Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields

Let K=Fq be a finite field with q elements and let X be a subset of a projective space Ps−1, over the field K, parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) to compute some of its invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code, arising from a connected graph, we are able to compute its length and to make our results more precise. If the graph is non-bipartite, we show an upper bound for the minimum distance.