  Minimal linear codes are algebraic objects which gained interest in the last
twenty years, due to their link with Massey's secret sharing schemes. In this
context, Ashikhmin and Barg provided a useful and a quite easy to handle
sufficient condition for a linear code to be minimal, which has been applied in
the construction of many minimal linear codes. In this paper, we generalize
some recent constructions of minimal linear codes which are not based on
Ashikhmin-Barg's condition. More combinatorial and geometric methods are
involved in our proofs. In particular, we present a family of codes arising
from particular blocking sets, which are well-studied combinatorial objects. In
this context, we will need to define cutting blocking sets and to prove some of
their relations with other notions in blocking sets' theory. At the end of the
paper, we provide one explicit family of cutting blocking sets and related
minimal linear codes, showing that infinitely many of its members do not
satisfy the Ashikhmin-Barg's condition.
