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

TL;DR

This paper develops a new coding theory for normal surfaces with ADE singularities, extending existing codes for projective nodal surfaces, and introduces concepts like generalized labeled codes and code shortening.

Contribution

It introduces a generalized coding framework for normal surfaces with ADE singularities, including new notions like ancestors and partial smoothings.

Findings

Established restrictions on code weights for these surfaces

Extended the concept of code shortening to generalized codes

Illustrated the theory with several examples

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.