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.

翻译：我们证明：对于一个不含成比例列的自对偶码，其关联点集是算术Gorenstein的当且仅当该码不可分解。这回答了Tohăneanu提出的问题。我们通过提供一种组合方法来实现这一证明，该方法通过对自对偶码的生成矩阵进行零一对称化来计算其Schur平方的维数。我们的方法还能计算Gorenstein亏量。由此，我们得到了代数闭域上算术Gorenstein自关联点集的组合特征描述。