Based on previous considerations published in J. theor. Biol., new analyses of the organization of the genetic system are reported in this paper. We show that theoretical considerations about the order observed in the genetic code table support the idea of a primitive self-aminoacylation process achieved by primordial tRNAs. The physico-chemical constraints connected with this process may explain why a primitive genetic system predominantly uses sequences with the codonic pattern RNN (R=purine; Y=pyrimidine; N=any of the four bases) to polymerize the amino acids into peptides through translation. These considerations lead us to propose the Translation --> Translation/Replication hypothesis, which may explain why only RNA sequences with the pattern RNY, instead of less restrictive RNN, are susceptible to amplification. Using these ideas, supported by properties of symmetry, features of the genetic code may be connected with the replication of specific RNA sequences in the RNA world.