  We develop three approaches of combinatorial flavour to study the structure
of minimal codes and cutting blocking sets in finite geometry, each of which
has a particular application. The first approach uses techniques from algebraic
combinatorics, describing the supports in a linear code via the Alon-F\"uredi
Theorem and the Combinatorial Nullstellensatz. The second approach combines
methods from coding theory and statistics to compare the mean and variance of
the nonzero weights in a minimal code. Finally, the third approach regards
minimal codes as cutting blocking sets and studies these using the theory of
spreads in finite geometry. Applying and combining these approaches with each
other, we derive several new bounds and constraints on the parameters of
minimal codes. Moreover, we obtain two new constructions of cutting blocking
sets of small cardinality in finite projective spaces. In turn, these allow us
to give explicit constructions of minimal codes having short length for the
given field and dimension.
