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 Σ1,Σ2⊂Σ* such that there is a (t-3) -subspace Γ⊂ Σ* (Σ1∪ Σ2∪ ℓ) with the property that for i= 1 , 2 , L is the projection of Σi from center Γ and there exists no collineation ϕ of Σ* such that Γ^ϕ= Γ and Σ1^ϕ=Σ2. 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.
On the equivalence of linear sets / Csajbok, Bence; Zanella, Corrado. - In: DESIGNS, CODES AND CRYPTOGRAPHY. - ISSN 0925-1022. - STAMPA. - 81:2(2016), pp. 269-281. [10.1007/s10623-015-0141-z]
On the equivalence of linear sets
Csajbok, Bence;
2016-01-01
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 Σ1,Σ2⊂Σ* such that there is a (t-3) -subspace Γ⊂ Σ* (Σ1∪ Σ2∪ ℓ) with the property that for i= 1 , 2 , L is the projection of Σi from center Γ and there exists no collineation ϕ of Σ* such that Γ^ϕ= Γ and Σ1^ϕ=Σ2. 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
