Some conjectures on codes

Variable-length codes are the bases of the free submonoids of a free monoid. There are some important longstanding open questions about the structure of finite maximal codes. In this paper we discuss this conjectures and their relations with factorizations of cyclic groups.

[1]  Christophe Reutenauer,et al.  Solution partielle de la conjecture de factorisation des codes , 1986 .

[2]  Marcel Paul Schützenberger,et al.  A conjecture on sets of differences of integer pairs , 1981, J. Comb. Theory, Ser. B.

[3]  Clelia de Felice.  On a complete set of operations for factorizing codes , 2006, RAIRO Theor. Informatics Appl..

[4]  Sándor Szabó.  Completing codes and the Rédei property of groups , 2006, Theor. Comput. Sci..

[5]  Christophe Reutenauer,et al.  Acyclic, connected and tree sets , 2013, Monatshefte für Mathematik.

[6]  Imre Simon,et al.  A Note on the Triangle Conjecture , 1982, J. Comb. Theory, Ser. A.

[7]  A. D. Sands,et al.  On the factorisation of finite abelian groups , 1957 .

[8]  Jean Berstel,et al.  Bifix codes and Sturmian words , 2010, ArXiv.

[9]  Dominique Perrin,et al.  Maximal bifix decoding , 2013, Discret. Math..

[10]  E. T..  An Introduction to the Theory of Numbers , 1946, Nature.

[11]  A. D. Sands,et al.  Factoring Groups into Subsets , 2009 .

[12]  Clelia De Felice.  On factorizing codes: Structural properties and related decision problems , 2007, Adv. Appl. Math..

[13]  M. Schützenberger,et al.  Une théorie algébrique du codage , 1956 .

[14]  Nguyen Huong Lam,et al.  Hajós Factorizations and Completion of Codes , 1997, Theor. Comput. Sci..

[15]  Clelia de Felice.  An enhanced property of factorizing codes , 2005, Theor. Comput. Sci..

[16]  Antonio Restivo,et al.  Some Results On Finite Maximal Codes , 1985, RAIRO Theor. Informatics Appl..

[17]  Dominique Perrin,et al.  Codes and Automata (Encyclopedia of Mathematics and its Applications) , 2009 .

[18]  Liang Zhang,et al.  Finite maximal codes and triangle conjecture , 2017, Discrete Mathematics.

[19]  Antonio Restivo,et al.  On codes having no finite completions , 1977, Discret. Math..

[20]  Clelia de Felice.  A note on the factorization conjecture , 2013, Acta Informatica.

[21]  Dominique Perrin,et al.  Bifix codes and interval exchanges , 2014, 1408.0389.

[22]  Antonio Restivo,et al.  Completing Codes , 1989, RAIRO Theor. Informatics Appl..

[23]  Clelia de Felice,et al.  An Application of Hajós Factorizations to Variable-Length Codes , 1996, Theor. Comput. Sci..

[24]  Clelia de Felice.  On Some Schützenberger Conjectures , 2001, Inf. Comput..

[25]  Dominique Perrin,et al.  The finite index basis property , 2013, 1305.0127.

[26]  C. Reutenauer,et al.  Noncommutative Rational Series with Applications , 2010 .

[27]  Clelia de Felice,et al.  A partial result about the factorization conjecture for finite variable-length codes , 1993, Discret. Math..

[28]  Clelia de Felice.  Finite Completions via Factorizing Codes , 2007, Int. J. Algebra Comput..

[29]  Peter W. Shor,et al.  A Counterexample to the Triangle Conjecture , 1985, J. Comb. Theory A.

[30]  Clelia de Felice.  Construction of a Family of Finite Maximal Codes , 1989, Theor. Comput. Sci..

[31]  Jean-Marie Boë.  Une Famille Remarquable de Codes Indecomposables , 1978, ICALP.

[32]  Aldo de Luca.  A Conjecture on Continued Fractions , 1998, Theor. Comput. Sci..

[33]  Antonio Restivo,et al.  A Family of Codes Commutatively Equivalent to Prefix Codes , 1981, Inf. Process. Lett..