A new coding theory, for normal surfaces, and ADE singularities, I
Fabrizio Catanese
Journal of Singularities
volume 30 (2026), 116-143
Received: 22 August 2025. In revised form: 5 March 2026
Abstract:
In this article we extend the theory of the binary codes (the strict code K and the extended code 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 inequalities. 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.
2020 Mathematical Subject Classification:
14C30, 14J28, 14J25, 14J70, 14N25, 16G99, 32G20, 32Q15
Author(s) information:
Fabrizio Catanese
Mathematisches Institut der Universität
Bayreuth, NW II
Universitätsstr. 30
95447 Bayreuth, Germany
email: Fabrizio.Catanese@uni-bayreuth.de