Error-correcting codes over the Mordell-Weil groups of extremal rational elliptic surfaces and the $E_8$ lattice
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917324372574208 |
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| author | Mizoguchi, Shun'ya Oikawa, Takumi |
| author_facet | Mizoguchi, Shun'ya Oikawa, Takumi |
| contents | We construct the $E_8$ lattice from classical error-correcting codes over the Mordell-Weil groups of rational elliptic surfaces that have a singularity lattice of rank 8 (maximal) for all cases of Oguiso-Shioda's classification. By the structure theorem of the Mordell-Weil lattice of rational elliptic surfaces, if the rank of the singularity lattice is maximal, then the Mordell-Weil group is a cyclic group or a direct sum of them. The singularity lattices are glued together by a code over their natural ring to form the $E_8$ lattice. Such constructions of the $E_8$ lattice from codes can be seen as a Lie algebraic extension and further generalization of known code lattice constructions such as Construction A and Construction A${}_{\rm C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08102 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Error-correcting codes over the Mordell-Weil groups of extremal rational elliptic surfaces and the $E_8$ lattice Mizoguchi, Shun'ya Oikawa, Takumi High Energy Physics - Theory We construct the $E_8$ lattice from classical error-correcting codes over the Mordell-Weil groups of rational elliptic surfaces that have a singularity lattice of rank 8 (maximal) for all cases of Oguiso-Shioda's classification. By the structure theorem of the Mordell-Weil lattice of rational elliptic surfaces, if the rank of the singularity lattice is maximal, then the Mordell-Weil group is a cyclic group or a direct sum of them. The singularity lattices are glued together by a code over their natural ring to form the $E_8$ lattice. Such constructions of the $E_8$ lattice from codes can be seen as a Lie algebraic extension and further generalization of known code lattice constructions such as Construction A and Construction A${}_{\rm C}$. |
| title | Error-correcting codes over the Mordell-Weil groups of extremal rational elliptic surfaces and the $E_8$ lattice |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2603.08102 |