A new coding theory, for normal surfaces, and ADE singularities, I

Fabrizio Catanese

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arXiv · 2508.16369

A new coding theory, for normal surfaces, and ADE singularities, I

Fabrizio Catanese

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Abstract

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.

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Fabrizio Catanese. 2025-08-22. A new coding theory, for normal surfaces, and ADE singularities, I. https://arxiv.org/abs/2508.16369

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