New invariants for incidence structures

Abstract

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 GF(q ^t) for the complementary incidence structure ${\mathcal{D}^*}$, where t may be any positive integer; the associated codes C = C(M) spanned by M over GF(q ^t); and the corresponding trace codes Tr(C(M)) over GF(q). The new invariant, namely the q-dimension ${{\rm dim}_q(\mathcal{D}^*)}$ of ${\mathcal{D}^*}$, 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 Tonchev (Des Codes Cryptogr 17:121–128, 1999). On the other hand, we consider embeddings of ${\mathcal{D}}$ into projective geometries ${\Pi = PG(n, q)}$, 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 ${\mathcal{D}^*}$ always coincides with the smallest value of N for which ${\mathcal{D}}$ may be embedded into the (N − 1)-dimensional projective geometry PG(N − 1, q). We also give a necessary and sufficient condition when actually an embedding into the affine geometry AG(N − 1, q) is possible. Several examples and applications will be discussed: designs with classical parameters, some Steiner designs, and some configurations.