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

Csajbók, Bence and Marino, Giuseppe and Polverino, Olga (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 GF(q)-linear sets of rank n of PG(1,q^n) is investigated, also in terms of the associated variety, projecting configurations,GF(q)-linear blocking sets of Rédei type and MRD-codes. We call an GF(q)-linear set L_U of rank n in PG(W,Fq^n) = PG(1,q^n) simple if for any n-dimensional GF(q)-subspace V of W, L_V is PGammaL(2, q^n)-equivalent to L_U only when U and V lie on the same orbit of GammaL(2,q^n). We prove that U = {(x,Tr(x)) : x \in GF(q^n)} defines a simple GF(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 GF(q)-linear sets of rank 4 are simple in PG(1,q^4).

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