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

Csajbók, Bence and Marino, G. and Polverino, O. (2018) Classes and equivalence of linear sets in PG(1, q(n)). JOURNAL OF COMBINATORIAL THEORY SERIES A, 157. pp. 402-426. ISSN 0097-3165

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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,]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.

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