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

摘要

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,]Fq-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-qn) = PG(1, q(n)) simple if for any n-dimensional F-q-subspace V of W, L-v is P Gamma L(2, q(n))-equivalent to L-U only when U and V lie on the same orbit of Gamma L(2, q(n)). We prove that U = {(x,Tr q(n)/q (x)): x is an element of F-qn defines a simple]Fq-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)). (C) 2018 Elsevier Inc. All rights reserved.