Sure, and you can add some correction bits to fix the lossy encoding of the autoencoder.

The problem is that you have to encode/send the DNN itself, otherwise your receiver won't know how to decode the data. If you are not smart, the added codelength of the DNN will likely blow away your savings. If you are smart, this leads to a whole formulation of machine learning called MDL:

https://en.wikipedia.org/wiki/Minimum_description_length