Expander graphs, strong blocking sets and minimal codes

Abstract

We give a new explicit construction of strong blocking sets in finite projective spaces using expander graphs and asymptotically good linear codes. Using the recently found equivalence between strong blocking sets and linear minimal codes, we give the first explicit construction of $\mathbb{F}_q$-linear minimal codes of length $n$ and dimension $k$ such that $n$ is at most a constant times $q k$. This solves one of the main open problems on minimal codes.