Codes over a weighted torus

We define the notion of weighted projective Reed–Muller codes over a subset X⊂P(w1,…,ws)X⊂P(w1,…,ws) of a weighted projective space over a finite field. We focus on the case when X is a projective weighted torus. We show that the vanishing ideal of X   is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of the numerical semigroup Q=〈w1,…,ws〉⊂NQ=〈w1,…,ws〉⊂N. We compute the index of regularity of the vanishing ideal of X in terms of the weights of the projective space and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed–Muller codes over a 1-dimensional weighted torus and prove they are maximum distance separable codes.