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A Note with Computer Exploration on the Triangle Conjecture

  • Christophe CorderoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)

Abstract

The triangle conjecture states that codes formed by words of the form \(a^i b a^j\) are either commutatively equivalent to a prefix code or not included in a finite maximal code. Thanks to computer exploration, we exhibit new examples of such non-commutatively prefix codes. In particular, we improve a lower bound in a bounding due to Shor and Hansel. We discuss in the rest of the article the possibility of those codes to be included in a finite maximal code.

Keywords

Codes Triangle conjecture Commutative equivalence conjecture 

Notes

Acknowledgements

The author wants to thank Dominique Perrin for introducing him to the commutatively prefix conjecture, also his Ph.D. supervisors Samuele Giraudo and Jean-Christophe Novelli.

References

  1. 1.
    Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata, vol. 129. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  2. 2.
    De Felice, C.: A note on the factorization conjecture. Acta Informatica 50(7–8), 381–402 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Hansel, G.: Baionnettes et cardinaux. Discrete Math. 39(3), 331–335 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Perrin, D., Schützenberger, M.-P.: Codes et sous-monoïdes possédant des mots neutres. In: Theoretical Computer Science. LNCS, vol. 48, pp. 270–281. Springer, Heidelberg (1977).  https://doi.org/10.1007/3-540-08138-0_23Google Scholar
  5. 5.
    Restivo, A., Salemi, S., Sportelli, T.: Completing codes. RAIRO-Theoret. Inf. Appl. 23(2), 135–147 (1989)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sands, A.D.: Replacement of factors by subgroups in the factorization of abelian groups. Bull. Lond. Math. Soc. 32(3), 297–304 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sardinas, A.A., Patterson, G.W.: A necessary and sufficient condition for unique decomposition of coded messages. In: Proceedings of the Institute of Radio Engineers, vol. 41, no. 3, pp. 425–425 (1953)Google Scholar
  8. 8.
    Shor, P.W.: A counterexample to the triangle conjecture. J. Comb. Theory Ser. A 38(1), 110–112 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhang, L., Shum, K.P.: Finite maximal codes and triangle conjecture. Discrete Math. 340(3), 541–549 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEMMarne-la-ValléeFrance

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