogr. 4 (1), 11–30 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    W. C. Shiu, “Decoding de Bruijn Arrays Constructed by FFMS Method,” Ars Combin. 47, 33–48 (1997).MathSciNetzbMATHGoogle Scholar
  14. 14.
    J. Tuliani, “De Bruijn Sequences with Efficient Decoding Algorithms,” Discrete Math. 226 (1–3), 313–336 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    V. Horan and B. Stevens, “Locating Patterns in the de Bruijn Torus,” Discrete Math. 339 (4), 1274–1282 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D. A. Makarov, “Construction of Easily Decodable Sub-de Bruijn Matrices with a 2 × 2 Window,” in Proceedings of 10th Young Scientists’ School on Discrete Mathematics and Its Applications Moscow, Russia, October 5–11, 2015 (Keldysh Inst. Appl. Math., Moscow, 2015), pp. 47–50. Available at http://keldysh.ru/dmschool/datastore/media/sbornikX.pdf (accessed Feb. 25, 2019).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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