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

We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields Q(ζp) (ζp=e2πip) for general prime p≥3. This code-lattice construction is a generalization of more familiar ones such as Construction AC 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 Zq with non-prime q. Corresponding Narain CFTs are found for codes embedded into quotient rings of root and weight lattices of ADE series, except E8 and Dk with k even. In a sense, this provides a unified description of the relationship between various QECCs over Fp (or Zq) and Narain CFTs. A further extension on constructing the E8 lattice from codes over the Mordell-Weil groups of extremal rational elliptic surfaces is also briefly discussed.