You're being downvited because, whilst this is a clever sounding idea, unfortunately it doesn't work like that.

The noise is random, so you can't modulate it because you can't predict what it will be. A probability distribution isn't enough; say you want to modulate by XORing with a random, uniformly distributed stream of zeros and ones. Knowing that distribution doesn't enable the receiver to extract the data.

(Fun fact: GPS signals are under the noise floor, but detectable because they are encoded with a predictable pseudorandom carrier which can be found even in the noise.)