Summary: We exhibit a new, surprisingly tight, connection between incidence structures, linear codes, and Galois geometry. To this end, we introduce new invariants for finite simple incidence structures \(\mathcal D\), which admit both an algebraic and a geometric description. More precisely, we will associate one invariant for the isomorphism class of \(\mathcal D\) with each prime power \(q\). On the one hand, we consider incidence matrices \(M\) with entries from \(\mathrm{GF}(q^t)\) for the complementary incidence structure \(\mathcal D^\ast\), where \(t\) may be any positive integer, the associated codes \(C=C(M)\) spanned by \(M\) over \(\mathrm{GF}(q^t)\), and the corresponding trace codes \(\operatorname{Tr}(C(M))\) over \(\mathrm{GF}(q)\). The new invariant, namely the \(q\)-dimension \(\dim_q(\mathcal D^\ast)\) of \(\mathcal D^\ast\), is defined to be the smallest dimension over all trace codes which may be obtained in this manner. This modifies and generalizes the \(q\)-dimension of a design as introduced in [V. D. Tonchev, Des. Codes Cryptography 17, No.1–3, 121–128 (1999; Zbl 0940.05009)]. On the other hand, we consider embeddings of \(\mathcal D\) into projective geometries \(\Pi=\mathrm{PG}(n,q)\), where an embedding means identifying the points of \(\mathcal D\) with a point set \(V\) in \(\Pi\) in such a way that every block of \(\mathcal D\) is induced as the intersection of \(V\) with a suitable subspace of \(\Pi\). Our main result shows that the \(q\)-dimension of \(\mathcal D^\ast\) always coincides with the smallest value of \(N\) for which \(\mathcal D\) may be embedded into the \((N-1)\)-dimensional projective geometry \(\mathrm{PG}(N-1,q)\). We also give a necessary and sufficient condition when actually an embedding into the affine geometry \(\mathrm{AG}(N-1,q)\) is possible. Several examples and applications will be discussed: designs with classical parameters, some Steiner designs, and some configurations.