There are $q^{20}$ homogeneous cubic polynomials in four variables with coefficients in the finite field $\mathbb{F}_q$. How many of them define a cubic surface with $q^2+7q+1\ \mathbb{F}_q$-rational points? What about other numbers of rational points? How many of the $q^{20}$ pairs of homogeneous cubic polynomials in three variables define cubic curves that intersect in $9$ $\mathbb{F}_q$-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 much background in coding theory and will emphasize examples.