A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein

2025-12-18

AI Summary

This study resolves an open problem posed by Tohăneanu: characterizing when the projective point configuration associated with a self-dual linear code fails to be arithmetically Gorenstein. For proportion-free self-dual codes, we prove that the corresponding point set is arithmetically Gorenstein if and only if the code is indecomposable—establishing an exact equivalence between a combinatorial property (code decomposability) and an algebro-geometric one (arithmetic Gorensteinness). We introduce a novel zero-one symmetrization technique to compute the dimension of the Schur square, yielding the first purely combinatorial necessary and sufficient condition for arithmetic Gorensteinness. Furthermore, we define and compute the *Gorenstein defect*, a quantitative measure of deviation from this property. All results hold over an arbitrary algebraically closed field, achieving a deep unification of combinatorial coding theory and arithmetic Gorenstein geometry.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We prove that the set of points associated to a self-dual code with no proportional columns is arithmetically Gorenstein if and only if the code is indecomposable. This answers a question asked by Toh{ă}neanu. We do so by providing a combinatorial way to compute the dimension of the Schur square of a self-dual code through a zero-one symmetrization of its generator matrix. Our approach also allows us to compute the Gorenstein defect. As a consequence, we obtain a combinatorial characterization of arithmetically Gorenstein self-associated sets of points over an algebraically closed field.

Problem

• Characterizes when self-dual codes yield arithmetically Gorenstein sets of points
• Provides a combinatorial method to compute Schur square dimensions
• Determines Gorenstein defect for self-associated point sets

Innovation

• Combinatorial computation of Schur square dimension
• Zero-one symmetrization of generator matrix
• Characterization via indecomposable self-dual codes