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

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.