Just because they do it in fiction doesn't make it possible in and of itself. If it is contrary to natural law, finding water that runs uphill should pull your attention in fiction, and if it's not explained in the story, it seems stupid, like maybe even the author wasn't being clever so much as he's apparently ignorant of natural law. I'm not saying you can't have or invent a reason - I'm saying if it's contrary to known science, you should explain it, or try, and by acknowledging it that way, you at least show you are not ignorant of the facts or just unintentionally did something stupid and for no reason.[\QUOTE]
Didn't mean to imply actually possible, just possible in-world. The teleporter, replicator, holodeck, time travel etc aren't actually explained, and I'm pretty sure if we used our science to try and explain it they'd fall apart (and often do). The puzzle was created by being well beyond the SF tech... I say he didn't explain it because he couldn't, not because there wasn't an explanation.
Geordie was looking at the code, IIRC, and say a pattern in there (the puzzle, I guess) that he recognized was so astronomically improbable that it obviously wasn't natural. This we could see and should be able to see. Another has proposed only the puzzle is on that level, and everything else this code does is on a much smaller level below the genetic code - like writing on electrons might be, and making those self replicating. I don't even begin to understand how that might be done, but that alone doesn't make it impossible, I guess. And not being a geneticist, I can't say for sure, but too much seems wrong, and some wild speculation abounds as to how that might work to drive (well, not evolution if its controlled) life toward humanoid form, as if evolution had an end product in mind, wasn't natural, and could do other extraordinary things, like reconfigure the hardware of a tricorder or other, well, I'm gonna call it nonsense. I didn't like that aspect of the story, or feel it was necessary to make the rest of the story work, so I gave it a lower grade. My POV. YMMV.
Ah, right... so this actually is possible (writing messages in genes). They're not on the genes like "on paper," they use gene sequences as a code that can be translated into a readable language (though you could do it the other way I guess) or by creating "molecular shapes/markers." People sign GMO genes this way.
The fact that no one has discovered this (in-world) before now is fairly easily explained too, given they've never defined how far along in genetics people are. There's tons of human gene code we have no idea about right now. Obviously they know enough to make genetically modified people... but then you only have to know stuff about what you're targeting, not what everything in the genome does. It's also made clear that most of the puzzle is from non-Federation worlds... if they didn't have enough samples that contained it, they would've never seen a pattern before now.
The guided evolution via genetic manipulation in the way they portray (with a specific end goal form in mind, and without further manipulation along the process, which we don't know about) does seem a bit far fetched... maybe. The natural process of evolution would have a deep impact on whatever genetics they put in there... but then, they were kind of humanoid blanks: their appearance seemed to be that of all their humanoid offspring minus distinguishing characteristics: bipedal locomotion, symmetric features consisting of 2 eyes, 2 ears, 2 nostrils, mouth, (I assume ten fingers and toes and sexual organs that are compatible (we can assume this between Klingons and Humans, Humans and Vulcans, and by extension Humans and Romulans)), same general proportions... so the real problem with it would be, "if you put this code into the earliest (or even just early) forms of life, why have other things evolved (on the same planet, in the same span) from that same life not developed to the same design and level?" There's very little evidence to support that (a few examples of "creatures that haven't evolved since the X period")... and would probably need the response of selectively breeding, picking and choosing which species to let evolve, or something like that.
Not everything gets (or needs) an immediate and thorough answer, part of leaving it open means there's a chance to tell a good story in the future.