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 182:45–58, 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 (\(C^3\)-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 \(C^3\)-codes can be partitioned into 27 equally sized equivalence classes by the action of a transformation group \(L \subseteq S_4\) which is isomorphic to the dihedral group. Here, we extend the above findings to circular codes over a finite alphabet of even cardinality \(|\Sigma |=2n\) for \(n \in {\mathbb {N}}\). We describe the corresponding group \(L_n\) using matrices and we investigate what classes of circular codes are split into equally sized equivalence classes under the natural equivalence relation induced by \(L_n\). Surprisingly, this is not always the case. All results and constructions are illustrated by examples.

在 1950 年代，克里克提出了所谓的无逗号密码的概念，以解决生物学家在研究将核苷酸碱基序列翻译成蛋白质的过程时遇到的移码问题。过了一会儿，事实证明，不幸的是，这个提议不符合生物学现实。然而，在 90 年代中期，在 J Theor Biol 182:45-58, 1996 中在自然界中发现了一种较弱的无逗号代码版本，即所谓的循环代码。循环代码允许在翻译过程中检索阅读框核糖体中的过程，令人惊讶的是，自然界中发现的循环密码在所有三种可能的阅读框中都是循环的（\（C^3\）-财产）。此外，它是最大的，因为它包含 20 个密码子并且是自我互补的，这意味着它由成对的密码子和相应的反密码子组成。在进一步的调查中，发现恰好有 216 个代码具有与 J Theor Biol 182:45-58 中最初发现的代码相同的强属性。使用代数方法，在 J Math Biol, 2004 中表明，216 个最大自互补\(C^3\)代码的类可以通过变换组的作用划分为 27 个大小相等的等价类\( L \subseteq S_4\)同构于二面体群。在这里，我们将上述发现扩展到偶数基数\(|\Sigma |=2n\)的有限字母表上的循环码：\(n \in {\mathbb {N}}\)。我们使用矩阵描述相应的组\(L_n\) ，并研究在\(L_n\)引起的自然等价关系下，哪些类循环码被分成大小相等的等价类。令人惊讶的是，情况并非总是如此。所有的结果和结构都通过例子来说明。