A hierarchy for circular codes
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 4, pp. 717-728.

We first prove an extremal property of the infinite Fibonacci * word f: the family of the palindromic prefixes {h n | n≥6} of f is not only a circular code but “almost” a comma-free one (see Prop. 12 in Sect. 4). We also extend to a more general situation the notion of a necklace introduced for the study of trinucleotides codes on the genetic alphabet, and we present a hierarchy relating two important classes of codes, the comma-free codes and the circular ones.

DOI : 10.1051/ita:2008002
Classification : 68R15, 94A45
Mots-clés : theory of codes, comma-free codes, circular codes 
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Pirillo, Giuseppe. A hierarchy for circular codes. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 4, pp. 717-728. doi : 10.1051/ita:2008002. http://www.numdam.org/articles/10.1051/ita:2008002/

[1] D.G. Arquès and C.J. Michel, A complementary circular code in the protein coding genes. J. Theor. Biol. 182 (1996) 45-58.

[2] D.G. Arquès and C.J. Michel, A circular code in the protein coding genes of mitochondria. J. Theor. Biol. 189 (1997) 273-290.

[3] J. Berstel, Mots de Fibonacci. Séminaire d'informatique théorique. LITP, Paris (1980-81) 57-78.

[4] J. Berstel and D. Perrin, Theory of codes. Academic Press (1985). | MR | Zbl

[5] J. Berstel and P. Seebold, Sturmian words, in Algebraic Combinatorics on words, edited by M. Lothaire. Cambridge University Press (2002). | MR

[6] F.H.C. Crick, J.S. Griffith and L.E. Orgel, Codes without commas. Proc. Natl. Acad. Sci. USA 43 (1957) 416-421. | MR

[7] A. De Luca, A combinatorial property of the Fibonacci words. Inform. Process. Lett. 12 (1981) 193-195. | MR | Zbl

[8] A. De Luca, Sturmian words: structure, combinatorics, and their arithmetics. Theoret. Comput. Sci. 183 (1997) 45-82. | MR | Zbl

[9] X. Droubay, Palindromes in the Fibonacci word. Inform. Process Lett. 55 (1995) 217-221. | MR | Zbl

[10] J. Justin and G. Pirillo, On some factorizations of infinite words by elements of codes. Inform. Process. Lett. 62 (1997) 289-294. | MR

[11] J. Karhumäki, On cube-free ω-words generated by binary morphism. Discrete Appl. Math. 5 (1983) 279-297. | MR | Zbl

[12] D.E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, Mass. (1968). | MR | Zbl

[13] D.E. Knuth, J.H. Morris and V.R. Pratt, Fast pattern matching in strings. SIAM J. Comput. 6 (1977) 323-350. | MR | Zbl

[14] M. Lothaire, Combinatorics on words. Addison-Wesley (1983). | MR | Zbl

[15] C.J. Michel, G. Pirillo and M.A. Pirillo, Varieties of comma-free codes. Comput. Math. Appl. (in press). | Zbl

[16] F. Mignosi and G. Pirillo, Repetitions in the Fibonacci infinite word. RAIRO-Theor. Inf. Appl. 26 (1992) 199-204. | EuDML | Numdam | MR | Zbl

[17] G. Pirillo, Infinite words and biprefix codes. Inform. Process Lett. 50 293-295 (1994). | MR | Zbl

[18] G. Pirillo, Fibonacci numbers and words. Discrete Math. 173 (1997) 197-207. | MR | Zbl

[19] G. Pirillo, Some factorizations of the Fibonacci word. Algebra Colloquium 6 (1999) 361-368. | MR | Zbl

[20] G. Pirillo, A characterization for a set of trinucleotides to be a circular code, In Determinism, Holism, and Complexity, edited by C. Pellegrini, P. Cerrai, P. Freguglia, V. Benci and G. Israel. Kluwer (2003).

[21] G. Pirillo and M.A. Pirillo, Growth function of self-complementary circular codes. Biology Forum 98 (2005) 97-110.

[22] P.P. Séébold, Propriétés combinatoires des mots infinis engendrés par certains morphismes. PhD. thesis, L.I.T.P., Paris. (1985).

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