Summary: By using results and techniques from commutative algebra such as the vanishing ideal of a set of points, its \(a\)-invariant, the Hilbert polynomial and series, as well as finite free resolutions and the canonical module, some results about Reed-Muller codes defined on a zero-dimensional complete intersection in the \(n\)-projective dimensional space are given. Several examples of this class of codes are presented in order to illustrate the ideas.