A new coding theory, for normal surfaces, and ADE singularities, I
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908499191005184 |
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| author | Catanese, Fabrizio |
| author_facet | Catanese, Fabrizio |
| contents | In this article we extend the theory of the binary codes (the strict code $\mathcal{K}$ and the extended code $\mathcal{K}'$), associated to a projective nodal surface, to a coding theory for normal surfaces, with special consideration of the surfaces with ADE (Rational Double Points) singularities.
We define a new theory of generalized labeled codes, establish in the geometric case basic restrictions for the weights of these codes, and some basic inequality. A crucial method that we establish is the extension of the concept of `code shortening' to the case of generalized codes: this is the algebraic counterpart of the geometric notion of a partial smoothing of the singular points, and leads to the concept of ancestors, which we illustrate through several examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16369 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new coding theory, for normal surfaces, and ADE singularities, I Catanese, Fabrizio Algebraic Geometry Complex Variables 14C30, 14J28, 14J25, 14J70, 14N25, 16G99, 32G20, 32Q15 In this article we extend the theory of the binary codes (the strict code $\mathcal{K}$ and the extended code $\mathcal{K}'$), associated to a projective nodal surface, to a coding theory for normal surfaces, with special consideration of the surfaces with ADE (Rational Double Points) singularities. We define a new theory of generalized labeled codes, establish in the geometric case basic restrictions for the weights of these codes, and some basic inequality. A crucial method that we establish is the extension of the concept of `code shortening' to the case of generalized codes: this is the algebraic counterpart of the geometric notion of a partial smoothing of the singular points, and leads to the concept of ancestors, which we illustrate through several examples. |
| title | A new coding theory, for normal surfaces, and ADE singularities, I |
| topic | Algebraic Geometry Complex Variables 14C30, 14J28, 14J25, 14J70, 14N25, 16G99, 32G20, 32Q15 |
| url | https://arxiv.org/abs/2508.16369 |