  We construct an infinite-dimensional analog of the HaPPY code as a growing
series of stabilizer codes defined respective to their Hilbert spaces. The
Hilbert spaces are related by isometric maps, which we define explicitly. We
construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY
code and further study the stabilizer of our code, which has an inherent
fractal structure. We use this result to study the dynamics of the code and map
a nontrivial bulk Hamiltonian to the boundary. We find that the image of the
mapping is scale invariant, but does not create any long-range entanglement in
the boundary, therefore failing to reproduce the features of a CFT. This result
shows the limits of the HaPPY code as a model of the AdS/CFT correspondence,
but also hints that the relevance of quantum error correction in quantum
gravity may not be limited to the CFT context.
