Error correcting codes and heterotic Narain CFTs

TL;DR

This paper explores how error correcting codes can be used to construct heterotic string Narain lattices, revealing new code-lattice correspondences and extending to codes over different finite fields.

Contribution

It introduces novel code constructions for heterotic Narain lattices, including over F_3 and F_5, and clarifies their relation to Euclidean lattices and NSR-fermions.

Findings

Constructed Narain lattices from binary codes via Construction A.

Extended code constructions to F_3 and F_5 for heterotic strings.

Clarified the link between Euclidean lattices, codes, and NSR-fermions.

Abstract

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.