  A conjecture of Berge suggests that every bridgeless cubic graph can have its
edges covered with at most five perfect matchings. Since three perfect
matchings suffice only when the graph in question is $3$-edge-colourable, the
rest of cubic graphs falls into two classes: those that can be covered with
four perfect matchings, and those that need at least five. Cubic graphs that
require more than four perfect matchings to cover their edges are particularly
interesting as potential counterexamples to several profound and long-standing
conjectures including the celebrated cycle double cover conjecture. However, so
far they have been extremely difficult to find.
  In this paper we build a theory that describes coverings with four perfect
match\-ings as flows whose flow values and outflow patterns form a
configuration of six lines spanned by four points of the 3-dimensional
projective space $\mathbb{P}_3(\mathbb{F}_2)$ in general position. This theory
provides powerful tools for investigation of graphs that do not admit such a
cover and offers a great variety of methods for their construction. As an
illustrative example we produce a rich family of snarks (nontrivial cubic
graphs with no $3$-edge-colouring) that cannot be covered with four perfect
matchings. The family contains all previously known graphs with this property.
