  Let $l,r \in \mathbb{Z}_{\geq 0}$. We mathematically study conformal field
theories with central charge $(\frac{l}{2},\frac{r}{2})$ which have the
symmetry of the $(l+r)$ Virasoro algebras of central charge $\frac{1}{2}$,
$\text{Vir}_{\frac{1}{2}}^{\oplus l}\oplus
{\overline{\text{Vir}_{\frac{1}{2}}^{\oplus r}}}$ in its holomorphic and
anti-holomorphic parts. Such conformal field theories are called $(l,r)$-code
conformal field theories in this paper. We introduce a notion of an
$(l,r)$-framed algebra which is a non-associative finite dimensional algebra
and show that the category of $(l,r)$-code conformal field theories and the
category of $(l,r)$-framed algebras are equivalent. Therefore, the construction
of code CFTs are reduced to the construction of the framed algebras.
  For each code $G \subset \mathbb{Z}_2^r$ satisfying $1^r=(1,1,\dots,1) \in
G$, we constructed an $(r,r)$-framed algebra $S_G$. The corresponding code CFT
$F_G$ defines a modular invariant conformal field theory. For example, when
$G=\langle 1^{r}\rangle$, the corresponding conformal field theory is the
$\mathrm{SO}(r)$-WZW model at level $1$ for $r \geq 4$ and the critical Ising
model (resp. the $\mathrm{SO}(3)$-WZW model at level $2$) for $r=1$ (resp.
$r=3$). We can also obtain a family of new conformal field theories by
considering nontrivial codes $G$. All operator product expansions and the
four-point correlation functions of these conformal field theories can be
computed combinatorially using the codes. Furthermore, by considering the
current-current deformation of code conformal field theories, we mathematically
construct a continuous family of conformal field theories.
