On the equivalence of linear sets

Abstract

Let L be a linear set of pseudoregulus type in a line ℓ in Σ * = PG(t − 1, q t ), t = 5 or t > 6. We provide examples of q-order canonical subgeometries Σ 1 , Σ 2 ⊂ Σ * such that there is a (t − 3)-subspace Γ ⊂ Σ * \ (Σ 1 ∪Σ 2 ∪ℓ) with the property that for i = 1, 2, L is the projection of Σ i from center Γ and there exists no collineation φ of Σ * such that Γ φ = Γ and Σ φ 1 = Σ 2 . Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.

Vertex properties of maximum scattered linear sets of PG(1,qn)

“…For our purpose it is important to look to the ΓL-class in a more geometric way. The following result has been stated in [4,Section 5.2] as a consequence of [10,Theorems 6 & 7]. Theorem 3.4.…”

“…A point set L of Λ is said to be an Two linear sets L U and L W of PG(r − 1, q n ) are said to be PΓL-equivalent if there is an element φ in PΓL(r, q n ) such that L φ U = L W . It may happen that two F q -linear sets L U and L W of PG(r − 1, q n ) are PΓL-equivalent even if the F q -vector subspaces U and W are not in the same orbit of ΓL(r, q n ) (see [10] and [4] for further details). In this paper we focus on maximum scattered F q -linear sets of PG(1, q n ), that is, F q -linear sets of rank n in PG(1, q n ) of size (q n − 1)/(q − 1).…”

Rank-metric codes, linear sets, and their duality

“…If C and C ′ are inequivalent subspaces but define PΓL-equivalent linear sets, their duals do not necessarily define PΓL-equivalent linear sets. Let C 1 = x, x q and C 2 = x, x q 2 be subspaces in PG(4, q 5 ), then C 1 and C 2 are inequivalent, but L x,x q = L x,x q 2 , and hence, L x,x q 2 ∈ L C 1 (see [3]). We have C ⊥ 1 = x q 2 , x q 3 , x q 4 and C ⊥ 2 = x q , x q 3 , x q 4 .…”

A Carlitz type result for linearized polynomials

“…In the applications it is crucial to have methods to decide whether two linear sets are equivalent or not. This can be a difficult problem and some results in this direction can be found in [12,9]. If L U and L V are two equivalent F q -linear sets of rank n in PG(1, q n ) and ϕ is an element of ΓL(2, q n ) which induces a collineation mapping L U to L V , then L U ϕ = L V .…”