Advertisement

Cryptography and Communications

, Volume 10, Issue 4, pp 667–683 | Cite as

A 2D non-overlapping code over a q-ary alphabet

  • Elena Barcucci
  • Antonio Bernini
  • Stefano Bilotta
  • Renzo Pinzani
Article
  • 115 Downloads

Abstract

We define a set of matrices over a finite alphabet where all possible overlaps between any two matrices are forbidden. The set is also enumerated by providing some recurrences counting particular classes of restricted words. Moreover, we analyze the asymptotic cardinality of the set according to the parameters related to the construction of the matrices.

Keywords

Bidimensional codes Non-overlapping matrices Restricted words 

Mathematics Subjects Classification (2010)

68R15 94B25 05A15 

Notes

Acknowledgements

This work has been partially supported by the PRIN project “Automi e linguaggi formali: aspetti matematici ed applicativi”, GNCS project “Strutture discrete con vincoli” and GNCS project “Codici di stringhe e matrici non sovrapponibili”. The authors would like to thank the anonymous referee for his valuable advices about the asymptotics.

References

  1. 1.
    Aigner, M.: Discrete Matemathics. American Mathematical Society (2007)Google Scholar
  2. 2.
    Anselmo, M., Giammarresi, D., Madonia, M.: Two-dimensional rational automata: A bridge unifying one- and two-dimensional language theory. Lect. Notes Comput. Sci. 7741, 133–145 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anselmo, M., Giammarresi, D., Madonia, M.: Structure and measure of a decidable class of two-dimensional codes. Lect. Notes Comput. Sci. 8977, 315–327 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anselmo, M., Giammarresi, D., Madonia, M.: Unbordered pictures: Properties and construction. Lect. Notes Comput. Sci. 9270, 45–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bacher, A., Bernini, A., Ferrari, L., Gunby, B., Pinzani, R., West, J.: The Dyck pattern poset. Discrete Math. 321, 12–23 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barcucci, E., Bernini, A., Bilotta, S., Pinzani, R.: Cross-bifix-free sets in two dimensions. Theoret. Comput Sci. (2015). doi: 10.1016/j.tcs.2015.08.032
  7. 7.
    Barcucci, E., Bernini, A., Bilotta, S., Pinzani, R.: Non-overlapping matrices. Theoret. Comput Sci. (2016). doi: 10.1016/j.tcs.2016.05.009
  8. 8.
    Barcucci, E., Bilotta, S., Pergola, E., Pinzani, R., Succi, J.: Cross-bifix-free generation via Motzkin paths . RAIRO Theor. Inform. Appl. 50, 81–91 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bajic, D., Loncar-Turukalo, T.: A simple suboptimal construction of cross-bifix-free codes. Cryptogr. Commun. 6, 27–37 (2014)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bernini, A., Bilotta, S., Pinzani, R., Sabri, A., Vajnovszki, V.: Prefix partitioned gray codes for particular cross-bifix-free sets. Cryptogr. Commun. 6, 359–369 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bernini, A., Ferrari, L., Pinzani, R., West, J.: Pattern-avoiding Dyck paths. Discrete Math. Theoret. Comput. Sci. FPSAC 2013, 683–694 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bernini, A., Ferrari, L., Steingrímsson, E.: The Möbius function of the consecutive pattern poset. Electron. J. Combin. 18, P146 (2011)zbMATHGoogle Scholar
  13. 13.
    Bilotta, S., Grazzini, E., Pergola, E., Pinzani, R.: Avoiding cross-bifix-free binary words. Acta Inform. 50, 157–173 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bilotta, S., Merlini, D., Pergola, E., Pinzani, R.: Pattern 1j+10j avoiding binary words. Fund. Inform. 117, 35–55 (2012)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bilotta, S., Pergola, E., Pinzani, R.: A new approach to cross-bifix-free sets. IEEE Trans. Inform. Theory 58, 4058–4063 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Björner, A.: The Möbius Function of Subword Order. Invariant Theory and Tableaux (Minneapolis, MN, 1988), 118124, IMA Vol. Math Appl., vol. 19. Springer, New York (1990)Google Scholar
  17. 17.
    Blackburn, S.: Non-overlapping codes. IEEE Trans. Inform. Theory 61, 4890–4894 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Burstein, A.: Enumeration of Words with Forbidden Patterns. PhD thesis, University of Pennsylvania (1998)Google Scholar
  19. 19.
    Chee, Y.M., Kiah, H.M., Purkayastha, P., Wang, C.: Cross-bifix-free codes within a constant factor of optimality. IEEE Trans. Inform. Theory 59, 4668–4674 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Er, M.C.: On generating the N-ary reflected Gray code. IEEE Trans. Comput. 33, 739–741 (1984)CrossRefzbMATHGoogle Scholar
  21. 21.
    Flajolet, P., Sedgewick, R.: Analytic Combinaotrics. Cambridge University Press (2009)Google Scholar
  22. 22.
    Gray, F.: Pulse code communication. U.S. Patent 2(632), 058 (1953)Google Scholar
  23. 23.
    Guibas, L.J., Odlyzko, A. M.: String overlaps, pattern matching and nontransitive games. J. Combin. Theory Ser. A 30, 183–208 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Klazar, M.: On abab-free and abba-free sets partitions. European J. Combin. 17, 53–68 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Knuth, D.: The Art of Computer Programming, vol. 1. Addison Wesley, Boston (1968)zbMATHGoogle Scholar
  26. 26.
    Lando, S.K.: Lecture on Generating Functions. American Mathematical Society (2003)Google Scholar
  27. 27.
    Levenshtein, V.I.: Maximum number of words in codes without overlaps. Prob. Inform. Transmission 6, 355–357 (1970)Google Scholar
  28. 28.
    Otto, F., Mráz, F.: Extended two-way ordered restarting automata for picture languages. Lecture Notes Comput. Sci. 8370, 541–552 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pradella, M., Cherubini, A., Crespi-Reghizzi, S.: A unifying approach to picture grammars. Inform. Comput. 209, 1246–1267 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Rowland, E.: Pattern avoidance in binary trees. J. Combin. Theory Ser. A 117, 741–758 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Sagan, B.E.: Pattern avoidance in set partitions. Ars Combin. 117, 79–96 (2010)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Spiegel, M.R.: Complex Variables. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  33. 33.
    Wolfram, D.A.: Solving generalized Fibonacci recurrences. Fibonacci Quart. 36, 129–145 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Informatica “U. Dini”Universitá degli Studi di FirenzeFirenzeItaly

Personalised recommendations