Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909449377021952 |
|---|---|
| author | Mizoguchi, Shun'ya Oikawa, Takumi |
| author_facet | Mizoguchi, Shun'ya Oikawa, Takumi |
| contents | We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q(ζ_p)$ $(ζ_p=e^{\frac{2πi}p})$ 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. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_12488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields Mizoguchi, Shun'ya Oikawa, Takumi High Energy Physics - Theory Mathematical Physics Quantum Physics We identify Narain conformal field theories (CFTs) that correspond to code lattices for quantum error-correcting codes (QECC) over integers of cyclotomic fields $Q(ζ_p)$ $(ζ_p=e^{\frac{2πi}p})$ 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. 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. |
| title | Unifying error-correcting code/Narain CFT correspondences via lattices over integers of cyclotomic fields |
| topic | High Energy Physics - Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2410.12488 |