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
)
(k-1)
-dimensional projective space over
F
q
\mathbb {F}_q
that have size at most
c
q
k
c q k
for some universal constant
c
c
. Since strong blocking sets have recently been shown to be equivalent to minimal linear codes, our construction gives the first explicit construction of
F
q
\mathbb {F}_q
-linear minimal codes of length
n
n
and dimension
k
k
, for every prime power
q
q
, for which
n
≤
c
q
k
n \leq c q k
. This solves one of the main open problems on minimal codes.