In this chapter a new mathematical theory of the genetic code is presented, based on a particular kind of number representation: a non-power binary representation. A mathematical model is constructed that allows the description of many known properties of the genetic code, such as the degeneracy distribution and the specific codon-amino acid assignation, and also of some new properties such as, for example, palindromic symmetry (a degeneracy preserving transformation), which is shown to be the highest level of a series of hierarchical symmetries. The role of chemical dichotomy classes, which varies between purine–pyrimidine, amino–keto, and strong–weak following the position of the bases in the codon frame, is shown. A new characterization of codons, obtained through the parity of the corresponding binary strings in the mathematical model, together with the associated symbolic structure acting on the …