Curves and surfaces over finite fields and coding theory

We explain an approach of Elkies to counting points on varieties over
finite fields. We can interpret a vector space of polynomials as a
linear subspace of (F_q)^N, a linear code, by the evaluation map.
Studying properties of this code and its dual gives information about
the distribution of rational points in the family of varieties defined
by these polynomials. We apply this approach to study rational points
on families of genus one curves and del Pezzo surfaces. No previous
familiarity with coding theory is required.