RNA triplets have their corresponding tRNAs and hence amino acids, this has been discovered in the past. And I know that the first coden be deciphered is UUU, which is known to be corresponding to phenylalanine, this was done by synthesizing a poly-U chain from uracil monomers by special enzyme (I would also like to know what this enzyme is...). However, when came to cases like AGC or AUG or any other coden which is not composed of single kind of nucleotide, how could the polymer chain be synthesized? it should be related to enzymes, but I would like to know the procedures to achieve this aim.

I think what Nirenberg and others did is that they synthesised poly chains like UGUGUGUG..... and they obtained the polypeptide chains with alternating amino acids (i.e. UGU- codes for cysteine and GUG- codes for Valine) and simillarly CACACACA..... and they obtained the codes for amino acids histidine (CAC) and threonine (ACA).

well, then how could one synthesize polymer like poly-(UG)? and what about coden of which is composed of three different bases?

psst... it's 'codon' not 'coden.'

Some of the later experiments involved the use of synthetic trinucleotides, radioactive aminoacyl-tRNAs and incubate them with ribosomes. After filtering which will only retain ribosomes and everything bound to it, the radioactivity was screened.