  The holographic principle suggests that the low energy effective field theory
of gravity, as used to describe perturbative quantum fields about some
background has far too many states. It is then natural that any quantum error
correcting code with such a quantum field theory as the code subspace is not
isometric. We discuss how this framework can naturally arise in an algebraic
QFT treatment of a family of CFT with a large-$N$ limit described by the single
trace sector. We show that an isometric code can be recovered in the $N
\rightarrow \infty$ limit when acting on fixed states in the code Hilbert
space. Asymptotically isometric codes come equipped with the notion of simple
operators and nets of causal wedges. While the causal wedges are additive, they
need not satisfy Haag duality, thus leading to the possibility of non-trivial
entanglement wedge reconstructions. Codes with complementary recovery are
defined as having extensions to Haag dual nets, where entanglement wedges are
well defined for all causal boundary regions. We prove an asymptotic version of
the information disturbance trade-off theorem and use this to show that
boundary theory causality is maintained by net extensions. We give a
characterization of the existence of an entanglement wedge extension via the
asymptotic equality of bulk and boundary relative entropy or modular flow.
While these codes are asymptotically exact, at fixed $N$ they can have large
errors on states that do not survive the large-$N$ limit. This allows us to fix
well known issues that arise when modeling gravity as an exact codes, while
maintaining the nice features expected of gravity, including, among other
things, the emergence of non-trivial von Neumann algebras of various types.
