  A strong blocking set in a finite projective space is a set of points that
intersects each hyperplane in a spanning set. We provide a new graph theoretic
construction of such sets: combining constant-degree expanders with
asymptotically good codes, we explicitly construct strong blocking sets in the
$(k-1)$-dimensional projective space over $\mathbb{F}_q$ that have size $O( q k
)$. Since strong blocking sets have recently been shown to be equivalent to
minimal linear codes, our construction gives the first explicit construction of
$\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$, for every
prime power $q$, for which $n = O (q k)$. This solves one of the main open
problems on minimal codes.
