Codes over a weighted torus

TL;DR

This paper introduces weighted projective Reed-Muller codes over a weighted projective torus, analyzing their algebraic structure and parameters, and demonstrates they are maximum distance separable codes.

Contribution

It defines these codes over weighted projective spaces, characterizes their vanishing ideals as lattice ideals, and computes their parameters, including the index of regularity and code optimality.

Findings

Vanishing ideal of the weighted torus is a lattice ideal.

Weighted projective Reed-Muller codes are maximum distance separable.

Parameters of the codes are explicitly computed for 1-dimensional cases.

Abstract

We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set 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 Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights 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.