Reminds me of that Star Trek TNG episode where they had to do a DNA scavenger hunt in order to locate an actual message (notably, not written in DNA) from the so-called "founders" of life in the universe.
I'll certainly concede that it would be possible to store messages in genetic code - since DNA can last for a million years or more in the right conditions. However, I do not think DNA used to transmit a message would actually be able to support life, for the simple reason that too much of it would be required to support an actual life-form. When you add the fact that DNA inside a living organism tends to change over time, the idea of passing messages down in the DNA of living organisms is simply too farfetched to take seriously. Such messages would either be very short-term (such as RNA codons used to make amino acids) or very long-term (DNA stored in places where it can't be changed).
Simply finding patterns in the genetic code doesn't indicate an intelligent designer any more than patterns in snowflakes or waves or ripples do. Remember the business with amazingly regular radio signals (with periods of seconds), that were hailed as being artificial that ended up being the natural emissions of pulsars? That's cause scientists didn't fasten on "they must be artificial" and refuse to consider any other explanation.
There's nothing wrong with proposing a non-natural explanation for something, as this paper does. However, it needs to be supported by evidence. And by that I don't mean simply claiming that DNA coding contains hallmarks of artificiality, which may not actually be true. Science requires evidence to support stuff like this, not merely making claims. It's good that they presented it in a peer-reviewed journal, but the next step is always to see if other people come to the same conclusions using the same data. As we saw with the people who claimed that their data showed that neutrinos could move faster than light, and then later had to retract this claim because of errors, a single paper is not, in and of itself, convincing.