We study $t$-designs of parameters $(n,k,λ)$ 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 $λ$ 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.