Skip to main content
SpringerLink
Log in

A Note with Computer Exploration on the Triangle Conjecture

  • Conference paper
  • First Online:
Language and Automata Theory and Applications (LATA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11417))

  • 443 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata, vol. 129. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  2. De Felice, C.: A note on the factorization conjecture. Acta Informatica 50(7–8), 381–402 (2013)

    Article  MathSciNet  Google Scholar 

  3. Hansel, G.: Baionnettes et cardinaux. Discrete Math. 39(3), 331–335 (1982)

    Article  MathSciNet  Google Scholar 

  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_23

    Google Scholar 

  5. Restivo, A., Salemi, S., Sportelli, T.: Completing codes. RAIRO-Theoret. Inf. Appl. 23(2), 135–147 (1989)

    Article  MathSciNet  Google Scholar 

  6. Sands, A.D.: Replacement of factors by subgroups in the factorization of abelian groups. Bull. Lond. Math. Soc. 32(3), 297–304 (2000)

    Article  MathSciNet  Google Scholar 

  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. Shor, P.W.: A counterexample to the triangle conjecture. J. Comb. Theory Ser. A 38(1), 110–112 (1985)

    Article  MathSciNet  Google Scholar 

  9. Zhang, L., Shum, K.P.: Finite maximal codes and triangle conjecture. Discrete Math. 340(3), 541–549 (2017)

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Université Paris-Est, LIGM (UMR 8049), CNRS, ENPC, ESIEE Paris, UPEM, 77454, Marne-la-Vallée, France

    Christophe Cordero

Authors
  1. Christophe Cordero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christophe Cordero .

Editor information

Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  2. Saint Petersburg State University, St. Petersburg, Russia

    Alexander Okhotin

  3. Ariel University, Ariel, Israel

    Dana Shapira

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Cordero, C. (2019). A Note with Computer Exploration on the Triangle Conjecture. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-13435-8_30

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-13434-1

  • Online ISBN: 978-3-030-13435-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation