{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T17:12:15Z","timestamp":1765041135108,"version":"3.37.3"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T00:00:00Z","timestamp":1612137600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T00:00:00Z","timestamp":1612137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Projekt DEAL"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Theory Biosci."],"published-print":{"date-parts":[[2021,2]]},"abstract":"Abstract<\/jats:title>In the 1950s, Crick proposed the concept of so-called comma-free codes as an answer to the frame-shift problem that biologists have encountered when studying the process of translating a sequence of nucleotide bases into a protein. A little later it turned out that this proposal unfortunately does not correspond to biological reality. However, in the mid-90s, a weaker version of comma-free codes, so-called circular codes, was discovered in nature in J Theor Biol 182:45\u201358, 1996. Circular codes allow to retrieve the reading frame during the translational process in the ribosome and surprisingly the circular code discovered in nature is even circular in all three possible reading-frames ($$C^3$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-property). Moreover, it is maximal in the sense that it contains 20 codons and is self-complementary which means that it consists of pairs of codons and corresponding anticodons. In further investigations, it was found that there are exactly 216 codes that have the same strong properties as the originally found code from J Theor Biol 182:45\u201358. Using an algebraic approach, it was shown in J Math Biol, 2004 that the class of 216 maximal self-complementary $$C^3$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-codes can be partitioned into 27 equally sized equivalence classes by the action of a transformation group $$L \\subseteq S_4$$<\/jats:tex-math>\n \n L<\/mml:mi>\n \u2286<\/mml:mo>\n \n S<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which is isomorphic to the dihedral group. Here, we extend the above findings to circular codes over a finite alphabet of even cardinality $$|\\Sigma |=2n$$<\/jats:tex-math>\n \n |<\/mml:mo>\n \u03a3<\/mml:mi>\n |<\/mml:mo>\n =<\/mml:mo>\n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$n \\in {\\mathbb {N}}$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We describe the corresponding group $$L_n$$<\/jats:tex-math>\n \n L<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using matrices and we investigate what classes of circular codes are split into equally sized equivalence classes under the natural equivalence relation induced by $$L_n$$<\/jats:tex-math>\n \n L<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Surprisingly, this is not always the case. All results and constructions are illustrated by examples.<\/jats:p>","DOI":"10.1007\/s12064-020-00337-z","type":"journal-article","created":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T11:04:30Z","timestamp":1612177470000},"page":"107-121","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Equivalence classes of circular codes induced by permutation groups"],"prefix":"10.1007","volume":"140","author":[{"given":"Fariba","family":"Fayazi","sequence":"first","affiliation":[]},{"given":"Elena","family":"Fimmel","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1689-3611","authenticated-orcid":false,"given":"Lutz","family":"Str\u00fcngmann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,2,1]]},"reference":[{"key":"337_CR1","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1006\/jtbi.1996.0142","volume":"182","author":"DG Arqu\u00e8s","year":"1996","unstructured":"Arqu\u00e8s DG, Michel CJ (1996) A complementary circular code in the protein coding genes. J Theor Biol 182:45\u201358","journal-title":"J Theor Biol"},{"key":"337_CR2","doi-asserted-by":"crossref","unstructured":"Baji\u0107 D, Stojanovi\u0107 J (2004) Distributed sequences and search process, In: 2004 IEEE International Conference on Communications IEEE 1, 2004, pp 514-518","DOI":"10.1109\/ICC.2004.1312542"},{"key":"337_CR3","first-page":"239","volume":"1","author":"AH Ball","year":"1976","unstructured":"Ball AH, Cummings LJ (1976) Extremal digraphs and comma-free codes. Ars Combinat 1:239\u2013251","journal-title":"Ars