The standard genetic code (SGC) describes how 64 trinucleotides (codons) encode 20 amino acids and the stop translation signal. Biochemical and statistical studies have shown that the standard genetic code is optimized to reduce the impact of errors caused by incorporation of wrong amino acids during translation. This is achieved by mapping codons that differ by only one nucleotide to the same amino acid or one with similar biochemical properties, so that if misincorporation occurs, the structure and function of the translated protein remain relatively unaltered. Some previous studies have extended the analysis of SGC optimality to the effect of frameshift errors on the conservation of amino acids. Here, we compare the optimality of the SGC with a set of circular codes, and in particular the $X$ circular code identified in genes, on the basis of various biochemical properties over all possible frameshift errors. We show that the $X$ circular code is more optimized to minimize the impact of frameshift errors than the SGC for the chosen amino acid properties. Furthermore, in the context of a problem that has been unresolved since 1996, we also demonstrate that the $X$ circular code has a frameshift optimality in its combinatorial class of 216 maximal self-complementary $C^3$ circular codes. To our knowledge, this is the first demonstration of the role of the $X$ circular code in mitigation of translation errors. These results lead us to discuss the potential role of the $X$ circular code in the evolution of the standard genetic code.

标准遗传密码（SGC）描述了64个三核苷酸（密码子）如何编码20个氨基酸和终止翻译信号。生化和统计研究表明，对标准遗传密码进行了优化，以减少因翻译过程中掺入错误氨基酸而导致的错误影响。这是通过将仅相差一个核苷酸的密码子映射到相同氨基酸或具有相似生化特性的密码子来实现的，因此，如果发生错误掺入，翻译蛋白的结构和功能仍保持相对不变。先前的一些研究已将SGC最优性分析扩展到移码错误对氨基酸保守性的影响。在这里，我们将SGC的最优性与一组循环代码进行比较，尤其是$X$在所有可能的移码错误的基础上，基于各种生化特性，在基因中识别出循环密码。我们表明$X$对于所选氨基酸特性，与SGC相比，循环码更优化以最大程度地减少移码错误的影响。此外，在自1996年以来未解决的问题的背景下，我们还证明了$X$ 循环码在组合类别中具有216个最大自补码的移码最优性 $C^3$循环编码。据我们所知，这是第一个证明$X$减少翻译错误的循环代码。这些结果使我们讨论了潜在的作用$X$ 循环编码中的标准遗传密码的演变。