Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields
Shun’ya Mizoguchi, Takumi Oikawa · Physics Letters B · 2025
We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields Q ( ζ p ) ( ζ p = e 2 π i p ) for general prime p ≥ 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.