# Forbidden codon combinations in error-detecting circular codes

## TL;DR

This paper introduces a new software tool to construct special genetic codes called circular codes, using mathematical theorems to exclude certain codon combinations.

## Contribution

A novel software tool for constructing self-complementary circular codes using mathematical theorems to exclude forbidden codon combinations.

## Key findings

- A systematic approach using two mathematical theorems to exclude codons in constructing circular codes.
- The software provides an efficient method for building self-complementary C³-codes of maximum size.
- The approach sheds light on the mathematical foundations of these codes.

## Abstract

Circular codes, which are considered as putative remnants of primaeval comma-free codes, have recently become a focal point of research. These codes constitute a secondary type of genetic code, primarily tasked with detecting and preserving the normal reading frame within protein-coding sequences. The identification of a universal code present across various species has sparked numerous theoretical and experimental inquiries. Among these, the exploration of the class of 216 self-complementary \documentclass[12pt]{minimal}
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\begin{document}$$C^3$$\end{document}C3-codes of maximum size 20 has garnered significant attention. However, the origin of the number 216 lacks a satisfactory explanation, and the mathematical construction of these codes remains elusive. This paper introduces a new software designed to facilitate the construction of self-complementary \documentclass[12pt]{minimal}
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\begin{document}$$C^3$$\end{document}C3-codes (of maximum size). The approach involves a systematic exclusion of codons, guided by two fundamental mathematical theorems. These theorems demonstrate how codons can be automatically excluded from consideration when imposing requirements such as self-complementarity, circularity or maximality. By leveraging these theorems, our software provides a novel and efficient means to construct these intriguing circular codes, shedding light on their mathematical foundations and contributing to a deeper understanding of their biological significance.