The weight distributions of linear sets in PG(1,q5)
Maarten De Boeck, Geertrui Van de Voorde · Finite Fields and Their Applications · 2022
In this paper, we study the weight distributions of F q -linear sets in 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 , or q 2 points of weight 2, (iii) s points of weight 2 where s ∈ [ q − 2 q + 1 , q + 2 q + 1 ] ∪ { 2 q , 2 q + 1 , 2 q + 2 , 3 q , 3 q + 1 , q 2 + 1 } . In particular, there are no 2-clubs in PG ( 1 , q 5 ) .