Equivalence classes of circular codes induced by permutation groups

Abstract

AbstractIn 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 182:45–58, 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 ($$C^3$$ C 3 -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 182:45–58. Using an algebraic approach, it was shown in J Math Biol, 2004 that the class of 216 maximal self-complementary $$C^3$$ C 3 -codes can be partitioned into 27 equally sized equivalence classes by the action of a transformation group $$L \subseteq S_4$$ L ⊆ S 4 which is isomorphic to the dihedral group. Here, we extend the above findings to circular codes over a finite alphabet of even cardinality $$|\Sigma |=2n$$ | Σ | = 2 n for $$n \in {\mathbb {N}}$$ n ∈ N . We describe the corresponding group $$L_n$$ L n using matrices and we investigate what classes of circular codes are split into equally sized equivalence classes under the natural equivalence relation induced by $$L_n$$ L n . Surprisingly, this is not always the case. All results and constructions are illustrated by examples.