The geometric dimension of some small configurations

Abstract

Recently, Jungnickel and Tonchev (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012) introduced new invariants for simple incidence structures ${\mathcal{D}}$, which admit both a coding theoretic and a geometric description. Geometrically, one considers 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 ${\Pi}$. Then the new invariant, the geometric dimension ${\mathrm{geomdim}_{q}\mathcal{D}}$ of ${\mathcal{D}}$, is the smallest value of n for which ${\mathcal{D}}$ may be embedded into the n-dimensional projective geometry PG(n, q). It is the aim of this paper to discuss a few additional general results regarding these invariants, and to determine them for some further examples, mainly some small configurations; this will answer some problems posed in (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012).