The genome (mathematically) contains a smaller amount of data than is actually required for an organism as complex as the phenotype produced with the help of it to develop.

Sure. This is well known. See here, where it says:

Epigenesis: The ontogenetic program is limited in the amount of information that can be stored, thereby rendering the complete specification of the organism impossible. A well-known example is that of the human brain with some 10^10 neurons and 10^14 connections, far too large a number to be completely specified in the four-character genome of length approximately 3 x 10^9. Therefore, upon reaching a certain level of complexity, there must emerge a different process that permits the individual to integrate the vast quantity of interactions with the outside world. This process is known as epigenesis, and primarily includes the nervous system, the immune system, and the endocrine system.