The weight distributions of linear sets in PG(1,q^5)

TL;DR

This paper analyzes the weight distributions of linear sets in projective line spaces over finite fields, providing a detailed classification of possible weights and showing the absence of 2-clubs in PG(1,q^5).

Contribution

It establishes the precise weight distribution for non-scattered linear sets of rank 5 in PG(1,q^5), including the non-existence of 2-clubs.

Findings

Classification of point weights in linear sets

Explicit range for the number of weight-2 points

Proof that 2-clubs do not exist in PG(1,q^5)

Abstract

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 $\mathrm{PG}(1,q^5)$.