Classes and equivalence of linear sets in PG(1,q^n)

Abstract The equivalence problem of F q -linear sets of rank n of PG ( 1 , q n ) is investigated, also in terms of the associated variety, projecting configurations, F q -linear blocking sets of Redei type and MRD-codes. We call an F q -linear set L U of rank n in PG ( W , F q n ) = PG ( 1 , q n ) simple if for any n -dimensional F q -subspace V of W , L V is P Γ L ( 2 , q n ) -equivalent to L U only when U and V lie on the same orbit of Γ L ( 2 , q n ) . We prove that U = { ( x , Tr q n / q ( x ) ) : x ∈ F q n } defines a simple F q -linear set for each n . We provide examples of non-simple linear sets not of pseudoregulus type for n > 4 and we prove that all F q -linear sets of rank 4 are simple in PG ( 1 , q 4 ) .