  We study $t$-designs of parameters $(n,k,\lambda)$ over finite fields as
group divisible designs and set systems admitting a transitive action of a
linear group encoded in an hypergraph $G$ whose vertex set of size $n$ is
partitioned into sets of size $k$ in such a way that every $t$-subset is
contained in at least $\lambda$ subsets of $G$. We relate the problem to the
representation theory of the general linear group $\GL(n,\mathbb{F}_{q})$ and
the constructions of AG codes over finite fields.
