  Bent functions, or equivalently, Hadamard difference sets in the elementary
Abelian group $(\gf(2^{2m}), +)$, have been employed to construct symmetric and
quasi-symmetric designs having the symmetric difference property. The main
objective of this paper is to use bent vectorial functions for a construction
of a two-parameter family of binary linear codes that do not satisfy the
conditions of the Assmus-Mattson theorem, but nevertheless hold $2$-designs. A
new coding-theoretic characterization of bent vectorial functions is presented.
