A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein
Gonzalo Rodríguez-Pajares, Diego Ruano, Flavio Salizzoni
TL;DR
The paper resolves when the point set associated to a self-dual code is arithmetically Gorenstein by proving that this occurs if and only if the code is indecomposable, linking Gorensteinness to the dimension of the Schur square via a zero-one symmetrization graph of the generator matrix. It provides a concrete, combinatorial criterion: dim(C^{(2)})=2k−1 (equivalently nb(C)=1) characterizes arithmetically Gorenstein sets, and nb(C) equals the number of connected components of a specific graph, enabling efficient computation of the Gorenstein defect. The results extend to arbitrary algebraically closed fields and give a complete characterization in the self-associated setting, with corollaries tying indecomposability to Gorensteinness and a detailed treatment of the proportional-columns case. The paper also establishes asymptotic behavior: almost all large self-dual codes are indecomposable and yield Gorenstein point sets, with explicit counting formulas for self-dual and indecomposable self-dual codes across finite fields.
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.