  Let X* be a subset of an affine space A^s, over a finite field K, which is
parameterized by the edges of a clutter. Let X and Y be the images of X* under
the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are
points in the projective spaces P^{s-1} and P^s respectively. For certain
clutters and for connected graphs, we were able to relate the algebraic
invariants and properties of the vanishing ideals I(X) and I(Y). In a number of
interesting cases, we compute its degree and regularity. For Hamiltonian
bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We give
optimal bounds for the regularity when the graph is bipartite. It is shown that
X* is an affine torus if and only if I(Y) is a complete intersection. We
present some applications to coding theory and show some bounds for the minimum
distance of parameterized linear codes for connected bipartite graphs.
