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

Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes