  This paper studies {\em strong blocking sets} in the $N$-dimensional finite
projective space $\mathrm{PG}(N,q)$. We first show that certain unions of
blocking sets cannot form strong blocking sets, which leads to a new lower
bound on the size of a strong blocking set in $\mathrm{PG}(N,q)$.
  Our second main result shows that, for $q>\frac{2}{ln(2)}(N+1)$, there exists
a subset of $2N-2$ lines of a Desarguesian line spread in $\mathrm{PG}(N,q)$,
$N$ odd, in {\em higgledy-piggledy arrangement}; thus giving rise to a strong
blocking set of size $(2N-2)(q+1)$.
