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 $\mathrm{ PG}(1,q^5)$.
Categories: math.CO