A statistical analysis with 12 288 autocorrelation functions applied in protein (coding) genes of prokaryotes and eukaryotes identifies three subsets of trinucleotides in their three frames: T₀=X₀{AAA, TTT} with X₀={AAC, AAT, ACC, ATC, ATT, CAG, CTC, CTG, GAA, GAC, GAG, GAT, GCC, GGC, GGT, GTA, GTC, GTT, TAC, TTC} in frame 0 (the reading frame established by the ATG start trinucleotide), T₁=X₁{CCC} in frame 1 and T₂=X₂{GGG} in frame 2 (the frames 1 and 2 being the frame 0 shifted by one and two nucleotides, respectively, to the right). These three subsets are identical in these two gene populations and have five important properties: (i) the property of maximal (20 trinucleotides) circular code for X₀ (resp. X₁, X₂) allowing to retrieve automatically the frame 0 (resp. 1, 2) in any region of the gene without start codon; (ii) the DNA complementarity property (e.g. (AAC)=GTT): (T₀)=T₀, (T₁)=T₂ and (T₂)=T₁ allowing the two paired reading frames of a DNA double helix simultaneously to code for amino acids; (iii) the circular permutation property (e.g. (AAC)=ACA): (X₀)=X₁ and (X₁)=X₂ implying that the two subsets X₁ and X₂ can be deduced from X₀; (iv) the rarity property with an occurrence probability of X₀=6×10⁻⁸; and (v) the concatenation properties in favour of an evolutionary code: a high frequency (27.5%) of misplaced trinucleotides in the shifted frames, a maximum (13 nucleotides) length of the minimal window to retrieve automatically the frame and an occurrence of the four types of nucleotides in the three trinucleotide sites. In Discussion, a simulation based on an independent mixing of the trinucleotides of T₀ allows to retrieve the two subsets T₁ and T₂. Then, the identified subsets T₀, T₁ and T₂ replaced in the 2-letter genetic alphabet {R, Y} (R=purine=A or G, Y=pyrimidine=C or T) allow to retrieve the RNY model (N=R or Y) and to explain previous works in the alphabet {R, Y}. Then, these three subsets are related to the genetic code. The trinucleotides of T₀ code for 13 amino acids: Ala, Asn, Asp, Gln, Glu, Gly, Ile, Leu, Lys, Phe, Thr, Tyr and Val. Finally, a strong correlation between the usage of the trinucleotides of T₀ in protein genes and the amino acid frequencies in proteins is observed as six among seven amino acids not coded by T₀, have as expected the lowest frequencies in proteins of both prokaryotes and eukaryotes.