DNA is a code, but it's not just a code. It is also a really long molecule that has to bend and fold up in certain ways.
There are error detecting codes, in a way. Protein is encoded by 3 base codes, and if you insert or delete bases not in a multiple of 3 it will be misaligned, then eventually likely encode a stop code and cause the bad protein to be truncated and likely removed via nonsense mediated decay.
