Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields

Abstract

We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q\left({\zeta }_p\right)$ $({\zeta }_p=e^{\frac{2\pi i}{p}})$ for general prime $p\ge 3$. This code-lattice construction is a generalization of more familiar ones such as Construction A${}_C$ for ternary codes and (after the generalization stated below) Construction A for binary codes, containing them as special cases. This code-lattice construction is redescribed in terms of root and weight lattices of Lie algebras, which allows to construct lattices for codes over rings $Z_q$ with non-prime q. Corresponding Narain CFTs are found for codes embedded into quotient rings of root and weight lattices of ADE series, except $E_8$ and $D_k$ with k even. In a sense, this provides a unified description of the relationship between various QECCs over $F_p$ (or $Z_q$) and Narain CFTs. A further extension on constructing the $E_8$ lattice from codes over the Mordell-Weil groups of extremal rational elliptic surfaces is also briefly discussed.