The minimum legth of a codon could be two. If it was only one of the 6 nitrogeneous bases in a codon, we'd only have 6 possible amino acids. If we have, though, a combination of two amino acids out of the 6 nitrogeneous bases, we would have then 36 possible combinations (6 possibilities for the first position × 6 possibilities for the second position) that would allow the existence of the 20 different amino acids.
In the human case, for example, with only 4 nitrogeneous bases, a combination of two amino acids would be insufficient (4×4=16) for the 20 amino acids.
