Abstract
Let \(\mathcal S\) be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and \({\bar B}\) be, respectively, an (n – 2)-dimensional subspace of an element of \(\mathcal S \) and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base \({\bar B}\) , and consider the point set B defined by
in the Barlotti–Cofman representation of PG(r, q n) in PG(rn, q) associated to the (n – 1)-spread \(\mathcal S\) . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on \({\bar B}\) , we prove that B is a minimal blocking set in PG(r, q n). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q n+2 + 1 in PG(r, q n), 3 ≤ r ≤ 6 and n ≥ 3, and of size q 4 + 1 in PG(r, q 2), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q 3 Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q 2 + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q 2) of size q 4 + 1, for any q a power of 3.
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Mazzocca, F., Polverino, O. & Storme, L. Blocking Sets in PG(r, q n). Des. Codes Cryptogr. 44, 97–113 (2007). https://doi.org/10.1007/s10623-007-9068-3
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DOI: https://doi.org/10.1007/s10623-007-9068-3