Codes from varieties over finite fields

There are q²⁰ homogeneous cubic polynomials in four variables with coefficients in the finite field F_q. How many of them define a cubic surface with q²+7q+1 F_q-rational points? What about other numbers of rational points? How many of the q²⁰ pairs of homogeneous cubic polynomials in three variables define cubic curves that intersect in 9 F_q-rational points? The goal of this talk is to explain how ideas from the theory of error-correcting codes can be used to study families of varieties over a fixed finite field. We will not assume any previous familiarity with coding theory. We will start from the basics and emphasize examples.