The geometric dimension of some small configurations

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© 2012 Springer Basel. Recently, Jungnickel and Tonchev (Des Codes Cryptogr, doi:10.1007/s10623-012-9636-z, 2012) introduced new invariants for simple incidence structures D, which admit both a coding theoretic and a geometric description. Geometrically, one considers embeddings of D into projective geometries Π = PG(n, q), where an embedding means identifying the points of D with a point set V in Π in such a way that every block of D is induced as the intersection of V with a suitable subspace of Π. Then the new invariant, the geometric dimension geomdimq D of D, is the smallest value of n for which 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).

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Journal of Geometry