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

We first prove an extremal property of the infinite Fibonacci * word f: the family of the palindromic prefixes {h n | n6} 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 : https://doi.org/10.1051/ita:2008002
Classification:  68R15,  94A45
Keywords: theory of codes, comma-free codes, circular codes
@article{ITA_2008__42_4_717_0,
     author = {Pirillo, Giuseppe},
     title = {A hierarchy for circular codes},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {4},
     year = {2008},
     pages = {717-728},
     doi = {10.1051/ita:2008002},
     zbl = {1155.68069},
     mrnumber = {2458703},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2008__42_4_717_0}
}
Pirillo, Giuseppe. A hierarchy for circular codes. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 4, pp. 717-728. doi : 10.1051/ita:2008002. http://www.numdam.org/item/ITA_2008__42_4_717_0/

[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 797069 | Zbl 0587.68066

[5] J. Berstel and P. Seebold, Sturmian words, in Algebraic Combinatorics on words, edited by M. Lothaire. Cambridge University Press (2002). | MR 1905123

[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 86734

[7] A. De Luca, A combinatorial property of the Fibonacci words. Inform. Process. Lett. 12 (1981) 193-195. | MR 632866 | Zbl 0468.20049

[8] A. De Luca, Sturmian words: structure, combinatorics, and their arithmetics. Theoret. Comput. Sci. 183 (1997) 45-82. | MR 1468450 | Zbl 0911.68098

[9] X. Droubay, Palindromes in the Fibonacci word. Inform. Process Lett. 55 (1995) 217-221. | MR 1351896 | Zbl 1004.68537

[10] J. Justin and G. Pirillo, On some factorizations of infinite words by elements of codes. Inform. Process. Lett. 62 (1997) 289-294. | MR 1463344

[11] J. Karhumäki, On cube-free ω-words generated by binary morphism. Discrete Appl. Math. 5 (1983) 279-297. | MR 690339 | Zbl 0505.03022

[12] D.E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, Mass. (1968). | MR 378456 | Zbl 1127.68068

[13] D.E. Knuth, J.H. Morris and V.R. Pratt, Fast pattern matching in strings. SIAM J. Comput. 6 (1977) 323-350. | MR 451916 | Zbl 0372.68005

[14] M. Lothaire, Combinatorics on words. Addison-Wesley (1983). | MR 675953 | Zbl 0514.20045

[15] C.J. Michel, G. Pirillo and M.A. Pirillo, Varieties of comma-free codes. Comput. Math. Appl. (in press). | Zbl 1195.92022 | Zbl pre05288062

[16] F. Mignosi and G. Pirillo, Repetitions in the Fibonacci infinite word. RAIRO-Theor. Inf. Appl. 26 (1992) 199-204. | Numdam | MR 1170322 | Zbl 0761.68078

[17] G. Pirillo, Infinite words and biprefix codes. Inform. Process Lett. 50 293-295 (1994). | MR 1286606 | Zbl 0810.68108

[18] G. Pirillo, Fibonacci numbers and words. Discrete Math. 173 (1997) 197-207. | MR 1468849 | Zbl 0882.68113

[19] G. Pirillo, Some factorizations of the Fibonacci word. Algebra Colloquium 6 (1999) 361-368. | MR 1809671 | Zbl 1167.68416

[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).