Mathematical fundamentals for the noise immunity of the genetic code

The biologists recognised this problem immediately after the detection of the non-overlapping structure of the genetic code, i.e., coding sequences are to be read in a unique way determined by their reading frame. But how does the reading head of the ribosome recognises an error in the grouping of codons, caused by e.g. insertion or deletion of a base, that can be fatal during the translation process and may result in nonfunctional proteins? In this chapter we will discuss possible solutions to the frameshift problem with a focus on the theory of so-called circular codes that were discovered in large gene populations of prokaryotes and eukaryotes in the early 90s. Circular codes allow to detect a frameshift of one or two positions and recently a beautiful theory of such codes has been developed using statistics, group theory and graph theory.