Error correcting codes and heterotic Narain CFTs

A bstract We study error correcting codes that construct the Narain lattices of heterotic strings as code lattices. We identify, in both E 8 × 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.