  Intersecting codes are linear codes where every two nonzero codewords have
non-trivially intersecting support. In this article we expand on the theory of
this family of codes, by showing that nondegenerate intersecting codes
correspond to sets of points (with multiplicites) in a projective space that
are not contained in two hyperplanes. This correspondence allows the use of
geometric arguments to demonstrate properties and provide constructions of
intersecting codes. We improve on existing bounds on their length and provide
explicit constructions of short intersecting codes. Finally, generalizing a
link between coding theory and the theory of the Davenport constant (a
combinatorial invariant of finite abelian groups), we provide new asymptotic
bounds on the weighted $2$-wise Davenport constant. These bounds then yield
results on factorizations in rings of algebraic integers and related
structures.
