A geometric characterisation of linear \(k\)-blocking sets
Let \(V\) be an \(r\)-dimensional vector space over GF\((q^t)\); such a vector space can also be considered as an \(rt\)-dimensional vector space over GF\((q)\). By a result of \textit{G. Lunardon} [Combinatorica 21, 571-581 (2001; Zbl 0996.51003)], a \(k\)-blocking set \(B\) in the projective space PG\((r-1,q^t)=\)PG\((V, \)GF\((q^t))\) is linear if and only if there exists a subset \(W\) of \(V\) such that: (a)~\(W\) is a GF\((q)\)-vector space of dimension \(kt+1\); (b)~a point of PG\((r-1,q^t)\) belongs to \(B\) if and only if it is defined by a vector of \(W\); (c)~\(\langle W\rangle_{q^t}=V\). A subgeometry \(\Sigma=\)PG\((U, \)GF\((q))\) of \(\Sigma^*=\)PG\((U^*, \)GF\((q^t))\) is a canonical subgeometry when \(U^*=\)GF\((q^t)\bigotimes U\). In the paper under review it is proved that any linear \(k\)-blocking set is either a canonical subgeometry or a projection of some canonical subgeometry.
scientific article; zbMATH DE number 1890537