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

TL;DR

This paper provides a combinatorial criterion to determine when a self-associated set of points, derived from a self-dual code, is arithmetically Gorenstein, linking algebraic geometry with coding theory.

Contribution

It introduces a combinatorial method to characterize arithmetically Gorenstein self-associated points via self-dual codes, answering a question by Toh{neanu}.

Findings

Set of points from self-dual codes is Gorenstein iff the code is indecomposable.

Provides a combinatorial way to compute the Gorenstein defect.

Characterizes arithmetically Gorenstein self-associated points combinatorially.

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{\u{a}}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.