Self-complementary C3-codes and bijective transformations

Lutz Strüngmann and Elena Fimmel

Institute for Applied Mathematics
University of Applied Sciences, Mannheim, Germany

Circular codes gained considerable attention as a weaker form of comma-free codes which were suggested by Crick as a solution for the so-called frame-shift problem: a sequence of codons can be translated only in a right frame into the right amino acids. The discovering of an universal code across species suggested many theoretical and experimental questions. However, there is a key aspect that relates circular codes to symmetries and transformations that remains to a large extent unexplored. In this talk we aim at addressing the issue by studying the symmetries and transformations that connect different circular codes. The main result is that the class of 216 C3 maximal self-complementary codes can be partitioned into 27 equivalence classes defined by a particular set of transformations. We show that such transformations can be put in a group theoretic framework with an intuitive geometric interpretation. More general mathematical results about symmetry transformations which are valid for any kind of circular codes are also presented. Furthermore, we use the classification in equivalence classes for studying the codon usage of circular codes in different organisms. The coverage of circular codes inside equivalence classes allows to characterize the subset of codes inside the set of 216 that can be obtained from the nucleotide occurrence in real sequences. Our results pave the way to the study of the biological consequences of the mathematical structure behind circular codes and contribute to shed light on the evolutionary steps that led to the observed symmetries of present codes.


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