Self-complementary circular codes are involved in pairing genetic processes. A maximal \(C^3\) self-complementary circular code X of trinucleotides was identified in genes of bacteria, archaea, eukaryotes, plasmids and viruses (Michel in Life 7(20):1–16 2017, J Theor Biol 380:156–177, 2015; Arquès and Michel in J Theor Biol 182:45–58 1996). In this paper, self-complementary circular codes are investigated using the graph theory approach recently formulated in Fimmel et al. (Philos Trans R Soc A 374:20150058, 2016). A directed graph \(\mathcal {G}(X)\) associated with any code X mirrors the properties of the code. In the present paper, we demonstrate a necessary condition for the self-complementarity of an arbitrary code X in terms of the graph theory. The same condition has been proven to be sufficient for codes which are circular and of large size \(\mid X \mid \ge 18\) trinucleotides, in particular for maximal circular codes (\(\mid X \mid = 20\) trinucleotides). For codes of small-size \(\mid X \mid \le 16\) trinucleotides, some very rare counterexamples have been constructed. Furthermore, the length and the structure of the longest paths in the graphs associated with the self-complementary circular codes are investigated. It has been proven that the longest paths in such graphs determine the reading frame for the self-complementary circular codes. By applying this result, the reading frame in any arbitrary sequence of trinucleotides is retrieved after at most 15 nucleotides, i.e., 5 consecutive trinucleotides, from the circular code X identified in genes. Thus, an X motif of a length of at least 15 nucleotides in an arbitrary sequence of trinucleotides (not necessarily all of them belonging to X) uniquely defines the reading (correct) frame, an important criterion for analyzing the X motifs in genes in the future.

中文翻译：

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编码理论中的自互补循环码。

配对遗传过程涉及自我互补的循环编码。在细菌，古细菌，真核生物，质粒和病毒的基因中鉴定出三核苷酸的最大\（C ^ 3 \）自互补循环码X（Michel in Life 7（20）：1-16-1 2017，J Theor Biol 380： 156–177，2015；Arquès和Michel在J Theor Biol 182：45–58 1996中）。在本文中，使用最近在Fimmel等人中提出的图论方法研究了自补圆码。（Philos Trans R Soc A 374：20150058，2016）。与任何代码X关联的有向图\（\数学{G}（X）\）反映了代码的属性。在本文中，我们证明了任意代码自我互补的必要条件X在图论方面。事实证明，相同条件对于圆形和大\（\ mid X \ mid \ ge 18 \）三核苷酸的环形码就足够了，特别是对于最大循环码（\（\ mid X \ mid = 20 \）三核苷酸）。对于小型\（\ mid X \ mid \ le 16 \）代码三核苷酸，已经构建了一些非常罕见的反例。此外，研究了与自补循环码相关的图中最长路径的长度和结构。业已证明，此类图中最长的路径决定了自互补循环码的阅读框架。通过应用该结果，从基因中鉴定的环状密码X至多15个核苷酸，即5个连续的三核苷酸之后，检索任意三核苷酸序列的阅读框。因此，任意三核苷酸序列中长度至少为15个核苷酸的X基序（不一定全部属于X）唯一地定义了读取（正确）框架，这是将来分析基因X基序的重要标准。