Combinat"},{"key":"337_CR4","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1017\/S0004972700025065","volume":"14","author":"AH Ball","year":"1976","unstructured":"Ball AH, Cummings LJ (1976) The comma-free codes with words of length two. Bull Austral Math Soc 14:249\u2013258","journal-title":"Bull Austral Math Soc"},{"key":"337_CR5","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/s00236-013-0176-4","volume":"50","author":"S Bilotta","year":"2013","unstructured":"Bilotta S, Grazzini E, Pergola E, Pinzani R (2013) Avoiding cross-bifix-free binary words. Acta Inform 50:157\u2013173","journal-title":"Acta Inform"},{"key":"337_CR6","doi-asserted-by":"publisher","first-page":"4058","DOI":"10.1109\/TIT.2012.2189479","volume":"58","author":"S Bilotta","year":"2012","unstructured":"Bilotta S, Pergola E, Pinzani R (2012) A new approach to cross-bifix-free sets. IEEE Trans Inform Theory 58:4058\u20134063","journal-title":"IEEE Trans Inform Theory"},{"key":"337_CR7","doi-asserted-by":"publisher","first-page":"4890","DOI":"10.1109\/TIT.2015.2456634","volume":"61","author":"SR Blackburn","year":"2015","unstructured":"Blackburn SR (2015) Non-overlapping codes. IEEE Trans Inf Theory 61:4890\u20134894","journal-title":"IEEE Trans Inf Theory"},{"key":"337_CR8","doi-asserted-by":"publisher","first-page":"4668","DOI":"10.1109\/TIT.2013.2252952","volume":"59","author":"YM Chee","year":"2013","unstructured":"Chee YM, Kiah HM, Purkayastha P, Wang C (2013) Cross-bifix-free codes within a constant factor of optimality. IEEE Trans Inform Theory 59:4668\u20134674","journal-title":"IEEE Trans Inform Theory"},{"issue":"5","key":"337_CR9","doi-asserted-by":"publisher","first-page":"416","DOI":"10.1073\/pnas.43.5.416","volume":"43","author":"FHC Crick","year":"1957","unstructured":"Crick FHC, Griffith JS, Orgel LE (1957) Codes without commas. Proc Natl Acad Sci USA 43(5):416\u2013421","journal-title":"Proc Natl Acad Sci USA"},{"key":"337_CR10","unstructured":"Cummings LJ (1976) Comma-free codes and incidence algebras, In: Combinatorial mathematics IV, Lecture Notes in Math. 560, Springer, Berlin, 1976, pp 1\u20136"},{"key":"337_CR11","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1007\/s00239-019-09925-0","volume":"88","author":"J Demongeot","year":"2020","unstructured":"Demongeot J, Seligmann H (2020) Pentamers with Non-redundant Frames: Bias for Natural Circular Code Codons. J Mol Evol 88:194\u2013201. https:\/\/doi.org\/10.1007\/s00239-019-09925-0","journal-title":"J Mol Evol"},{"key":"337_CR12","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/j.gene.2019.03.069","volume":"705","author":"J Demongeot","year":"2019","unstructured":"Demongeot J, Seligmann H (2019) Spontaneous evolution of circular codes in theoretical minimal RNA rings. Gene 705:95\u2013102. https:\/\/doi.org\/10.1016\/j.gene.2019.03.069","journal-title":"Gene"},{"key":"337_CR13","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1109\/TIT.1965.1053766","volume":"11","author":"W Eastman","year":"1965","unstructured":"Eastman W (1965) On the construction of comma-free codes. IEEE Trans Inf Theory 11:263\u2013267","journal-title":"IEEE Trans Inf Theory"},{"key":"337_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/s11538-020-00724-z","author":"E Fimmel","year":"2020","unstructured":"Fimmel E, Starman M, Str\u00fcngmann L (2020) Tessera Codes in the Evolution of the Genetic Code. Math Biol Bull Circular. https:\/\/doi.org\/10.1007\/s11538-020-00724-z","journal-title":"Math Biol Bull Circular"},{"issue":"1","key":"337_CR15","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s12064-018-0259-4","volume":"137","author":"E Fimmel","year":"2018","unstructured":"Fimmel E, Michel CJ, Starman M, Str\u00fcngmann L (2018) Self-complementary circular codes in coding theory. Theory Biosci 137(1):51\u201365. https:\/\/doi.org\/10.1007\/s12064-018-0259-4","journal-title":"Theory Biosci"},{"key":"337_CR16","unstructured":"Fimmel E, Michel CJ, Pirot F, Sereni J, Str\u00fcngmann L (2019) Comma-free codes over finite alphabets, 2019, submitted"},{"issue":"8","key":"337_CR17","doi-asserted-by":"publisher","first-page":"1796","DOI":"10.1007\/s11538-017-0307-0","volume":"79","author":"E Fimmel","year":"2017","unstructured":"Fimmel E, Michel CJ, Str\u00fcngmann L (2017) Strong Comma-Free Codes in Genetic Information. Bull Math Biol 79(8):1796\u20131819. https:\/\/doi.org\/10.1007\/s11538-017-0307-0","journal-title":"Bull Math Biol"},{"key":"337_CR18","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1016\/j.mbs.2017.10.001","volume":"294","author":"E Fimmel","year":"2017","unstructured":"Fimmel E, Michel CJ, Str\u00fcngmann L (2017) Diletter circular codes over finite alphabets. Math Biosci. 