  A \textit{k}-arc in the projective space ${\rm PG}(n,q)$ is a set of $k$
projective points such that no subcollection of $n+1$ points is contained in a
hyperplane. In this paper, we construct new $60$-arcs and $110$-arcs in ${\rm
PG}(4,q)$ that do not arise from rational or elliptic curves. We introduce
computational methods that, when given a set $\mathcal{P}$ of projective points
in the projective space of dimension $n$ over an algebraic number field
$\mathcal{Q}(\xi)$, determines a complete list of primes $p$ for which the
reduction modulo $p$ of $\mathcal{P}$ to the projective space ${\rm PG}(n,p^h)$
may fail to be a $k$-arc. Using these methods, we prove that there are
infinitely many primes $p$ such that ${\rm PG}(4,p)$ contains a ${\rm
PSL}(2,11)$-invariant $110$-arc, where ${\rm PSL}(2,11)$ is given in one of its
natural irreducible representations as a subgroup of ${\rm PGL}(5,p)$.
Similarly, we show that there exist ${\rm PSL}(2,11)$-invariant $110$-arcs in
${\rm PG}(4,p^2)$ and ${\rm PSL}(2,11)$-invariant $60$-arcs in ${\rm PG}(4,p)$
for infinitely many primes $p$.
