Abstract
Let Ω and \({\bar B}\) be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and \({{\bar B}\not\subset \Sigma}\). Denote by K the cone of vertex Ω and base \({\bar B}\) and consider the point set B defined by
in the André, Bruck-Bose representation of PG(2,q n) in PG(2n,q) associated to a regular spread \({\cal S}\) of PG(2n−1,q). We are interested in finding conditions on \({\bar B}\) and Ω in order to force the set B to be a minimal blocking set in PG(2,q n) . Our interest is motivated by the following observation. Assume a Property α of the pair (Ω,\({\bar B}\)) forces B to turn out a minimal blocking set. Then one can try to find new classes of minimal blocking sets working with the list of all known pairs (Ω, \({\bar B}\)) with Property α. With this in mind, we deal with the problem in the case Ω is a subspace of PG(2n−1,q) and \({\bar B}\) a blocking set in a subspace of PG(2n,q); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in PG(2,q n), generalizing the main constructions of [14]. For example, for q = 3h, we get large blocking sets of size q n + 2 + 1 (n≥ 5) and of size greater than q n+2 + q n−6 (n≥ 6). As an application, a characterization of Buekenhout-Metz unitals in PG(2,q 2k) is also given.
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Mazzocca, F., Polverino, O. Blocking sets in PG(2, q n) from cones of PG(2n, q). J Algebr Comb 24, 61–81 (2006). https://doi.org/10.1007/s10801-006-9102-y
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DOI: https://doi.org/10.1007/s10801-006-9102-y