294:120\u2013129","journal-title":"Math Biosci."},{"key":"337_CR19","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2015.0058","author":"E Fimmel","year":"2016","unstructured":"Fimmel E, Michel CJ, Str\u00fcngmann L (2016) n-Nucleotide circular codes in graph theory. Phil Trans R Soc A. https:\/\/doi.org\/10.1098\/rsta.2015.0058","journal-title":"Phil Trans R Soc A"},{"key":"337_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/s00285-014-0806-7","author":"E Fimmel","year":"2014","unstructured":"Fimmel E, Gonzalez DL, Giannerini S, Str\u00fcngmann L (2014) Circular codes, symmetries and transformations. J Math Biol. https:\/\/doi.org\/10.1007\/s00285-014-0806-7","journal-title":"J Math Biol"},{"key":"337_CR21","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1016\/j.biosystems.2017.09.007","volume":"164","author":"E Fimmel","year":"2018","unstructured":"Fimmel E, Str\u00fcngmann L (2018) Mathematical Fundamentals for the noise immunity of the genetic code. BioSystems 164:186\u2013198","journal-title":"BioSystems"},{"key":"337_CR22","doi-asserted-by":"publisher","first-page":"202","DOI":"10.4153\/CJM-1958-023-9","volume":"10","author":"SW Golomb","year":"1958","unstructured":"Golomb SW, Gordon B, Welch LR (1958a) Comma-free codes. Can J Math 10:202\u2013209","journal-title":"Can J Math"},{"key":"337_CR23","first-page":"1","volume":"23","author":"SW Golomb","year":"1958","unstructured":"Golomb SW, Delbruck M, Welch LR (1958b) Construction and properties of comma-free codes. Biologiske Meddelelser, Kongelige Danske Videnskabernes Selskab 23:1\u201334","journal-title":"Biologiske Meddelelser, Kongelige Danske Videnskabernes Selskab"},{"key":"337_CR24","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1137\/0135034","volume":"35","author":"LJ Guibas","year":"1978","unstructured":"Guibas LJ, Odlyzko AM (1978) Maximal prefix-synchronized codes. SIAM J Appl Math 35:401\u2013418","journal-title":"SIAM J Appl Math"},{"key":"337_CR25","volume-title":"Hall: The theory of groups","author":"M Hall Jr","year":"1970","unstructured":"Hall M Jr (1970) Hall: The theory of groups. The Macmillan Company, New York"},{"issue":"1","key":"337_CR26","doi-asserted-by":"publisher","first-page":"8","DOI":"10.1511\/1998.17.3338","volume":"86","author":"B Hayes","year":"1998","unstructured":"Hayes B (1998) The invention of the genetic code. Am Sci 86(1):8\u201314","journal-title":"Am Sci"},{"key":"337_CR27","unstructured":"Keller V (2014) Fehlerkorrigierende genetische Codes, Bachelor-thesis, University of Applied Sciences Mannheim"},{"key":"337_CR28","unstructured":"Lemegne L (2015) Trinucleotide maximal circular self-complementary genetic codes and their symmetries, Bachelor-thesis, University of Applied Sciences Mannheim"},{"key":"337_CR29","first-page":"125","volume":"12","author":"VI Leven\u0161te\u012dn","year":"1964","unstructured":"Leven\u0161te\u012dn VI (1964) Decoding automata which are invariant with respect to the initial state, in Russian. Problemy Kibernet 12:125\u2013136","journal-title":"Problemy Kibernet"},{"issue":"4","key":"337_CR30","first-page":"88","volume":"6","author":"VN Leven\u0161te\u012dn","year":"1970","unstructured":"Leven\u0161te\u012dn VN (1970) The maximal number of words in codes without overlap. Problemy Pereda\u010di Informacii 6(4):88\u201390 in Russian","journal-title":"Problemy Pereda\u010di Informacii"},{"key":"337_CR31","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1002\/jcd.10071","volume":"12","author":"VI Levenshtein","year":"2004","unstructured":"Levenshtein VI (2004) Combinatorial problems motivated by comma-free codes. J Combin Des 12:184\u2013196","journal-title":"J Combin Des"},{"key":"337_CR32","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1038\/nature13314","volume":"509","author":"D Malyshev","year":"2014","unstructured":"Malyshev D, Dhami K, Lavergne T et al (2014) A semi-synthetic organism with an expanded genetic alphabet. Nature 509:385\u2013388. https:\/\/doi.org\/10.1038\/nature13314","journal-title":"Nature"},{"key":"337_CR33","doi-asserted-by":"publisher","first-page":"989","DOI":"10.1016\/j.camwa.2006.12.091","volume":"55","author":"CJ Michel","year":"2008","unstructured":"Michel CJ, Pirillo G, Pirillo MA (2008) Varieties of comma free codes. Comp Math Appl 55:989\u2013996","journal-title":"Comp Math Appl"},{"key":"337_CR34","unstructured":"Michel CJ, Pirillo G (2011) Strong trinucleotide circular codes. Int J Combin 2011, Article ID 659567, pp 1-14"},{"key":"337_CR35","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/j.biosystems.2014.02.002","volume":"118","author":"CJ Michel","year":"2014","unstructured":"Michel CJ, Seligmann H (2014) Bijective transformation circular codes and nucleotide exchanging RNA transcription. Biosystems 118:39\u201350","journal-title":"Biosystems"},{"key":"337_CR36","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/j.jtbi.2014.03.029","volume":"355","author":"CJ Michel","year":"2014","unstructured":"Michel CJ (2014) A genetic scale of reading frame coding. J Theor Biol 355:83\u201394","journal-title":"J Theor Biol"},{"key":"337_CR37","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1016\/j.jtbi.2015.04.009","volume":"380","author":"CJ Michel","year":"2015","unstructured":"Michel CJ (2015) The maximal $$C^3$$-self-complementary trinucleotide circular code $$X$$in genes of bacteria, eukaryotes, plasmids and viruses. J Theor Biol 380:156\u2013177","journal-title":"J Theor Biol"},{"key":"337_CR38","unstructured":"Michel CJ, Pirillo G (2013) Dinucleotide circular codes, ISRN Biomathematics. Article ID 538631:1\u20138"},{"key":"337_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.biosystems.2020.104201","volume":"197","author":"CJ Michel","year":"2020","unstructured":"Michel CJ (2020) The maximality of circular codes in genes statistically verified. Biosystems 197:1\u20137","journal-title":"Biosystems"},{"issue":"2","key":"337_CR40","first-page":"1","volume":"7, 20","author":"CJ Michel","year":"2017","unstructured":"Michel CJ (2017) The maximal $$C^3$$ self-complementary trinucleotide circular code $$X$$ in genes of bacteria, archaea, eukaryotes, plasmids and viruses. Life 7, 20(2):1\u201316","journal-title":"Life"},{"key":"337_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4176-8","volume-title":"An introduction to the theory of groups","author":"JJ Rotman","year":"1995","unstructured":"Rotman JJ (1995) An introduction to the theory of groups. Springer, Berlin"},{"key":"337_CR42","doi-asserted-by":"publisher","first-page":"300","DOI":"10.1109\/TIT.1969.1054292","volume":"15","author":"R Scholtz","year":"1969","unstructured":"Scholtz R (1969) Maximal and variable word-length comma-free codes. IEEE Trans Inf Theory 15:300\u2013306","journal-title":"IEEE Trans Inf Theory"},{"key":"337_CR43","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2016.04.007","author":"H Seligman","year":"2016","unstructured":"Seligman H (2016) Swinger RNA self-hybridization and mitochondrial non-canonical swinger transcription, transcription systematically exchanging nucleotides. J Theor Biol. https:\/\/doi.org\/10.1016\/j.jtbi.2016.04.007","journal-title":"J Theor Biol"},{"key":"337_CR44","doi-asserted-by":"publisher","first-page":"513","DOI":"10.4153\/CJM-1987-023-7","volume":"39","author":"B Tang","year":"1987","unstructured":"Tang B, Golomb SW, Graham RL (1987) A new result on comma-free codes of even word-length. Canad J Math 39:513\u2013526","journal-title":"Canad J Math"}],"container-title":["Theory in Biosciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12064-020-00337-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s12064-020-00337-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s12064-020-00337-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,2,21]],"date-time":"2021-02-21T06:06:59Z","timestamp":1613887619000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s12064-020-00337-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["337"],"URL":"https:\/\/doi.org\/10.1007\/s12064-020-00337-z","relation":{},"ISSN":["1431-7613","1611-7530"],"issn-type":[{"type":"print","value":"1431-7613"},{"type":"electronic","value":"1611-7530"}],"subject":[],"published":{"date-parts":[[2021,2]]},"assertion":[{"value":"22 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 December 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 February 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}