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Let K be a finite field with q elements and let X be a subset of a projectivespace P^{s-1}, over the field K, which is parameterized by Laurent monomials.Let I(X) be the vanishing ideal of X. Some of the main contributions of thispaper are in determining the structure of I(X) and some of their invariants. Itis shown that I(X) is a lattice ideal. We introduce the notion of aparameterized code arising from X and present algebraic methods to compute andstudy its dimension, length and minimum distance. For a parameterized codearising from a connected graph we are able to compute its length and to makeour results more precise. If the graph is non-bipartite, we show an upper boundfor the minimum distance. We also study the underlying geometric structure ofX.

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