A classification of 20-trinucleotide circular codes

Trinucleotide comma-free codes and trinucleotide circular codes are two important classes of codes in code theory and theoretical biology. A trinucleotide circular code containing exactly 20 elements is called here a 20-trinucleotide circular code. In this paper, solving a combinatorial problem of hard computational complexity, we extend and improve our results of C.J. Michel, G. Pirillo, and M.A. Pirillo (2008) [14] concerning the small class of 528 self-complementary 20-trinucleotide circular codes, to the complete class of the 20-trinucleotide circular codes which contains 12,964,440 elements. A surprising relation with the symmetric group Σ4 appears but it remains unexplained so far.