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(qt) 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(qt); 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.