  There is a rich connection between classical error-correcting codes,
Euclidean lattices, and chiral conformal field theories. Here we show that
quantum error-correcting codes, those of the stabilizer type, are related to
Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual
stabilizer codes can be associated with even self-dual Lorentzian lattices, and
thus define Narain CFTs. We dub the resulting theories code CFTs and study
their properties. T-duality transformations of a code CFT, at the level of the
underlying code, reduce to code equivalences. By means of such equivalences,
any stabilizer code can be reduced to a graph code. We can therefore represent
code CFTs by graphs. We study code CFTs with small central charge $c=n\leq 12$,
and find many interesting examples. Among them is a non-chiral $E_8$ theory,
which is based on the root lattice of $E_8$ understood as an even self-dual
Lorentzian lattice. By analyzing all graphs with $n\leq 8$ nodes we find many
pairs and triples of physically distinct isospectral theories. We also
construct numerous modular invariant functions satisfying all the basic
properties expected of the CFT partition function, yet which are not partition
functions of any known CFTs. We consider the ensemble average over all code
theories, calculate the corresponding partition function, and discuss its
possible holographic interpretation. The paper is written in a self-contained
manner, and includes an extensive pedagogical introduction and many explicit
examples.
