For projective Reed--Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for projective Reed--Muller-type codes corresponding to Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. The theory of Gorenstein vanishing ideals is examined using indicator functions. For projective evaluation codes, we give local duality criteria 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：针对投影Reed-Muller型码，我们基于消逝理想的v-数与Hilbert函数给出了一个对偶判据。作为应用，我们为对应Gorenstein消逝理想的投影Reed-Muller型码提供了显式对偶性，推广了消逝理想是完全交情形的已知结论。利用指示函数研究了Gorenstein消逝理想的理论。对于投影评估码，我们给出了仿射评估码启发下的局部对偶判据。我们证明了如何利用评估点集的标准指示函数计算第r阶广义汉明重量函数的正则性指标。