  In this paper, we study the weight distributions of $\mathbb{F}_q$-linear
sets in $\mathrm{PG}(1,q^5)$. Our main theorem proves that a linear set $S$ of
rank $5$, which is not scattered has the following weight distribution for its
points with weight larger than 1: (i) one point of weight $4$ or $5$, (ii) one
point of weight $3$ and $0$, $q$, $q^2$ points of weight two, (iii) $s$ points
of weight $2$ where $s\in
[q-2\sqrt{q}+1,q+2\sqrt{q}+1]\cup\{2q,2q+1,2q+2,3q,3q+1,q^2+1\}$. In
particular, there are no $2$-clubs in $\mathrm{PG}(1,q^5)$.

Combinatorics