← 研踪 PaperTrailarXiv:2512.16766 · v2
A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein
首次提交 2025-12-19 01:01
版本更新 2026-10-27 19:37
雷达观测 2026-10-29 00:35
摘要
We give a combinatorial proof 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 Tohăneanu. 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 a systematic 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. We also show that, for any finite field, almost all self-dual codes of large length are indecomposable, and hence their associated sets of points are arithmetically Gorenstein.
作者备注：15 pages
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v2 · 2026-10-27 19:37 · 最新观测版本
A combinatorial description of when a self-associated set of points fails to be arithmetically Gorenstein
We give a combinatorial proof 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 Tohăneanu. 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 a systematic 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. We also show that, for any finite field, almost all self-dual codes of large length are indecomposable, and hence their associated sets of points are arithmetically Gorenstein.
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