For projective Reed--Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed--Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed-Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the $r$-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.

翻译：针对射影Reed-Muller型码，我们基于消逝理想的v-数与Hilbert函数给出了整体对偶性判据。作为应用，我们建立了Gorenstein消逝理想上射影Reed-Muller型码的整体对偶定理，推广了消逝理想为完全交的已知情形。我们利用正则性与奇偶校验矩阵，分类了Gorenstein理想上的自对偶Reed-Muller型码。受仿射评估码对偶性的启发，我们为射影评估码建立了一个对偶定理。进一步，我们展示了如何利用评估点集的标准指示函数计算第r阶广义Hamming权函数的正则指标。