1. DNA can be treated like a code for many purposes and meets some definitions of a code.
2. Most codes humans interact with are human-generated. (Although you can get a one-time pad from any long random sequence, and my computer generated my private key for me, and even if computers could have minds, mine definitely doesn't.)
3. You can't get more data out of a syllogism than you put in. If you say one of the characteristics of a code is having come from a mind, then when you say that DNA is a code, you are already saying that it has come from a mind.