Error correcting codes and heterotic Narain CFTs

We study error correcting codes that construct the Narain lattices of heterotic strings as code lattices. We identify, in both $E_8\times E_8$ and Spin$(32)/Z_2$ heterotic strings, a pair of a binary code and a set of the corresponding metric, B field, and background gauge field, such that the lattice constructed from the binary code by Construction A coincides with the Narain lattice. We also construct heterotic Narain lattices using codes over $F_3$ and $F_5$ by Construction A${}_C$ and "Construction A${}_g$" with $g=SU(5)$, respectively. As a bi-product, we also clarify the relationship between codes that construct Euclidean even self-dual lattices and NSR-fermions, where the $Z_2$ inversion structure of the generator matrices plays a significant role.