It has been argued that the sum of usage probabilities for codons, belonging to quartets, that have as third nucleotide C or A, is independent of the biological species for vertebrates. The comparison between the theoretical correlation matrix derived from these sum rules and the experimentally computed matrix for 26 species shows a satisfactory agreement. The Shannon entropy, weakly depending on the biological species, gives further support. Suppression of codons containing the dinucleotides CG or AU is put in evidence.