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

• URL: http://arxiv.org/abs/2410.12488v1
• Date: Wed, 16 Oct 2024 12:08:04 GMT
• Title: Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields
• Authors: Shun'ya Mizoguchi, Takumi Oikawa,
• Abstract summary: We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q(zeta_p)$ $(zeta_p=efrac2pi ip)$ for general prime $pgeq 3$.
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• Abstract: We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q(\zeta_p)$ $(\zeta_p=e^{\frac{2\pi i}p})$ for general prime $p\geq 3$. 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.

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