Under certain conditions, a binary linear code can be lifted to a lattice, which can in turn be lifted to a vertex operator algebra (VOA). For example, the extended Hamming code lifts to the E_8 lattice, which lifts to a lattice VOA. In a somewhat more complicated construction, the extended Golay code lifts to the Leech lattice, which can be used to construct the monster module, an orbifold of the Leech lattice VOA. We investigate such towers by considering the associated sub-objects fixed by lifts of code automorphisms. We identify replicable functions which occur as quotients of theta functions of fixed sublattices and consider further ways in which the structure at the code (resp. lattice) level determines behavior at the lattice (resp. VOA) level of the tower. This is joint work with Lea Beneish, Jen Berg, Hussain Kadhem, Eva Goedhart, and Allechar Serrano Lopez.