Blocking sets in ${PG}(2,q^n)$ from cones of ${PG}(2n,q)$

Mazzocca, Francesco ; Polverino, Olga

Journal of Algebraic Combinatorics, Tome 24 (2006) no. 1, pp. 61-81.

Résumé

Summary: Let $\Omega $and [ `$( B)] {\bar B}$ be a subset of $\Sigma = P$G($2 n - 1, q$) and a subset of $P$G($2 n, q$) respectively, with $\Sigma \subset P$G($2 n, q$) and [ `$( B)$] Ë S $\bar B$not$\subset $Sigma . Denote by $K$ the cone of vertex $\Omega $and base [ `$( B)] {\bar B}$ and consider the point set $B$ defined by $B=( K$ S) È $X$ Ĩ § : $X$ Ç $K \textonesuperior $ Æ, B=$big(KsetminusSigmabig) cup$\{X$in$\S\, : \, X$capKneq$\emptyset\}, in the Andr\'e, Bruck-Bose representation of $P$G$(2, q ^ n)$ in $P$G($2 n, q$) associated to a regular spread $S calS$ of $P$G($2 n - 1, q$). We are interested in finding conditions on [ `$( B)] barB$ and $Omega$in order to force the set $B$ to be a minimal blocking set in $P$G$(2, q ^ n)$ . Our interest is motivated by the following observation. Assume a Property $alpha$of the pair $(Omega, [ `( B)] barB )$ 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 $(Omega, [ `( B)] barB )$ with Property $alpha$. With this in mind, we deal with the problem in the case $Omega$is a subspace of $P$G($2 n - 1, q$) and [ `$( B)] barB$ a blocking set in a subspace of $P$G($2 n, q$); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in $P$G$(2, q ^ n)$, generalizing the main constructions of [14]. For example, for $q = 3 ^ h$, we get large blocking sets of size $q ^ n + 2 + 1 ( ngeq5)$ and of size greater than $q ^ n+2 + q ^ n - 6 ( ngeq6)$. As an application, a characterization of Buekenhout-Metz unitals in $P$G$(2, q ^2 k)$ is also given.$

Détail

Keywords: keywords blocking set, André/bruck-Bose representation, ovoid