Let K be a finite field with q elements and let X be a subset of a projective
space of dimension s-1, over the field K, which is parameterized by Laurent
monomials. We introduce the class of parameterized linear codes arising from X
and present algebraic methods to compute their dimensions and lengths. Using
tools from commutative algebra, along with the theory of lattices and finite
fields, we study the structure of the graded ideal I(X) generated by the
homogeneous polynomials of K[t1,...,ts] that vanish on X. It is shown that I(X)
is a lattice ideal.