On saturating sets in small projective geometries
A set of points \(S\) of the projective space \(PG(r,q)\) is said to be \(\rho\)-saturating if for every point \(x\) of \(PG(r,q)\), there exist \(\rho+1\) points in \(S\) that generate a subspace in which \(x\) lies. One of the interesting facts on \(\rho\)-saturating sets is that \(\rho\)-saturating sets in projective geometry correspond to linear codes in coding theory having covering radius \(R=\rho+1\); thus implying that these \(\rho\)-saturating sets are important for both finite geometry and coding theory. The authors concentrate in this article on the parameter \(k(r,q,\rho)\) which denotes the cardinality of a smallest possible \(\rho\)-saturating set \(S\) of \(PG(r,q)\). They survey what is known on \(k(r,q,1)\), present a new upper bound \(k(5,q,2)\leq 3q+1\) for \(q \neq 3\), and a new upper bound on \(k(\rho+1,p^m,\rho)\). The article also contains upper bounds on \(k(r,q,2) \) for \(r=3,4,5\) and \(q \leq 16\).
scientific article; zbMATH DE number 1512031
scientific article