Multigraded Hilbert functions and toric complete intersection codes

Let X be a complete n-dimensional simplicial toric variety with homogeneous coordinate ring S. We study the multigraded Hilbert function HY of reduced 0-dimensional subschemes Y in X. We provide explicit formulas and prove non-decreasing and stabilization properties of HY when Y is a 0-dimensional complete intersection in X. We apply our results to computing the dimension of some evaluation codes on 0-dimensional complete intersection in simplicial toric varieties.